Theorem splval 14728

Description: Value of the substring replacement operator. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by AV, 11-May-2020.) (Revised by AV, 15-Oct-2022.)

Assertion

Ref	Expression
splval	⊢ ((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ 𝑊 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌)) → (𝑆 splice ⟨𝐹, 𝑇, 𝑅⟩) = (((𝑆 prefix 𝐹) ++ 𝑅) ++ (𝑆 substr ⟨𝑇, (♯‘𝑆)⟩)))

Proof of Theorem splval

Dummy variables 𝑠 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-splice 14727	. . 3 ⊢ splice = (𝑠 ∈ V, 𝑏 ∈ V ↦ (((𝑠 prefix (1st ‘(1st ‘𝑏))) ++ (2nd ‘𝑏)) ++ (𝑠 substr ⟨(2nd ‘(1st ‘𝑏)), (♯‘𝑠)⟩)))
2	1	a1i 11	. 2 ⊢ ((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ 𝑊 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌)) → splice = (𝑠 ∈ V, 𝑏 ∈ V ↦ (((𝑠 prefix (1st ‘(1st ‘𝑏))) ++ (2nd ‘𝑏)) ++ (𝑠 substr ⟨(2nd ‘(1st ‘𝑏)), (♯‘𝑠)⟩))))
3		simprl 770	. . . . 5 ⊢ (((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ 𝑊 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌)) ∧ (𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩)) → 𝑠 = 𝑆)
4		2fveq3 6897	. . . . . . 7 ⊢ (𝑏 = ⟨𝐹, 𝑇, 𝑅⟩ → (1st ‘(1st ‘𝑏)) = (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)))
5	4	adantl 481	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → (1st ‘(1st ‘𝑏)) = (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)))
6		ot1stg 8002	. . . . . . 7 ⊢ ((𝐹 ∈ 𝑊 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌) → (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)) = 𝐹)
7	6	adantl 481	. . . . . 6 ⊢ ((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ 𝑊 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌)) → (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)) = 𝐹)
8	5, 7	sylan9eqr 2790	. . . . 5 ⊢ (((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ 𝑊 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌)) ∧ (𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩)) → (1st ‘(1st ‘𝑏)) = 𝐹)
9	3, 8	oveq12d 7433	. . . 4 ⊢ (((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ 𝑊 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌)) ∧ (𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩)) → (𝑠 prefix (1st ‘(1st ‘𝑏))) = (𝑆 prefix 𝐹))
10		fveq2 6892	. . . . . 6 ⊢ (𝑏 = ⟨𝐹, 𝑇, 𝑅⟩ → (2nd ‘𝑏) = (2nd ‘⟨𝐹, 𝑇, 𝑅⟩))
11	10	adantl 481	. . . . 5 ⊢ ((𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → (2nd ‘𝑏) = (2nd ‘⟨𝐹, 𝑇, 𝑅⟩))
12		ot3rdg 8004	. . . . . . 7 ⊢ (𝑅 ∈ 𝑌 → (2nd ‘⟨𝐹, 𝑇, 𝑅⟩) = 𝑅)
13	12	3ad2ant3 1133	. . . . . 6 ⊢ ((𝐹 ∈ 𝑊 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌) → (2nd ‘⟨𝐹, 𝑇, 𝑅⟩) = 𝑅)
14	13	adantl 481	. . . . 5 ⊢ ((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ 𝑊 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌)) → (2nd ‘⟨𝐹, 𝑇, 𝑅⟩) = 𝑅)
15	11, 14	sylan9eqr 2790	. . . 4 ⊢ (((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ 𝑊 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌)) ∧ (𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩)) → (2nd ‘𝑏) = 𝑅)
16	9, 15	oveq12d 7433	. . 3 ⊢ (((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ 𝑊 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌)) ∧ (𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩)) → ((𝑠 prefix (1st ‘(1st ‘𝑏))) ++ (2nd ‘𝑏)) = ((𝑆 prefix 𝐹) ++ 𝑅))
17		2fveq3 6897	. . . . . . 7 ⊢ (𝑏 = ⟨𝐹, 𝑇, 𝑅⟩ → (2nd ‘(1st ‘𝑏)) = (2nd ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)))
18	17	adantl 481	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → (2nd ‘(1st ‘𝑏)) = (2nd ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)))
19		ot2ndg 8003	. . . . . . 7 ⊢ ((𝐹 ∈ 𝑊 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌) → (2nd ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)) = 𝑇)
20	19	adantl 481	. . . . . 6 ⊢ ((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ 𝑊 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌)) → (2nd ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)) = 𝑇)
21	18, 20	sylan9eqr 2790	. . . . 5 ⊢ (((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ 𝑊 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌)) ∧ (𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩)) → (2nd ‘(1st ‘𝑏)) = 𝑇)
22	3	fveq2d 6896	. . . . 5 ⊢ (((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ 𝑊 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌)) ∧ (𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩)) → (♯‘𝑠) = (♯‘𝑆))
23	21, 22	opeq12d 4878	. . . 4 ⊢ (((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ 𝑊 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌)) ∧ (𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩)) → ⟨(2nd ‘(1st ‘𝑏)), (♯‘𝑠)⟩ = ⟨𝑇, (♯‘𝑆)⟩)
24	3, 23	oveq12d 7433	. . 3 ⊢ (((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ 𝑊 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌)) ∧ (𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩)) → (𝑠 substr ⟨(2nd ‘(1st ‘𝑏)), (♯‘𝑠)⟩) = (𝑆 substr ⟨𝑇, (♯‘𝑆)⟩))
25	16, 24	oveq12d 7433	. 2 ⊢ (((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ 𝑊 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌)) ∧ (𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩)) → (((𝑠 prefix (1st ‘(1st ‘𝑏))) ++ (2nd ‘𝑏)) ++ (𝑠 substr ⟨(2nd ‘(1st ‘𝑏)), (♯‘𝑠)⟩)) = (((𝑆 prefix 𝐹) ++ 𝑅) ++ (𝑆 substr ⟨𝑇, (♯‘𝑆)⟩)))
26		elex 3489	. . 3 ⊢ (𝑆 ∈ 𝑉 → 𝑆 ∈ V)
27	26	adantr 480	. 2 ⊢ ((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ 𝑊 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌)) → 𝑆 ∈ V)
28		otex 5462	. . 3 ⊢ ⟨𝐹, 𝑇, 𝑅⟩ ∈ V
29	28	a1i 11	. 2 ⊢ ((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ 𝑊 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌)) → ⟨𝐹, 𝑇, 𝑅⟩ ∈ V)
30		ovexd 7450	. 2 ⊢ ((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ 𝑊 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌)) → (((𝑆 prefix 𝐹) ++ 𝑅) ++ (𝑆 substr ⟨𝑇, (♯‘𝑆)⟩)) ∈ V)
31	2, 25, 27, 29, 30	ovmpod 7568	1 ⊢ ((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ 𝑊 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌)) → (𝑆 splice ⟨𝐹, 𝑇, 𝑅⟩) = (((𝑆 prefix 𝐹) ++ 𝑅) ++ (𝑆 substr ⟨𝑇, (♯‘𝑆)⟩)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099  Vcvv 3470  ⟨cop 4631  ⟨cotp 4633  ‘cfv 6543  (class class class)co 7415   ∈ cmpo 7417  1^st c1st 7986  2^nd c2nd 7987  ♯chash 14316   ++ cconcat 14547   substr csubstr 14617   prefix cpfx 14647   splice csplice 14726

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-sep 5294  ax-nul 5301  ax-pr 5424  ax-un 7735

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  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 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-sbc 3776  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-ot 4634  df-uni 4905  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7418  df-oprab 7419  df-mpo 7420  df-1st 7988  df-2nd 7989  df-splice 14727

This theorem is referenced by:  splid  14730  spllen  14731  splfv1  14732  splfv2a  14733  splval2  14734  gsumspl  18790  efgredleme  19692  efgredlemc  19694  efgcpbllemb  19704  frgpuplem  19721  splfv3  32674  cycpmco2f1  32840  cycpmco2rn  32841  cycpmco2lem2  32843  cycpmco2lem3  32844  cycpmco2lem4  32845  cycpmco2lem5  32846  cycpmco2lem6  32847  cycpmco2  32849