Theorem pgpfac1lem1 20031

Description: Lemma for pgpfac1 20037. (Contributed by Mario Carneiro, 27-Apr-2016.)

Hypotheses

Ref	Expression
pgpfac1.k	⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺))
pgpfac1.s	⊢ 𝑆 = (𝐾‘{𝐴})
pgpfac1.b	⊢ 𝐵 = (Base‘𝐺)
pgpfac1.o	⊢ 𝑂 = (od‘𝐺)
pgpfac1.e	⊢ 𝐸 = (gEx‘𝐺)
pgpfac1.z	⊢ 0 = (0g‘𝐺)
pgpfac1.l	⊢ ⊕ = (LSSum‘𝐺)
pgpfac1.p	⊢ (𝜑 → 𝑃 pGrp 𝐺)
pgpfac1.g	⊢ (𝜑 → 𝐺 ∈ Abel)
pgpfac1.n	⊢ (𝜑 → 𝐵 ∈ Fin)
pgpfac1.oe	⊢ (𝜑 → (𝑂‘𝐴) = 𝐸)
pgpfac1.u	⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))
pgpfac1.au	⊢ (𝜑 → 𝐴 ∈ 𝑈)
pgpfac1.w	⊢ (𝜑 → 𝑊 ∈ (SubGrp‘𝐺))
pgpfac1.i	⊢ (𝜑 → (𝑆 ∩ 𝑊) = { 0 })
pgpfac1.ss	⊢ (𝜑 → (𝑆 ⊕ 𝑊) ⊆ 𝑈)
pgpfac1.2	⊢ (𝜑 → ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ (𝑆 ⊕ 𝑊) ⊊ 𝑤))

Assertion

Ref	Expression
pgpfac1lem1	⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) = 𝑈)

Distinct variable groups:   𝑤,𝐴   𝑤, ⊕   𝑤,𝑃   𝑤,𝐺   𝑤,𝑈   𝑤,𝐶   𝑤,𝑆   𝑤,𝑊   𝜑,𝑤   𝑤,𝐾

Allowed substitution hints:   𝐵(𝑤)   𝐸(𝑤)   𝑂(𝑤)   0 (𝑤)

Proof of Theorem pgpfac1lem1

Step	Hyp	Ref	Expression
1		pgpfac1.ss	. . . 4 ⊢ (𝜑 → (𝑆 ⊕ 𝑊) ⊆ 𝑈)
2	1	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → (𝑆 ⊕ 𝑊) ⊆ 𝑈)
3		pgpfac1.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ Abel)
4		ablgrp 19740	. . . . . 6 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
5		pgpfac1.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐺)
6	5	subgacs 19116	. . . . . 6 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘𝐵))
7		acsmre 17632	. . . . . 6 ⊢ ((SubGrp‘𝐺) ∈ (ACS‘𝐵) → (SubGrp‘𝐺) ∈ (Moore‘𝐵))
8	3, 4, 6, 7	4syl 19	. . . . 5 ⊢ (𝜑 → (SubGrp‘𝐺) ∈ (Moore‘𝐵))
9	8	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → (SubGrp‘𝐺) ∈ (Moore‘𝐵))
10		eldifi 4125	. . . . . 6 ⊢ (𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊)) → 𝐶 ∈ 𝑈)
11	10	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → 𝐶 ∈ 𝑈)
12	11	snssd 4813	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → {𝐶} ⊆ 𝑈)
13		pgpfac1.u	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))
14	13	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → 𝑈 ∈ (SubGrp‘𝐺))
15		pgpfac1.k	. . . . 5 ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺))
16	15	mrcsscl 17600	. . . 4 ⊢ (((SubGrp‘𝐺) ∈ (Moore‘𝐵) ∧ {𝐶} ⊆ 𝑈 ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝐾‘{𝐶}) ⊆ 𝑈)
17	9, 12, 14, 16	syl3anc 1369	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → (𝐾‘{𝐶}) ⊆ 𝑈)
18		pgpfac1.s	. . . . . . 7 ⊢ 𝑆 = (𝐾‘{𝐴})
19	5	subgss 19082	. . . . . . . . . 10 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ 𝐵)
20	13, 19	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ⊆ 𝐵)
21		pgpfac1.au	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
22	20, 21	sseldd 3981	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
23	15	mrcsncl 17592	. . . . . . . 8 ⊢ (((SubGrp‘𝐺) ∈ (Moore‘𝐵) ∧ 𝐴 ∈ 𝐵) → (𝐾‘{𝐴}) ∈ (SubGrp‘𝐺))
24	8, 22, 23	syl2anc 583	. . . . . . 7 ⊢ (𝜑 → (𝐾‘{𝐴}) ∈ (SubGrp‘𝐺))
25	18, 24	eqeltrid 2833	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺))
26		pgpfac1.w	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ (SubGrp‘𝐺))
27		pgpfac1.l	. . . . . . 7 ⊢ ⊕ = (LSSum‘𝐺)
28	27	lsmsubg2 19814	. . . . . 6 ⊢ ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑊 ∈ (SubGrp‘𝐺)) → (𝑆 ⊕ 𝑊) ∈ (SubGrp‘𝐺))
29	3, 25, 26, 28	syl3anc 1369	. . . . 5 ⊢ (𝜑 → (𝑆 ⊕ 𝑊) ∈ (SubGrp‘𝐺))
30	29	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → (𝑆 ⊕ 𝑊) ∈ (SubGrp‘𝐺))
31	20	sselda 3980	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑈) → 𝐶 ∈ 𝐵)
32	10, 31	sylan2 592	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → 𝐶 ∈ 𝐵)
33	15	mrcsncl 17592	. . . . 5 ⊢ (((SubGrp‘𝐺) ∈ (Moore‘𝐵) ∧ 𝐶 ∈ 𝐵) → (𝐾‘{𝐶}) ∈ (SubGrp‘𝐺))
34	9, 32, 33	syl2anc 583	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → (𝐾‘{𝐶}) ∈ (SubGrp‘𝐺))
35	27	lsmlub 19619	. . . 4 ⊢ (((𝑆 ⊕ 𝑊) ∈ (SubGrp‘𝐺) ∧ (𝐾‘{𝐶}) ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (((𝑆 ⊕ 𝑊) ⊆ 𝑈 ∧ (𝐾‘{𝐶}) ⊆ 𝑈) ↔ ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) ⊆ 𝑈))
36	30, 34, 14, 35	syl3anc 1369	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → (((𝑆 ⊕ 𝑊) ⊆ 𝑈 ∧ (𝐾‘{𝐶}) ⊆ 𝑈) ↔ ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) ⊆ 𝑈))
37	2, 17, 36	mpbi2and 711	. 2 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) ⊆ 𝑈)
38	27	lsmub1 19612	. . . . . 6 ⊢ (((𝑆 ⊕ 𝑊) ∈ (SubGrp‘𝐺) ∧ (𝐾‘{𝐶}) ∈ (SubGrp‘𝐺)) → (𝑆 ⊕ 𝑊) ⊆ ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})))
39	30, 34, 38	syl2anc 583	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → (𝑆 ⊕ 𝑊) ⊆ ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})))
40	27	lsmub2 19613	. . . . . . 7 ⊢ (((𝑆 ⊕ 𝑊) ∈ (SubGrp‘𝐺) ∧ (𝐾‘{𝐶}) ∈ (SubGrp‘𝐺)) → (𝐾‘{𝐶}) ⊆ ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})))
41	30, 34, 40	syl2anc 583	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → (𝐾‘{𝐶}) ⊆ ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})))
42	32	snssd 4813	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → {𝐶} ⊆ 𝐵)
43	9, 15, 42	mrcssidd 17605	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → {𝐶} ⊆ (𝐾‘{𝐶}))
44		snssg 4788	. . . . . . . 8 ⊢ (𝐶 ∈ 𝐵 → (𝐶 ∈ (𝐾‘{𝐶}) ↔ {𝐶} ⊆ (𝐾‘{𝐶})))
45	32, 44	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → (𝐶 ∈ (𝐾‘{𝐶}) ↔ {𝐶} ⊆ (𝐾‘{𝐶})))
46	43, 45	mpbird 257	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → 𝐶 ∈ (𝐾‘{𝐶}))
47	41, 46	sseldd 3981	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → 𝐶 ∈ ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})))
48		eldifn 4126	. . . . . 6 ⊢ (𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊)) → ¬ 𝐶 ∈ (𝑆 ⊕ 𝑊))
49	48	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → ¬ 𝐶 ∈ (𝑆 ⊕ 𝑊))
50	39, 47, 49	ssnelpssd 4110	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → (𝑆 ⊕ 𝑊) ⊊ ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})))
51	27	lsmub1 19612	. . . . . . . . 9 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑊 ∈ (SubGrp‘𝐺)) → 𝑆 ⊆ (𝑆 ⊕ 𝑊))
52	25, 26, 51	syl2anc 583	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ⊆ (𝑆 ⊕ 𝑊))
53	22	snssd 4813	. . . . . . . . . . 11 ⊢ (𝜑 → {𝐴} ⊆ 𝐵)
54	8, 15, 53	mrcssidd 17605	. . . . . . . . . 10 ⊢ (𝜑 → {𝐴} ⊆ (𝐾‘{𝐴}))
55	54, 18	sseqtrrdi 4031	. . . . . . . . 9 ⊢ (𝜑 → {𝐴} ⊆ 𝑆)
56		snssg 4788	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑈 → (𝐴 ∈ 𝑆 ↔ {𝐴} ⊆ 𝑆))
57	21, 56	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 ∈ 𝑆 ↔ {𝐴} ⊆ 𝑆))
58	55, 57	mpbird 257	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝑆)
59	52, 58	sseldd 3981	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ (𝑆 ⊕ 𝑊))
60	59	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → 𝐴 ∈ (𝑆 ⊕ 𝑊))
61	39, 60	sseldd 3981	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → 𝐴 ∈ ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})))
62		psseq1 4085	. . . . . . . 8 ⊢ (𝑤 = ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) → (𝑤 ⊊ 𝑈 ↔ ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) ⊊ 𝑈))
63		eleq2 2818	. . . . . . . 8 ⊢ (𝑤 = ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) → (𝐴 ∈ 𝑤 ↔ 𝐴 ∈ ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶}))))
64	62, 63	anbi12d 631	. . . . . . 7 ⊢ (𝑤 = ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) → ((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) ↔ (((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) ⊊ 𝑈 ∧ 𝐴 ∈ ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})))))
65		psseq2 4086	. . . . . . . 8 ⊢ (𝑤 = ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) → ((𝑆 ⊕ 𝑊) ⊊ 𝑤 ↔ (𝑆 ⊕ 𝑊) ⊊ ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶}))))
66	65	notbid 318	. . . . . . 7 ⊢ (𝑤 = ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) → (¬ (𝑆 ⊕ 𝑊) ⊊ 𝑤 ↔ ¬ (𝑆 ⊕ 𝑊) ⊊ ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶}))))
67	64, 66	imbi12d 344	. . . . . 6 ⊢ (𝑤 = ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) → (((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ (𝑆 ⊕ 𝑊) ⊊ 𝑤) ↔ ((((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) ⊊ 𝑈 ∧ 𝐴 ∈ ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶}))) → ¬ (𝑆 ⊕ 𝑊) ⊊ ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})))))
68		pgpfac1.2	. . . . . . 7 ⊢ (𝜑 → ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ (𝑆 ⊕ 𝑊) ⊊ 𝑤))
69	68	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ (𝑆 ⊕ 𝑊) ⊊ 𝑤))
70	3	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → 𝐺 ∈ Abel)
71	27	lsmsubg2 19814	. . . . . . 7 ⊢ ((𝐺 ∈ Abel ∧ (𝑆 ⊕ 𝑊) ∈ (SubGrp‘𝐺) ∧ (𝐾‘{𝐶}) ∈ (SubGrp‘𝐺)) → ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) ∈ (SubGrp‘𝐺))
72	70, 30, 34, 71	syl3anc 1369	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) ∈ (SubGrp‘𝐺))
73	67, 69, 72	rspcdva 3610	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → ((((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) ⊊ 𝑈 ∧ 𝐴 ∈ ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶}))) → ¬ (𝑆 ⊕ 𝑊) ⊊ ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶}))))
74	61, 73	mpan2d 693	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → (((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) ⊊ 𝑈 → ¬ (𝑆 ⊕ 𝑊) ⊊ ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶}))))
75	50, 74	mt2d 136	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → ¬ ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) ⊊ 𝑈)
76		npss 4108	. . 3 ⊢ (¬ ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) ⊊ 𝑈 ↔ (((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) ⊆ 𝑈 → ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) = 𝑈))
77	75, 76	sylib 217	. 2 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → (((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) ⊆ 𝑈 → ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) = 𝑈))
78	37, 77	mpd 15	1 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) = 𝑈)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099  ∀wral 3058   ∖ cdif 3944   ∩ cin 3946   ⊆ wss 3947   ⊊ wpss 3948  {csn 4629   class class class wbr 5148  ‘cfv 6548  (class class class)co 7420  Fincfn 8964  Basecbs 17180  0_gc0g 17421  Moorecmre 17562  mrClscmrc 17563  ACScacs 17565  Grpcgrp 18890  SubGrpcsubg 19075  odcod 19479  gExcgex 19480   pGrp cpgp 19481  LSSumclsm 19589  Abelcabl 19736

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 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-cnex 11195  ax-resscn 11196  ax-1cn 11197  ax-icn 11198  ax-addcl 11199  ax-addrcl 11200  ax-mulcl 11201  ax-mulrcl 11202  ax-mulcom 11203  ax-addass 11204  ax-mulass 11205  ax-distr 11206  ax-i2m1 11207  ax-1ne0 11208  ax-1rid 11209  ax-rnegex 11210  ax-rrecex 11211  ax-cnre 11212  ax-pre-lttri 11213  ax-pre-lttrn 11214  ax-pre-ltadd 11215  ax-pre-mulgt0 11216

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 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-riota 7376  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7871  df-1st 7993  df-2nd 7994  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-er 8725  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-pnf 11281  df-mnf 11282  df-xr 11283  df-ltxr 11284  df-le 11285  df-sub 11477  df-neg 11478  df-nn 12244  df-2 12306  df-sets 17133  df-slot 17151  df-ndx 17163  df-base 17181  df-ress 17210  df-plusg 17246  df-0g 17423  df-mre 17566  df-mrc 17567  df-acs 17569  df-mgm 18600  df-sgrp 18679  df-mnd 18695  df-submnd 18741  df-grp 18893  df-minusg 18894  df-subg 19078  df-cntz 19268  df-lsm 19591  df-cmn 19737  df-abl 19738

This theorem is referenced by:  pgpfac1lem2  20032  pgpfac1lem3  20034