Theorem sseqval 33375

Description: Value of the strong sequence builder function. The set 𝑊 represents here the words of length greater than or equal to the lenght of the initial sequence 𝑀. (Contributed by Thierry Arnoux, 21-Apr-2019.)

Hypotheses

Ref	Expression
sseqval.1	⊢ (𝜑 → 𝑆 ∈ V)
sseqval.2	⊢ (𝜑 → 𝑀 ∈ Word 𝑆)
sseqval.3	⊢ 𝑊 = (Word 𝑆 ∩ (◡♯ “ (ℤ≥‘(♯‘𝑀))))
sseqval.4	⊢ (𝜑 → 𝐹:𝑊⟶𝑆)

Assertion

Ref	Expression
sseqval	⊢ (𝜑 → (𝑀seqstr𝐹) = (𝑀 ∪ (lastS ∘ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ0 × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)})))))

Distinct variable groups:   𝑥,𝑦,𝐹   𝑥,𝑀,𝑦   𝜑,𝑥,𝑦

Allowed substitution hints:   𝑆(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem sseqval

Dummy variables 𝑓 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-sseq 33371	. . 3 ⊢ seqstr = (𝑚 ∈ V, 𝑓 ∈ V ↦ (𝑚 ∪ (lastS ∘ seq(♯‘𝑚)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝑓‘𝑥)”⟩)), (ℕ0 × {(𝑚 ++ ⟨“(𝑓‘𝑚)”⟩)})))))
2	1	a1i 11	. 2 ⊢ (𝜑 → seqstr = (𝑚 ∈ V, 𝑓 ∈ V ↦ (𝑚 ∪ (lastS ∘ seq(♯‘𝑚)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝑓‘𝑥)”⟩)), (ℕ0 × {(𝑚 ++ ⟨“(𝑓‘𝑚)”⟩)}))))))
3		simprl 769	. . 3 ⊢ ((𝜑 ∧ (𝑚 = 𝑀 ∧ 𝑓 = 𝐹)) → 𝑚 = 𝑀)
4	3	fveq2d 6892	. . . . 5 ⊢ ((𝜑 ∧ (𝑚 = 𝑀 ∧ 𝑓 = 𝐹)) → (♯‘𝑚) = (♯‘𝑀))
5		simp1rr 1239	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑚 = 𝑀 ∧ 𝑓 = 𝐹)) ∧ 𝑥 ∈ V ∧ 𝑦 ∈ V) → 𝑓 = 𝐹)
6	5	fveq1d 6890	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑚 = 𝑀 ∧ 𝑓 = 𝐹)) ∧ 𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑓‘𝑥) = (𝐹‘𝑥))
7	6	s1eqd 14547	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑚 = 𝑀 ∧ 𝑓 = 𝐹)) ∧ 𝑥 ∈ V ∧ 𝑦 ∈ V) → ⟨“(𝑓‘𝑥)”⟩ = ⟨“(𝐹‘𝑥)”⟩)
8	7	oveq2d 7421	. . . . . 6 ⊢ (((𝜑 ∧ (𝑚 = 𝑀 ∧ 𝑓 = 𝐹)) ∧ 𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ++ ⟨“(𝑓‘𝑥)”⟩) = (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩))
9	8	mpoeq3dva 7482	. . . . 5 ⊢ ((𝜑 ∧ (𝑚 = 𝑀 ∧ 𝑓 = 𝐹)) → (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝑓‘𝑥)”⟩)) = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)))
10		simprr 771	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 = 𝑀 ∧ 𝑓 = 𝐹)) → 𝑓 = 𝐹)
11	10, 3	fveq12d 6895	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 = 𝑀 ∧ 𝑓 = 𝐹)) → (𝑓‘𝑚) = (𝐹‘𝑀))
12	11	s1eqd 14547	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑚 = 𝑀 ∧ 𝑓 = 𝐹)) → ⟨“(𝑓‘𝑚)”⟩ = ⟨“(𝐹‘𝑀)”⟩)
13	3, 12	oveq12d 7423	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 = 𝑀 ∧ 𝑓 = 𝐹)) → (𝑚 ++ ⟨“(𝑓‘𝑚)”⟩) = (𝑀 ++ ⟨“(𝐹‘𝑀)”⟩))
14	13	sneqd 4639	. . . . . 6 ⊢ ((𝜑 ∧ (𝑚 = 𝑀 ∧ 𝑓 = 𝐹)) → {(𝑚 ++ ⟨“(𝑓‘𝑚)”⟩)} = {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)})
15	14	xpeq2d 5705	. . . . 5 ⊢ ((𝜑 ∧ (𝑚 = 𝑀 ∧ 𝑓 = 𝐹)) → (ℕ0 × {(𝑚 ++ ⟨“(𝑓‘𝑚)”⟩)}) = (ℕ0 × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)}))
16	4, 9, 15	seqeq123d 13971	. . . 4 ⊢ ((𝜑 ∧ (𝑚 = 𝑀 ∧ 𝑓 = 𝐹)) → seq(♯‘𝑚)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝑓‘𝑥)”⟩)), (ℕ0 × {(𝑚 ++ ⟨“(𝑓‘𝑚)”⟩)})) = seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ0 × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)})))
17	16	coeq2d 5860	. . 3 ⊢ ((𝜑 ∧ (𝑚 = 𝑀 ∧ 𝑓 = 𝐹)) → (lastS ∘ seq(♯‘𝑚)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝑓‘𝑥)”⟩)), (ℕ0 × {(𝑚 ++ ⟨“(𝑓‘𝑚)”⟩)}))) = (lastS ∘ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ0 × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)}))))
18	3, 17	uneq12d 4163	. 2 ⊢ ((𝜑 ∧ (𝑚 = 𝑀 ∧ 𝑓 = 𝐹)) → (𝑚 ∪ (lastS ∘ seq(♯‘𝑚)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝑓‘𝑥)”⟩)), (ℕ0 × {(𝑚 ++ ⟨“(𝑓‘𝑚)”⟩)})))) = (𝑀 ∪ (lastS ∘ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ0 × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)})))))
19		sseqval.2	. . 3 ⊢ (𝜑 → 𝑀 ∈ Word 𝑆)
20		elex 3492	. . 3 ⊢ (𝑀 ∈ Word 𝑆 → 𝑀 ∈ V)
21	19, 20	syl 17	. 2 ⊢ (𝜑 → 𝑀 ∈ V)
22		sseqval.4	. . 3 ⊢ (𝜑 → 𝐹:𝑊⟶𝑆)
23		sseqval.3	. . . 4 ⊢ 𝑊 = (Word 𝑆 ∩ (◡♯ “ (ℤ≥‘(♯‘𝑀))))
24		sseqval.1	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ V)
25		wrdexg 14470	. . . . 5 ⊢ (𝑆 ∈ V → Word 𝑆 ∈ V)
26		inex1g 5318	. . . . 5 ⊢ (Word 𝑆 ∈ V → (Word 𝑆 ∩ (◡♯ “ (ℤ≥‘(♯‘𝑀)))) ∈ V)
27	24, 25, 26	3syl 18	. . . 4 ⊢ (𝜑 → (Word 𝑆 ∩ (◡♯ “ (ℤ≥‘(♯‘𝑀)))) ∈ V)
28	23, 27	eqeltrid 2837	. . 3 ⊢ (𝜑 → 𝑊 ∈ V)
29	22, 28	fexd 7225	. 2 ⊢ (𝜑 → 𝐹 ∈ V)
30		df-lsw 14509	. . . . . 6 ⊢ lastS = (𝑥 ∈ V ↦ (𝑥‘((♯‘𝑥) − 1)))
31	30	funmpt2 6584	. . . . 5 ⊢ Fun lastS
32	31	a1i 11	. . . 4 ⊢ (𝜑 → Fun lastS)
33		seqex 13964	. . . . 5 ⊢ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ0 × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)})) ∈ V
34	33	a1i 11	. . . 4 ⊢ (𝜑 → seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ0 × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)})) ∈ V)
35		cofunexg 7931	. . . 4 ⊢ ((Fun lastS ∧ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ0 × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)})) ∈ V) → (lastS ∘ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ0 × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)}))) ∈ V)
36	32, 34, 35	syl2anc 584	. . 3 ⊢ (𝜑 → (lastS ∘ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ0 × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)}))) ∈ V)
37		unexg 7732	. . 3 ⊢ ((𝑀 ∈ V ∧ (lastS ∘ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ0 × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)}))) ∈ V) → (𝑀 ∪ (lastS ∘ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ0 × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)})))) ∈ V)
38	21, 36, 37	syl2anc 584	. 2 ⊢ (𝜑 → (𝑀 ∪ (lastS ∘ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ0 × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)})))) ∈ V)
39	2, 18, 21, 29, 38	ovmpod 7556	1 ⊢ (𝜑 → (𝑀seqstr𝐹) = (𝑀 ∪ (lastS ∘ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ0 × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)})))))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  Vcvv 3474   ∪ cun 3945   ∩ cin 3946  {csn 4627   × cxp 5673  ^◡ccnv 5674   “ cima 5678   ∘ ccom 5679  Fun wfun 6534  ⟶wf 6536  ‘cfv 6540  (class class class)co 7405   ∈ cmpo 7407  1c1 11107   − cmin 11440  ℕ₀cn0 12468  ℤ_≥cuz 12818  seqcseq 13962  ♯chash 14286  Word cword 14460  lastSclsw 14508   ++ cconcat 14516  ⟨“cs1 14541  seq_strcsseq 33370

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 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-inf2 9632  ax-cnex 11162  ax-1cn 11164  ax-addcl 11166

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-map 8818  df-nn 12209  df-n0 12469  df-seq 13963  df-word 14461  df-lsw 14509  df-s1 14542  df-sseq 33371

This theorem is referenced by:  sseqfv1  33376  sseqfn  33377  sseqf  33379  sseqfv2  33381