Theorem splval 14668

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 14667	. . 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 6836	. . . . . . 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 7944	. . . . . . 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 7373	. . . 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 6831	. . . . . 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 7946	. . . . . . 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 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 7373	. . 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 6836	. . . . . . 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 7945	. . . . . . 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 6835	. . . . 5 ⊢ (((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ 𝑊 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌)) ∧ (𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩)) → (♯‘𝑠) = (♯‘𝑆))
23	21, 22	opeq12d 4834	. . . 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 7373	. . 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 7373	. 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 3459	. . 3 ⊢ (𝑆 ∈ 𝑉 → 𝑆 ∈ V)
27	26	adantr 480	. 2 ⊢ ((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ 𝑊 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌)) → 𝑆 ∈ V)
28		otex 5410	. . 3 ⊢ ⟨𝐹, 𝑇, 𝑅⟩ ∈ V
29	28	a1i 11	. 2 ⊢ ((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ 𝑊 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌)) → ⟨𝐹, 𝑇, 𝑅⟩ ∈ V)
30		ovexd 7390	. 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 7507	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 1086   = wceq 1541   ∈ wcel 2113  Vcvv 3438  ⟨cop 4583  ⟨cotp 4585  ‘cfv 6489  (class class class)co 7355   ∈ cmpo 7357  1^st c1st 7928  2^nd c2nd 7929  ♯chash 14247   ++ cconcat 14487   substr csubstr 14558   prefix cpfx 14588   splice csplice 14666

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374  ax-un 7677

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-sbc 3739  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-ot 4586  df-uni 4861  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-iota 6445  df-fun 6491  df-fv 6497  df-ov 7358  df-oprab 7359  df-mpo 7360  df-1st 7930  df-2nd 7931  df-splice 14667

This theorem is referenced by:  splid  14670  spllen  14671  splfv1  14672  splfv2a  14673  splval2  14674  gsumspl  18762  efgredleme  19665  efgredlemc  19667  efgcpbllemb  19677  frgpuplem  19694  splfv3  32950  cycpmco2f1  33104  cycpmco2rn  33105  cycpmco2lem2  33107  cycpmco2lem3  33108  cycpmco2lem4  33109  cycpmco2lem5  33110  cycpmco2lem6  33111  cycpmco2  33113