Theorem sseqval 31867

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 31863	. . 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 771	. . 3 ⊢ ((𝜑 ∧ (𝑚 = 𝑀 ∧ 𝑓 = 𝐹)) → 𝑚 = 𝑀)
4	3	fveq2d 6663	. . . . 5 ⊢ ((𝜑 ∧ (𝑚 = 𝑀 ∧ 𝑓 = 𝐹)) → (♯‘𝑚) = (♯‘𝑀))
5		simp1rr 1237	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑚 = 𝑀 ∧ 𝑓 = 𝐹)) ∧ 𝑥 ∈ V ∧ 𝑦 ∈ V) → 𝑓 = 𝐹)
6	5	fveq1d 6661	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑚 = 𝑀 ∧ 𝑓 = 𝐹)) ∧ 𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑓‘𝑥) = (𝐹‘𝑥))
7	6	s1eqd 13995	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑚 = 𝑀 ∧ 𝑓 = 𝐹)) ∧ 𝑥 ∈ V ∧ 𝑦 ∈ V) → ⟨“(𝑓‘𝑥)”⟩ = ⟨“(𝐹‘𝑥)”⟩)
8	7	oveq2d 7167	. . . . . 6 ⊢ (((𝜑 ∧ (𝑚 = 𝑀 ∧ 𝑓 = 𝐹)) ∧ 𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ++ ⟨“(𝑓‘𝑥)”⟩) = (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩))
9	8	mpoeq3dva 7226	. . . . 5 ⊢ ((𝜑 ∧ (𝑚 = 𝑀 ∧ 𝑓 = 𝐹)) → (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝑓‘𝑥)”⟩)) = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)))
10		simprr 773	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 = 𝑀 ∧ 𝑓 = 𝐹)) → 𝑓 = 𝐹)
11	10, 3	fveq12d 6666	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 = 𝑀 ∧ 𝑓 = 𝐹)) → (𝑓‘𝑚) = (𝐹‘𝑀))
12	11	s1eqd 13995	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑚 = 𝑀 ∧ 𝑓 = 𝐹)) → ⟨“(𝑓‘𝑚)”⟩ = ⟨“(𝐹‘𝑀)”⟩)
13	3, 12	oveq12d 7169	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 = 𝑀 ∧ 𝑓 = 𝐹)) → (𝑚 ++ ⟨“(𝑓‘𝑚)”⟩) = (𝑀 ++ ⟨“(𝐹‘𝑀)”⟩))
14	13	sneqd 4535	. . . . . 6 ⊢ ((𝜑 ∧ (𝑚 = 𝑀 ∧ 𝑓 = 𝐹)) → {(𝑚 ++ ⟨“(𝑓‘𝑚)”⟩)} = {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)})
15	14	xpeq2d 5555	. . . . 5 ⊢ ((𝜑 ∧ (𝑚 = 𝑀 ∧ 𝑓 = 𝐹)) → (ℕ0 × {(𝑚 ++ ⟨“(𝑓‘𝑚)”⟩)}) = (ℕ0 × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)}))
16	4, 9, 15	seqeq123d 13420	. . . 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 5703	. . 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 4070	. 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 3429	. . 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 13916	. . . . 5 ⊢ (𝑆 ∈ V → Word 𝑆 ∈ V)
26		inex1g 5190	. . . . 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 2857	. . 3 ⊢ (𝜑 → 𝑊 ∈ V)
29		fex 6981	. . 3 ⊢ ((𝐹:𝑊⟶𝑆 ∧ 𝑊 ∈ V) → 𝐹 ∈ V)
30	22, 28, 29	syl2anc 588	. 2 ⊢ (𝜑 → 𝐹 ∈ V)
31		df-lsw 13955	. . . . . 6 ⊢ lastS = (𝑥 ∈ V ↦ (𝑥‘((♯‘𝑥) − 1)))
32	31	funmpt2 6375	. . . . 5 ⊢ Fun lastS
33	32	a1i 11	. . . 4 ⊢ (𝜑 → Fun lastS)
34		seqex 13413	. . . . 5 ⊢ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ0 × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)})) ∈ V
35	34	a1i 11	. . . 4 ⊢ (𝜑 → seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ0 × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)})) ∈ V)
36		cofunexg 7655	. . . 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)
37	33, 35, 36	syl2anc 588	. . 3 ⊢ (𝜑 → (lastS ∘ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ0 × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)}))) ∈ V)
38		unexg 7471	. . 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)
39	21, 37, 38	syl2anc 588	. 2 ⊢ (𝜑 → (𝑀 ∪ (lastS ∘ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ0 × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)})))) ∈ V)
40	2, 18, 21, 30, 39	ovmpod 7298	1 ⊢ (𝜑 → (𝑀seqstr𝐹) = (𝑀 ∪ (lastS ∘ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ0 × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)})))))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1085   = wceq 1539   ∈ wcel 2112  Vcvv 3410   ∪ cun 3857   ∩ cin 3858  {csn 4523   × cxp 5523  ^◡ccnv 5524   “ cima 5528   ∘ ccom 5529  Fun wfun 6330  ⟶wf 6332  ‘cfv 6336  (class class class)co 7151   ∈ cmpo 7153  1c1 10569   − cmin 10901  ℕ₀cn0 11927  ℤ_≥cuz 12275  seqcseq 13411  ♯chash 13733  Word cword 13906  lastSclsw 13954   ++ cconcat 13962  ⟨“cs1 13989  seq_strcsseq 31862

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5235  ax-pr 5299  ax-un 7460  ax-inf2 9130  ax-cnex 10624  ax-1cn 10626  ax-addcl 10628

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-reu 3078  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-pss 3878  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-tp 4528  df-op 4530  df-uni 4800  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5431  df-eprel 5436  df-po 5444  df-so 5445  df-fr 5484  df-we 5486  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6127  df-ord 6173  df-on 6174  df-lim 6175  df-suc 6176  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7581  df-wrecs 7958  df-recs 8019  df-rdg 8057  df-map 8419  df-nn 11668  df-n0 11928  df-seq 13412  df-word 13907  df-lsw 13955  df-s1 13990  df-sseq 31863

This theorem is referenced by:  sseqfv1  31868  sseqfn  31869  sseqf  31871  sseqfv2  31873