Theorem splvalpfxOLD 13820

Description: Obsolete proof of splval 13821 as of 12-Oct-2022. (Contributed by AV, 11-May-2020.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
splvalpfxOLD	⊢ ((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ ℕ0 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌)) → (𝑆 spliceOLD ⟨𝐹, 𝑇, 𝑅⟩) = (((𝑆 prefix 𝐹) ++ 𝑅) ++ (𝑆 substr ⟨𝑇, (♯‘𝑆)⟩)))

Proof of Theorem splvalpfxOLD

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

Step	Hyp	Ref	Expression
1		df-spliceOLD 13819	. . 3 ⊢ spliceOLD = (𝑠 ∈ V, 𝑏 ∈ V ↦ (((𝑠 substr ⟨0, (1st ‘(1st ‘𝑏))⟩) ++ (2nd ‘𝑏)) ++ (𝑠 substr ⟨(2nd ‘(1st ‘𝑏)), (♯‘𝑠)⟩)))
2	1	a1i 11	. 2 ⊢ ((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ ℕ0 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌)) → spliceOLD = (𝑠 ∈ V, 𝑏 ∈ V ↦ (((𝑠 substr ⟨0, (1st ‘(1st ‘𝑏))⟩) ++ (2nd ‘𝑏)) ++ (𝑠 substr ⟨(2nd ‘(1st ‘𝑏)), (♯‘𝑠)⟩))))
3		simprl 788	. . . . . 6 ⊢ (((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ ℕ0 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌)) ∧ (𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩)) → 𝑠 = 𝑆)
4		2fveq3 6415	. . . . . . . . 9 ⊢ (𝑏 = ⟨𝐹, 𝑇, 𝑅⟩ → (1st ‘(1st ‘𝑏)) = (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)))
5	4	adantl 474	. . . . . . . 8 ⊢ ((𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → (1st ‘(1st ‘𝑏)) = (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)))
6		ot1stg 7414	. . . . . . . . 9 ⊢ ((𝐹 ∈ ℕ0 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌) → (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)) = 𝐹)
7	6	adantl 474	. . . . . . . 8 ⊢ ((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ ℕ0 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌)) → (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)) = 𝐹)
8	5, 7	sylan9eqr 2854	. . . . . . 7 ⊢ (((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ ℕ0 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌)) ∧ (𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩)) → (1st ‘(1st ‘𝑏)) = 𝐹)
9	8	opeq2d 4599	. . . . . 6 ⊢ (((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ ℕ0 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌)) ∧ (𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩)) → ⟨0, (1st ‘(1st ‘𝑏))⟩ = ⟨0, 𝐹⟩)
10	3, 9	oveq12d 6895	. . . . 5 ⊢ (((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ ℕ0 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌)) ∧ (𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩)) → (𝑠 substr ⟨0, (1st ‘(1st ‘𝑏))⟩) = (𝑆 substr ⟨0, 𝐹⟩))
11		simp1 1167	. . . . . . . 8 ⊢ ((𝐹 ∈ ℕ0 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌) → 𝐹 ∈ ℕ0)
12	11	anim2i 611	. . . . . . 7 ⊢ ((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ ℕ0 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌)) → (𝑆 ∈ 𝑉 ∧ 𝐹 ∈ ℕ0))
13	12	adantr 473	. . . . . 6 ⊢ (((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ ℕ0 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌)) ∧ (𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩)) → (𝑆 ∈ 𝑉 ∧ 𝐹 ∈ ℕ0))
14		pfxval 13713	. . . . . 6 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐹 ∈ ℕ0) → (𝑆 prefix 𝐹) = (𝑆 substr ⟨0, 𝐹⟩))
15	13, 14	syl 17	. . . . 5 ⊢ (((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ ℕ0 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌)) ∧ (𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩)) → (𝑆 prefix 𝐹) = (𝑆 substr ⟨0, 𝐹⟩))
16	10, 15	eqtr4d 2835	. . . 4 ⊢ (((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ ℕ0 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌)) ∧ (𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩)) → (𝑠 substr ⟨0, (1st ‘(1st ‘𝑏))⟩) = (𝑆 prefix 𝐹))
17		fveq2 6410	. . . . . 6 ⊢ (𝑏 = ⟨𝐹, 𝑇, 𝑅⟩ → (2nd ‘𝑏) = (2nd ‘⟨𝐹, 𝑇, 𝑅⟩))
18	17	adantl 474	. . . . 5 ⊢ ((𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → (2nd ‘𝑏) = (2nd ‘⟨𝐹, 𝑇, 𝑅⟩))
19		ot3rdg 7416	. . . . . . 7 ⊢ (𝑅 ∈ 𝑌 → (2nd ‘⟨𝐹, 𝑇, 𝑅⟩) = 𝑅)
20	19	3ad2ant3 1166	. . . . . 6 ⊢ ((𝐹 ∈ ℕ0 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌) → (2nd ‘⟨𝐹, 𝑇, 𝑅⟩) = 𝑅)
21	20	adantl 474	. . . . 5 ⊢ ((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ ℕ0 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌)) → (2nd ‘⟨𝐹, 𝑇, 𝑅⟩) = 𝑅)
22	18, 21	sylan9eqr 2854	. . . 4 ⊢ (((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ ℕ0 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌)) ∧ (𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩)) → (2nd ‘𝑏) = 𝑅)
23	16, 22	oveq12d 6895	. . 3 ⊢ (((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ ℕ0 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌)) ∧ (𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩)) → ((𝑠 substr ⟨0, (1st ‘(1st ‘𝑏))⟩) ++ (2nd ‘𝑏)) = ((𝑆 prefix 𝐹) ++ 𝑅))
24		2fveq3 6415	. . . . . . 7 ⊢ (𝑏 = ⟨𝐹, 𝑇, 𝑅⟩ → (2nd ‘(1st ‘𝑏)) = (2nd ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)))
25	24	adantl 474	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → (2nd ‘(1st ‘𝑏)) = (2nd ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)))
26		ot2ndg 7415	. . . . . . 7 ⊢ ((𝐹 ∈ ℕ0 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌) → (2nd ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)) = 𝑇)
27	26	adantl 474	. . . . . 6 ⊢ ((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ ℕ0 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌)) → (2nd ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)) = 𝑇)
28	25, 27	sylan9eqr 2854	. . . . 5 ⊢ (((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ ℕ0 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌)) ∧ (𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩)) → (2nd ‘(1st ‘𝑏)) = 𝑇)
29	3	fveq2d 6414	. . . . 5 ⊢ (((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ ℕ0 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌)) ∧ (𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩)) → (♯‘𝑠) = (♯‘𝑆))
30	28, 29	opeq12d 4600	. . . 4 ⊢ (((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ ℕ0 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌)) ∧ (𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩)) → ⟨(2nd ‘(1st ‘𝑏)), (♯‘𝑠)⟩ = ⟨𝑇, (♯‘𝑆)⟩)
31	3, 30	oveq12d 6895	. . 3 ⊢ (((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ ℕ0 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌)) ∧ (𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩)) → (𝑠 substr ⟨(2nd ‘(1st ‘𝑏)), (♯‘𝑠)⟩) = (𝑆 substr ⟨𝑇, (♯‘𝑆)⟩))
32	23, 31	oveq12d 6895	. 2 ⊢ (((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ ℕ0 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌)) ∧ (𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩)) → (((𝑠 substr ⟨0, (1st ‘(1st ‘𝑏))⟩) ++ (2nd ‘𝑏)) ++ (𝑠 substr ⟨(2nd ‘(1st ‘𝑏)), (♯‘𝑠)⟩)) = (((𝑆 prefix 𝐹) ++ 𝑅) ++ (𝑆 substr ⟨𝑇, (♯‘𝑆)⟩)))
33		elex 3399	. . 3 ⊢ (𝑆 ∈ 𝑉 → 𝑆 ∈ V)
34	33	adantr 473	. 2 ⊢ ((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ ℕ0 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌)) → 𝑆 ∈ V)
35		otex 5123	. . 3 ⊢ ⟨𝐹, 𝑇, 𝑅⟩ ∈ V
36	35	a1i 11	. 2 ⊢ ((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ ℕ0 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌)) → ⟨𝐹, 𝑇, 𝑅⟩ ∈ V)
37		ovexd 6911	. 2 ⊢ ((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ ℕ0 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌)) → (((𝑆 prefix 𝐹) ++ 𝑅) ++ (𝑆 substr ⟨𝑇, (♯‘𝑆)⟩)) ∈ V)
38	2, 32, 34, 36, 37	ovmpt2d 7021	1 ⊢ ((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ ℕ0 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌)) → (𝑆 spliceOLD ⟨𝐹, 𝑇, 𝑅⟩) = (((𝑆 prefix 𝐹) ++ 𝑅) ++ (𝑆 substr ⟨𝑇, (♯‘𝑆)⟩)))

Syntax hints:   → wi 4   ∧ wa 385   ∧ w3a 1108   = wceq 1653   ∈ wcel 2157  Vcvv 3384  ⟨cop 4373  ⟨cotp 4375  ‘cfv 6100  (class class class)co 6877   ↦ cmpt2 6879  1^st c1st 7398  2^nd c2nd 7399  0cc0 10223  ℕ₀cn0 11577  ♯chash 13367   ++ cconcat 13587   substr csubstr 13661   prefix cpfx 13710   spliceOLD cspliceold 13817

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2776  ax-sep 4974  ax-nul 4982  ax-pow 5034  ax-pr 5096  ax-un 7182

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2785  df-cleq 2791  df-clel 2794  df-nfc 2929  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3386  df-sbc 3633  df-dif 3771  df-un 3773  df-in 3775  df-ss 3782  df-nul 4115  df-if 4277  df-sn 4368  df-pr 4370  df-op 4374  df-ot 4376  df-uni 4628  df-br 4843  df-opab 4905  df-mpt 4922  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-iota 6063  df-fun 6102  df-fv 6108  df-ov 6880  df-oprab 6881  df-mpt2 6882  df-1st 7400  df-2nd 7401  df-pfx 13711  df-spliceOLD 13819

This theorem is referenced by: (None)