Theorem splval 14812

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 14811	. . 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 783	. . . . 5 ⊢ (((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ 𝑊 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌)) ∧ (𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩)) → 𝑠 = 𝑆)
4		2fveq3 6893	. . . . . . 7 ⊢ (𝑏 = ⟨𝐹, 𝑇, 𝑅⟩ → (1st ‘(1st ‘𝑏)) = (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)))
5	4	adantl 487	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → (1st ‘(1st ‘𝑏)) = (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)))
6		ot1stg 8009	. . . . . . 7 ⊢ ((𝐹 ∈ 𝑊 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌) → (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)) = 𝐹)
7	6	adantl 487	. . . . . 6 ⊢ ((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ 𝑊 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌)) → (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)) = 𝐹)
8	5, 7	sylan9eqr 2823	. . . . 5 ⊢ (((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ 𝑊 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌)) ∧ (𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩)) → (1st ‘(1st ‘𝑏)) = 𝐹)
9	3, 8	oveq12d 7441	. . . 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 6888	. . . . . 6 ⊢ (𝑏 = ⟨𝐹, 𝑇, 𝑅⟩ → (2nd ‘𝑏) = (2nd ‘⟨𝐹, 𝑇, 𝑅⟩))
11	10	adantl 487	. . . . 5 ⊢ ((𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → (2nd ‘𝑏) = (2nd ‘⟨𝐹, 𝑇, 𝑅⟩))
12		ot3rdg 8011	. . . . . . 7 ⊢ (𝑅 ∈ 𝑌 → (2nd ‘⟨𝐹, 𝑇, 𝑅⟩) = 𝑅)
13	12	3ad2ant3 1153	. . . . . 6 ⊢ ((𝐹 ∈ 𝑊 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌) → (2nd ‘⟨𝐹, 𝑇, 𝑅⟩) = 𝑅)
14	13	adantl 487	. . . . 5 ⊢ ((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ 𝑊 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌)) → (2nd ‘⟨𝐹, 𝑇, 𝑅⟩) = 𝑅)
15	11, 14	sylan9eqr 2823	. . . 4 ⊢ (((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ 𝑊 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌)) ∧ (𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩)) → (2nd ‘𝑏) = 𝑅)
16	9, 15	oveq12d 7441	. . 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 6893	. . . . . . 7 ⊢ (𝑏 = ⟨𝐹, 𝑇, 𝑅⟩ → (2nd ‘(1st ‘𝑏)) = (2nd ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)))
18	17	adantl 487	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → (2nd ‘(1st ‘𝑏)) = (2nd ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)))
19		ot2ndg 8010	. . . . . . 7 ⊢ ((𝐹 ∈ 𝑊 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌) → (2nd ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)) = 𝑇)
20	19	adantl 487	. . . . . 6 ⊢ ((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ 𝑊 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌)) → (2nd ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)) = 𝑇)
21	18, 20	sylan9eqr 2823	. . . . 5 ⊢ (((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ 𝑊 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌)) ∧ (𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩)) → (2nd ‘(1st ‘𝑏)) = 𝑇)
22	3	fveq2d 6892	. . . . 5 ⊢ (((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ 𝑊 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌)) ∧ (𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩)) → (♯‘𝑠) = (♯‘𝑆))
23	21, 22	opeq12d 4851	. . . 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 7441	. . 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 7441	. 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 3479	. . 3 ⊢ (𝑆 ∈ 𝑉 → 𝑆 ∈ V)
27	26	adantr 486	. 2 ⊢ ((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ 𝑊 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌)) → 𝑆 ∈ V)
28		otex 5452	. . 3 ⊢ ⟨𝐹, 𝑇, 𝑅⟩ ∈ V
29	28	a1i 11	. 2 ⊢ ((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ 𝑊 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌)) → ⟨𝐹, 𝑇, 𝑅⟩ ∈ V)
30		ovexd 7458	. 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 7575	1 ⊢ ((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ 𝑊 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌)) → (𝑆 splice ⟨𝐹, 𝑇, 𝑅⟩) = (((𝑆 prefix 𝐹) ++ 𝑅) ++ (𝑆 substr ⟨𝑇, (♯‘𝑆)⟩)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  Vcvv 3458  ⟨cop 4600  ⟨cotp 4602  ‘cfv 6543  (class class class)co 7423   ∈ cmpo 7425  1^st c1st 7993  2^nd c2nd 7994  ♯chash 14386   ++ cconcat 14627   substr csubstr 14700   prefix cpfx 14732   splice csplice 14810

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pr 5409  ax-un 7745

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-ot 4603  df-uni 4878  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-iota 6499  df-fun 6545  df-fv 6551  df-ov 7426  df-oprab 7427  df-mpo 7428  df-1st 7995  df-2nd 7996  df-splice 14811

This theorem is used by:  splid  14814  spllen  14815  splfv1  14816  splfv2a  14817  splval2  14818  gsumspl  18934  efgredleme  19844  efgredlemc  19846  efgcpbllemb  19856  frgpuplem  19873  splfv3  33309  cycpmco2f1  33475  cycpmco2rn  33476  cycpmco2lem2  33478  cycpmco2lem3  33479  cycpmco2lem4  33480  cycpmco2lem5  33481  cycpmco2lem6  33482  cycpmco2  33484