Theorem splval 14651

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 14650	. . 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 769	. . . . 5 ⊢ (((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ 𝑊 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌)) ∧ (𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩)) → 𝑠 = 𝑆)
4		2fveq3 6852	. . . . . . 7 ⊢ (𝑏 = ⟨𝐹, 𝑇, 𝑅⟩ → (1st ‘(1st ‘𝑏)) = (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)))
5	4	adantl 482	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → (1st ‘(1st ‘𝑏)) = (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)))
6		ot1stg 7940	. . . . . . 7 ⊢ ((𝐹 ∈ 𝑊 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌) → (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)) = 𝐹)
7	6	adantl 482	. . . . . 6 ⊢ ((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ 𝑊 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌)) → (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)) = 𝐹)
8	5, 7	sylan9eqr 2793	. . . . 5 ⊢ (((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ 𝑊 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌)) ∧ (𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩)) → (1st ‘(1st ‘𝑏)) = 𝐹)
9	3, 8	oveq12d 7380	. . . 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 6847	. . . . . 6 ⊢ (𝑏 = ⟨𝐹, 𝑇, 𝑅⟩ → (2nd ‘𝑏) = (2nd ‘⟨𝐹, 𝑇, 𝑅⟩))
11	10	adantl 482	. . . . 5 ⊢ ((𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → (2nd ‘𝑏) = (2nd ‘⟨𝐹, 𝑇, 𝑅⟩))
12		ot3rdg 7942	. . . . . . 7 ⊢ (𝑅 ∈ 𝑌 → (2nd ‘⟨𝐹, 𝑇, 𝑅⟩) = 𝑅)
13	12	3ad2ant3 1135	. . . . . 6 ⊢ ((𝐹 ∈ 𝑊 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌) → (2nd ‘⟨𝐹, 𝑇, 𝑅⟩) = 𝑅)
14	13	adantl 482	. . . . 5 ⊢ ((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ 𝑊 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌)) → (2nd ‘⟨𝐹, 𝑇, 𝑅⟩) = 𝑅)
15	11, 14	sylan9eqr 2793	. . . 4 ⊢ (((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ 𝑊 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌)) ∧ (𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩)) → (2nd ‘𝑏) = 𝑅)
16	9, 15	oveq12d 7380	. . 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 6852	. . . . . . 7 ⊢ (𝑏 = ⟨𝐹, 𝑇, 𝑅⟩ → (2nd ‘(1st ‘𝑏)) = (2nd ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)))
18	17	adantl 482	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → (2nd ‘(1st ‘𝑏)) = (2nd ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)))
19		ot2ndg 7941	. . . . . . 7 ⊢ ((𝐹 ∈ 𝑊 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌) → (2nd ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)) = 𝑇)
20	19	adantl 482	. . . . . 6 ⊢ ((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ 𝑊 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌)) → (2nd ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)) = 𝑇)
21	18, 20	sylan9eqr 2793	. . . . 5 ⊢ (((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ 𝑊 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌)) ∧ (𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩)) → (2nd ‘(1st ‘𝑏)) = 𝑇)
22	3	fveq2d 6851	. . . . 5 ⊢ (((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ 𝑊 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌)) ∧ (𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩)) → (♯‘𝑠) = (♯‘𝑆))
23	21, 22	opeq12d 4843	. . . 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 7380	. . 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 7380	. 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 3464	. . 3 ⊢ (𝑆 ∈ 𝑉 → 𝑆 ∈ V)
27	26	adantr 481	. 2 ⊢ ((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ 𝑊 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌)) → 𝑆 ∈ V)
28		otex 5427	. . 3 ⊢ ⟨𝐹, 𝑇, 𝑅⟩ ∈ V
29	28	a1i 11	. 2 ⊢ ((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ 𝑊 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌)) → ⟨𝐹, 𝑇, 𝑅⟩ ∈ V)
30		ovexd 7397	. 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 7512	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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  Vcvv 3446  ⟨cop 4597  ⟨cotp 4599  ‘cfv 6501  (class class class)co 7362   ∈ cmpo 7364  1^st c1st 7924  2^nd c2nd 7925  ♯chash 14240   ++ cconcat 14470   substr csubstr 14540   prefix cpfx 14570   splice csplice 14649

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3406  df-v 3448  df-sbc 3743  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-ot 4600  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-iota 6453  df-fun 6503  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-1st 7926  df-2nd 7927  df-splice 14650

This theorem is referenced by:  splid  14653  spllen  14654  splfv1  14655  splfv2a  14656  splval2  14657  gsumspl  18668  efgredleme  19539  efgredlemc  19541  efgcpbllemb  19551  frgpuplem  19568  splfv3  31882  cycpmco2f1  32043  cycpmco2rn  32044  cycpmco2lem2  32046  cycpmco2lem3  32047  cycpmco2lem4  32048  cycpmco2lem5  32049  cycpmco2lem6  32050  cycpmco2  32052