Theorem fsumrlim 15783

Description: Limit of a finite sum of converging sequences. Note that 𝐶(𝑘) is a collection of functions with implicit parameter 𝑘, each of which converges to 𝐷(𝑘) as 𝑛 ⇝ +∞. (Contributed by Mario Carneiro, 22-May-2016.)

Hypotheses

Ref	Expression
fsumrlim.1	⊢ (𝜑 → 𝐴 ⊆ ℝ)
fsumrlim.2	⊢ (𝜑 → 𝐵 ∈ Fin)
fsumrlim.3	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐶 ∈ 𝑉)
fsumrlim.4	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ 𝐶) ⇝𝑟 𝐷)

Assertion

Ref	Expression
fsumrlim	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝐵 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝐵 𝐷)

Distinct variable groups:   𝑥,𝑘,𝐴   𝐵,𝑘,𝑥   𝜑,𝑘,𝑥

Allowed substitution hints:   𝐶(𝑥,𝑘)   𝐷(𝑥,𝑘)   𝑉(𝑥,𝑘)

Proof of Theorem fsumrlim

Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssid 4000	. 2 ⊢ 𝐵 ⊆ 𝐵
2		fsumrlim.2	. . 3 ⊢ (𝜑 → 𝐵 ∈ Fin)
3		sseq1 4003	. . . . . 6 ⊢ (𝑤 = ∅ → (𝑤 ⊆ 𝐵 ↔ ∅ ⊆ 𝐵))
4		sumeq1 15661	. . . . . . . . 9 ⊢ (𝑤 = ∅ → Σ𝑘 ∈ 𝑤 𝐶 = Σ𝑘 ∈ ∅ 𝐶)
5		sum0 15693	. . . . . . . . 9 ⊢ Σ𝑘 ∈ ∅ 𝐶 = 0
6	4, 5	eqtrdi 2783	. . . . . . . 8 ⊢ (𝑤 = ∅ → Σ𝑘 ∈ 𝑤 𝐶 = 0)
7	6	mpteq2dv 5244	. . . . . . 7 ⊢ (𝑤 = ∅ → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) = (𝑥 ∈ 𝐴 ↦ 0))
8		sumeq1 15661	. . . . . . . 8 ⊢ (𝑤 = ∅ → Σ𝑘 ∈ 𝑤 𝐷 = Σ𝑘 ∈ ∅ 𝐷)
9		sum0 15693	. . . . . . . 8 ⊢ Σ𝑘 ∈ ∅ 𝐷 = 0
10	8, 9	eqtrdi 2783	. . . . . . 7 ⊢ (𝑤 = ∅ → Σ𝑘 ∈ 𝑤 𝐷 = 0)
11	7, 10	breq12d 5155	. . . . . 6 ⊢ (𝑤 = ∅ → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑤 𝐷 ↔ (𝑥 ∈ 𝐴 ↦ 0) ⇝𝑟 0))
12	3, 11	imbi12d 344	. . . . 5 ⊢ (𝑤 = ∅ → ((𝑤 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑤 𝐷) ↔ (∅ ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ 0) ⇝𝑟 0)))
13	12	imbi2d 340	. . . 4 ⊢ (𝑤 = ∅ → ((𝜑 → (𝑤 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑤 𝐷)) ↔ (𝜑 → (∅ ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ 0) ⇝𝑟 0))))
14		sseq1 4003	. . . . . 6 ⊢ (𝑤 = 𝑦 → (𝑤 ⊆ 𝐵 ↔ 𝑦 ⊆ 𝐵))
15		sumeq1 15661	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → Σ𝑘 ∈ 𝑤 𝐶 = Σ𝑘 ∈ 𝑦 𝐶)
16	15	mpteq2dv 5244	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) = (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶))
17		sumeq1 15661	. . . . . . 7 ⊢ (𝑤 = 𝑦 → Σ𝑘 ∈ 𝑤 𝐷 = Σ𝑘 ∈ 𝑦 𝐷)
18	16, 17	breq12d 5155	. . . . . 6 ⊢ (𝑤 = 𝑦 → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑤 𝐷 ↔ (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑦 𝐷))
19	14, 18	imbi12d 344	. . . . 5 ⊢ (𝑤 = 𝑦 → ((𝑤 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑤 𝐷) ↔ (𝑦 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑦 𝐷)))
20	19	imbi2d 340	. . . 4 ⊢ (𝑤 = 𝑦 → ((𝜑 → (𝑤 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑤 𝐷)) ↔ (𝜑 → (𝑦 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑦 𝐷))))
21		sseq1 4003	. . . . . 6 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝑤 ⊆ 𝐵 ↔ (𝑦 ∪ {𝑧}) ⊆ 𝐵))
22		sumeq1 15661	. . . . . . . 8 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → Σ𝑘 ∈ 𝑤 𝐶 = Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶)
23	22	mpteq2dv 5244	. . . . . . 7 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) = (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶))
24		sumeq1 15661	. . . . . . 7 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → Σ𝑘 ∈ 𝑤 𝐷 = Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐷)
25	23, 24	breq12d 5155	. . . . . 6 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑤 𝐷 ↔ (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ⇝𝑟 Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐷))
26	21, 25	imbi12d 344	. . . . 5 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ((𝑤 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑤 𝐷) ↔ ((𝑦 ∪ {𝑧}) ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ⇝𝑟 Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐷)))
27	26	imbi2d 340	. . . 4 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ((𝜑 → (𝑤 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑤 𝐷)) ↔ (𝜑 → ((𝑦 ∪ {𝑧}) ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ⇝𝑟 Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐷))))
28		sseq1 4003	. . . . . 6 ⊢ (𝑤 = 𝐵 → (𝑤 ⊆ 𝐵 ↔ 𝐵 ⊆ 𝐵))
29		sumeq1 15661	. . . . . . . 8 ⊢ (𝑤 = 𝐵 → Σ𝑘 ∈ 𝑤 𝐶 = Σ𝑘 ∈ 𝐵 𝐶)
30	29	mpteq2dv 5244	. . . . . . 7 ⊢ (𝑤 = 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) = (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝐵 𝐶))
31		sumeq1 15661	. . . . . . 7 ⊢ (𝑤 = 𝐵 → Σ𝑘 ∈ 𝑤 𝐷 = Σ𝑘 ∈ 𝐵 𝐷)
32	30, 31	breq12d 5155	. . . . . 6 ⊢ (𝑤 = 𝐵 → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑤 𝐷 ↔ (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝐵 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝐵 𝐷))
33	28, 32	imbi12d 344	. . . . 5 ⊢ (𝑤 = 𝐵 → ((𝑤 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑤 𝐷) ↔ (𝐵 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝐵 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝐵 𝐷)))
34	33	imbi2d 340	. . . 4 ⊢ (𝑤 = 𝐵 → ((𝜑 → (𝑤 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑤 𝐷)) ↔ (𝜑 → (𝐵 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝐵 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝐵 𝐷))))
35		fsumrlim.1	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
36		0cn 11230	. . . . . 6 ⊢ 0 ∈ ℂ
37		rlimconst 15514	. . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ 0 ∈ ℂ) → (𝑥 ∈ 𝐴 ↦ 0) ⇝𝑟 0)
38	35, 36, 37	sylancl 585	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 0) ⇝𝑟 0)
39	38	a1d 25	. . . 4 ⊢ (𝜑 → (∅ ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ 0) ⇝𝑟 0))
40		ssun1 4168	. . . . . . . . . 10 ⊢ 𝑦 ⊆ (𝑦 ∪ {𝑧})
41		sstr 3986	. . . . . . . . . 10 ⊢ ((𝑦 ⊆ (𝑦 ∪ {𝑧}) ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵) → 𝑦 ⊆ 𝐵)
42	40, 41	mpan 689	. . . . . . . . 9 ⊢ ((𝑦 ∪ {𝑧}) ⊆ 𝐵 → 𝑦 ⊆ 𝐵)
43	42	imim1i 63	. . . . . . . 8 ⊢ ((𝑦 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑦 𝐷) → ((𝑦 ∪ {𝑧}) ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑦 𝐷))
44		sumex 15660	. . . . . . . . . . . . . 14 ⊢ Σ𝑘 ∈ 𝑦 ⦋𝑤 / 𝑥⦌𝐶 ∈ V
45	44	a1i 11	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑦 𝐷) ∧ 𝑤 ∈ 𝐴) → Σ𝑘 ∈ 𝑦 ⦋𝑤 / 𝑥⦌𝐶 ∈ V)
46		simprr 772	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑦 ∪ {𝑧}) ⊆ 𝐵)
47	46	unssbd 4184	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → {𝑧} ⊆ 𝐵)
48		vex 3473	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑧 ∈ V
49	48	snss 4785	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ 𝐵 ↔ {𝑧} ⊆ 𝐵)
50	47, 49	sylibr 233	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → 𝑧 ∈ 𝐵)
51	50	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → 𝑧 ∈ 𝐵)
52		fsumrlim.3	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐶 ∈ 𝑉)
53	52	anass1rs 654	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑉)
54		fsumrlim.4	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ 𝐶) ⇝𝑟 𝐷)
55	53, 54	rlimmptrcl 15578	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℂ)
56	55	an32s 651	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℂ)
57	56	adantllr 718	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℂ)
58	57	ralrimiva 3141	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → ∀𝑘 ∈ 𝐵 𝐶 ∈ ℂ)
59		nfcsb1v 3914	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐶
60	59	nfel1 2914	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ
61		csbeq1a 3903	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑧 → 𝐶 = ⦋𝑧 / 𝑘⦌𝐶)
62	61	eleq1d 2813	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑧 → (𝐶 ∈ ℂ ↔ ⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ))
63	60, 62	rspc 3595	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ 𝐵 → (∀𝑘 ∈ 𝐵 𝐶 ∈ ℂ → ⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ))
64	51, 58, 63	sylc 65	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → ⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ)
65	64	ralrimiva 3141	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → ∀𝑥 ∈ 𝐴 ⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ)
66	65	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑦 𝐷) → ∀𝑥 ∈ 𝐴 ⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ)
67		nfcsb1v 3914	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥⦋𝑤 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶
68	67	nfel1 2914	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥⦋𝑤 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ
69		csbeq1a 3903	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑤 → ⦋𝑧 / 𝑘⦌𝐶 = ⦋𝑤 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶)
70	69	eleq1d 2813	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑤 → (⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ ↔ ⦋𝑤 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ))
71	68, 70	rspc 3595	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 ⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ → ⦋𝑤 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ))
72	66, 71	mpan9 506	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑦 𝐷) ∧ 𝑤 ∈ 𝐴) → ⦋𝑤 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ)
73	72	elexd 3490	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑦 𝐷) ∧ 𝑤 ∈ 𝐴) → ⦋𝑤 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶 ∈ V)
74		nfcv 2898	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑤Σ𝑘 ∈ 𝑦 𝐶
75		nfcv 2898	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥𝑦
76		nfcsb1v 3914	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥⦋𝑤 / 𝑥⦌𝐶
77	75, 76	nfsum 15663	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥Σ𝑘 ∈ 𝑦 ⦋𝑤 / 𝑥⦌𝐶
78		csbeq1a 3903	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑤 → 𝐶 = ⦋𝑤 / 𝑥⦌𝐶)
79	78	sumeq2sdv 15676	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑤 → Σ𝑘 ∈ 𝑦 𝐶 = Σ𝑘 ∈ 𝑦 ⦋𝑤 / 𝑥⦌𝐶)
80	74, 77, 79	cbvmpt 5253	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) = (𝑤 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 ⦋𝑤 / 𝑥⦌𝐶)
81		simpr 484	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑦 𝐷) → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑦 𝐷)
82	80, 81	eqbrtrrid 5178	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑦 𝐷) → (𝑤 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 ⦋𝑤 / 𝑥⦌𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑦 𝐷)
83		nfcv 2898	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑤⦋𝑧 / 𝑘⦌𝐶
84	83, 67, 69	cbvmpt 5253	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝐴 ↦ ⦋𝑧 / 𝑘⦌𝐶) = (𝑤 ∈ 𝐴 ↦ ⦋𝑤 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶)
85	54	ralrimiva 3141	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ∀𝑘 ∈ 𝐵 (𝑥 ∈ 𝐴 ↦ 𝐶) ⇝𝑟 𝐷)
86	85	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → ∀𝑘 ∈ 𝐵 (𝑥 ∈ 𝐴 ↦ 𝐶) ⇝𝑟 𝐷)
87		nfcv 2898	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑘𝐴
88	87, 59	nfmpt 5249	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘(𝑥 ∈ 𝐴 ↦ ⦋𝑧 / 𝑘⦌𝐶)
89		nfcv 2898	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘 ⇝𝑟
90		nfcsb1v 3914	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐷
91	88, 89, 90	nfbr 5189	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘(𝑥 ∈ 𝐴 ↦ ⦋𝑧 / 𝑘⦌𝐶) ⇝𝑟 ⦋𝑧 / 𝑘⦌𝐷
92	61	mpteq2dv 5244	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑧 → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ ⦋𝑧 / 𝑘⦌𝐶))
93		csbeq1a 3903	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑧 → 𝐷 = ⦋𝑧 / 𝑘⦌𝐷)
94	92, 93	breq12d 5155	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑧 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ⇝𝑟 𝐷 ↔ (𝑥 ∈ 𝐴 ↦ ⦋𝑧 / 𝑘⦌𝐶) ⇝𝑟 ⦋𝑧 / 𝑘⦌𝐷))
95	91, 94	rspc 3595	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ 𝐵 → (∀𝑘 ∈ 𝐵 (𝑥 ∈ 𝐴 ↦ 𝐶) ⇝𝑟 𝐷 → (𝑥 ∈ 𝐴 ↦ ⦋𝑧 / 𝑘⦌𝐶) ⇝𝑟 ⦋𝑧 / 𝑘⦌𝐷))
96	50, 86, 95	sylc 65	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑥 ∈ 𝐴 ↦ ⦋𝑧 / 𝑘⦌𝐶) ⇝𝑟 ⦋𝑧 / 𝑘⦌𝐷)
97	96	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑦 𝐷) → (𝑥 ∈ 𝐴 ↦ ⦋𝑧 / 𝑘⦌𝐶) ⇝𝑟 ⦋𝑧 / 𝑘⦌𝐷)
98	84, 97	eqbrtrrid 5178	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑦 𝐷) → (𝑤 ∈ 𝐴 ↦ ⦋𝑤 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶) ⇝𝑟 ⦋𝑧 / 𝑘⦌𝐷)
99	45, 73, 82, 98	rlimadd 15613	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑦 𝐷) → (𝑤 ∈ 𝐴 ↦ (Σ𝑘 ∈ 𝑦 ⦋𝑤 / 𝑥⦌𝐶 + ⦋𝑤 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶)) ⇝𝑟 (Σ𝑘 ∈ 𝑦 𝐷 + ⦋𝑧 / 𝑘⦌𝐷))
100		simprl 770	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → ¬ 𝑧 ∈ 𝑦)
101		disjsn 4711	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝑦)
102	100, 101	sylibr 233	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑦 ∩ {𝑧}) = ∅)
103	102	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → (𝑦 ∩ {𝑧}) = ∅)
104		eqidd 2728	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → (𝑦 ∪ {𝑧}) = (𝑦 ∪ {𝑧}))
105	2	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → 𝐵 ∈ Fin)
106	105, 46	ssfid 9285	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑦 ∪ {𝑧}) ∈ Fin)
107	106	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → (𝑦 ∪ {𝑧}) ∈ Fin)
108	46	sselda 3978	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝑘 ∈ 𝐵)
109	108	adantlr 714	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝑘 ∈ 𝐵)
110	109, 57	syldan 590	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝐶 ∈ ℂ)
111	103, 104, 107, 110	fsumsplit 15713	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶 = (Σ𝑘 ∈ 𝑦 𝐶 + Σ𝑘 ∈ {𝑧}𝐶))
112		nfcv 2898	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑤𝐶
113		nfcsb1v 3914	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑘⦋𝑤 / 𝑘⦌𝐶
114		csbeq1a 3903	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑤 → 𝐶 = ⦋𝑤 / 𝑘⦌𝐶)
115	112, 113, 114	cbvsumi 15669	. . . . . . . . . . . . . . . . . 18 ⊢ Σ𝑘 ∈ {𝑧}𝐶 = Σ𝑤 ∈ {𝑧}⦋𝑤 / 𝑘⦌𝐶
116		csbeq1 3892	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 = 𝑧 → ⦋𝑤 / 𝑘⦌𝐶 = ⦋𝑧 / 𝑘⦌𝐶)
117	116	sumsn 15718	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑧 ∈ 𝐵 ∧ ⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ) → Σ𝑤 ∈ {𝑧}⦋𝑤 / 𝑘⦌𝐶 = ⦋𝑧 / 𝑘⦌𝐶)
118	51, 64, 117	syl2anc 583	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → Σ𝑤 ∈ {𝑧}⦋𝑤 / 𝑘⦌𝐶 = ⦋𝑧 / 𝑘⦌𝐶)
119	115, 118	eqtrid 2779	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → Σ𝑘 ∈ {𝑧}𝐶 = ⦋𝑧 / 𝑘⦌𝐶)
120	119	oveq2d 7430	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → (Σ𝑘 ∈ 𝑦 𝐶 + Σ𝑘 ∈ {𝑧}𝐶) = (Σ𝑘 ∈ 𝑦 𝐶 + ⦋𝑧 / 𝑘⦌𝐶))
121	111, 120	eqtrd 2767	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶 = (Σ𝑘 ∈ 𝑦 𝐶 + ⦋𝑧 / 𝑘⦌𝐶))
122	121	mpteq2dva 5242	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) = (𝑥 ∈ 𝐴 ↦ (Σ𝑘 ∈ 𝑦 𝐶 + ⦋𝑧 / 𝑘⦌𝐶)))
123	122	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑦 𝐷) → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) = (𝑥 ∈ 𝐴 ↦ (Σ𝑘 ∈ 𝑦 𝐶 + ⦋𝑧 / 𝑘⦌𝐶)))
124		nfcv 2898	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑤(Σ𝑘 ∈ 𝑦 𝐶 + ⦋𝑧 / 𝑘⦌𝐶)
125		nfcv 2898	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥 +
126	77, 125, 67	nfov 7444	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥(Σ𝑘 ∈ 𝑦 ⦋𝑤 / 𝑥⦌𝐶 + ⦋𝑤 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶)
127	79, 69	oveq12d 7432	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑤 → (Σ𝑘 ∈ 𝑦 𝐶 + ⦋𝑧 / 𝑘⦌𝐶) = (Σ𝑘 ∈ 𝑦 ⦋𝑤 / 𝑥⦌𝐶 + ⦋𝑤 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶))
128	124, 126, 127	cbvmpt 5253	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝐴 ↦ (Σ𝑘 ∈ 𝑦 𝐶 + ⦋𝑧 / 𝑘⦌𝐶)) = (𝑤 ∈ 𝐴 ↦ (Σ𝑘 ∈ 𝑦 ⦋𝑤 / 𝑥⦌𝐶 + ⦋𝑤 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶))
129	123, 128	eqtrdi 2783	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑦 𝐷) → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) = (𝑤 ∈ 𝐴 ↦ (Σ𝑘 ∈ 𝑦 ⦋𝑤 / 𝑥⦌𝐶 + ⦋𝑤 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶)))
130		eqidd 2728	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑦 ∪ {𝑧}) = (𝑦 ∪ {𝑧}))
131		rlimcl 15473	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐶) ⇝𝑟 𝐷 → 𝐷 ∈ ℂ)
132	54, 131	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐷 ∈ ℂ)
133	132	adantlr 714	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑘 ∈ 𝐵) → 𝐷 ∈ ℂ)
134	108, 133	syldan 590	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝐷 ∈ ℂ)
135	102, 130, 106, 134	fsumsplit 15713	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐷 = (Σ𝑘 ∈ 𝑦 𝐷 + Σ𝑘 ∈ {𝑧}𝐷))
136		nfcv 2898	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑤𝐷
137		nfcsb1v 3914	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘⦋𝑤 / 𝑘⦌𝐷
138		csbeq1a 3903	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑤 → 𝐷 = ⦋𝑤 / 𝑘⦌𝐷)
139	136, 137, 138	cbvsumi 15669	. . . . . . . . . . . . . . . 16 ⊢ Σ𝑘 ∈ {𝑧}𝐷 = Σ𝑤 ∈ {𝑧}⦋𝑤 / 𝑘⦌𝐷
140	133	ralrimiva 3141	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → ∀𝑘 ∈ 𝐵 𝐷 ∈ ℂ)
141	90	nfel1 2914	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐷 ∈ ℂ
142	93	eleq1d 2813	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑧 → (𝐷 ∈ ℂ ↔ ⦋𝑧 / 𝑘⦌𝐷 ∈ ℂ))
143	141, 142	rspc 3595	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ 𝐵 → (∀𝑘 ∈ 𝐵 𝐷 ∈ ℂ → ⦋𝑧 / 𝑘⦌𝐷 ∈ ℂ))
144	50, 140, 143	sylc 65	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → ⦋𝑧 / 𝑘⦌𝐷 ∈ ℂ)
145		csbeq1 3892	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = 𝑧 → ⦋𝑤 / 𝑘⦌𝐷 = ⦋𝑧 / 𝑘⦌𝐷)
146	145	sumsn 15718	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ 𝐵 ∧ ⦋𝑧 / 𝑘⦌𝐷 ∈ ℂ) → Σ𝑤 ∈ {𝑧}⦋𝑤 / 𝑘⦌𝐷 = ⦋𝑧 / 𝑘⦌𝐷)
147	50, 144, 146	syl2anc 583	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → Σ𝑤 ∈ {𝑧}⦋𝑤 / 𝑘⦌𝐷 = ⦋𝑧 / 𝑘⦌𝐷)
148	139, 147	eqtrid 2779	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → Σ𝑘 ∈ {𝑧}𝐷 = ⦋𝑧 / 𝑘⦌𝐷)
149	148	oveq2d 7430	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (Σ𝑘 ∈ 𝑦 𝐷 + Σ𝑘 ∈ {𝑧}𝐷) = (Σ𝑘 ∈ 𝑦 𝐷 + ⦋𝑧 / 𝑘⦌𝐷))
150	135, 149	eqtrd 2767	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐷 = (Σ𝑘 ∈ 𝑦 𝐷 + ⦋𝑧 / 𝑘⦌𝐷))
151	150	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑦 𝐷) → Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐷 = (Σ𝑘 ∈ 𝑦 𝐷 + ⦋𝑧 / 𝑘⦌𝐷))
152	99, 129, 151	3brtr4d 5174	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑦 𝐷) → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ⇝𝑟 Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐷)
153	152	ex 412	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑦 𝐷 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ⇝𝑟 Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐷))
154	153	expr 456	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) → ((𝑦 ∪ {𝑧}) ⊆ 𝐵 → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑦 𝐷 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ⇝𝑟 Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐷)))
155	154	a2d 29	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) → (((𝑦 ∪ {𝑧}) ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑦 𝐷) → ((𝑦 ∪ {𝑧}) ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ⇝𝑟 Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐷)))
156	43, 155	syl5 34	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) → ((𝑦 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑦 𝐷) → ((𝑦 ∪ {𝑧}) ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ⇝𝑟 Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐷)))
157	156	expcom 413	. . . . . 6 ⊢ (¬ 𝑧 ∈ 𝑦 → (𝜑 → ((𝑦 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑦 𝐷) → ((𝑦 ∪ {𝑧}) ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ⇝𝑟 Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐷))))
158	157	a2d 29	. . . . 5 ⊢ (¬ 𝑧 ∈ 𝑦 → ((𝜑 → (𝑦 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑦 𝐷)) → (𝜑 → ((𝑦 ∪ {𝑧}) ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ⇝𝑟 Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐷))))
159	158	adantl 481	. . . 4 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → ((𝜑 → (𝑦 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑦 𝐷)) → (𝜑 → ((𝑦 ∪ {𝑧}) ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ⇝𝑟 Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐷))))
160	13, 20, 27, 34, 39, 159	findcard2s 9183	. . 3 ⊢ (𝐵 ∈ Fin → (𝜑 → (𝐵 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝐵 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝐵 𝐷)))
161	2, 160	mpcom 38	. 2 ⊢ (𝜑 → (𝐵 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝐵 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝐵 𝐷))
162	1, 161	mpi 20	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝐵 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝐵 𝐷)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099  ∀wral 3056  Vcvv 3469  ⦋csb 3889   ∪ cun 3942   ∩ cin 3943   ⊆ wss 3944  ∅c0 4318  {csn 4624   class class class wbr 5142   ↦ cmpt 5225  (class class class)co 7414  Fincfn 8957  ℂcc 11130  ℝcr 11131  0cc0 11132   + caddc 11135   ⇝_𝑟 crli 15455  Σcsu 15658

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-inf2 9658  ax-cnex 11188  ax-resscn 11189  ax-1cn 11190  ax-icn 11191  ax-addcl 11192  ax-addrcl 11193  ax-mulcl 11194  ax-mulrcl 11195  ax-mulcom 11196  ax-addass 11197  ax-mulass 11198  ax-distr 11199  ax-i2m1 11200  ax-1ne0 11201  ax-1rid 11202  ax-rnegex 11203  ax-rrecex 11204  ax-cnre 11205  ax-pre-lttri 11206  ax-pre-lttrn 11207  ax-pre-ltadd 11208  ax-pre-mulgt0 11209  ax-pre-sup 11210

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-er 8718  df-pm 8841  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-sup 9459  df-oi 9527  df-card 9956  df-pnf 11274  df-mnf 11275  df-xr 11276  df-ltxr 11277  df-le 11278  df-sub 11470  df-neg 11471  df-div 11896  df-nn 12237  df-2 12299  df-3 12300  df-n0 12497  df-z 12583  df-uz 12847  df-rp 13001  df-fz 13511  df-fzo 13654  df-seq 13993  df-exp 14053  df-hash 14316  df-cj 15072  df-re 15073  df-im 15074  df-sqrt 15208  df-abs 15209  df-clim 15458  df-rlim 15459  df-sum 15659

This theorem is referenced by:  climfsum  15792  logexprlim  27151  signsplypnf  34172