Theorem sseqval 31324

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 31319	. . 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 759	. . 3 ⊢ ((𝜑 ∧ (𝑚 = 𝑀 ∧ 𝑓 = 𝐹)) → 𝑚 = 𝑀)
4	3	fveq2d 6500	. . . . 5 ⊢ ((𝜑 ∧ (𝑚 = 𝑀 ∧ 𝑓 = 𝐹)) → (♯‘𝑚) = (♯‘𝑀))
5		simp1rr 1220	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑚 = 𝑀 ∧ 𝑓 = 𝐹)) ∧ 𝑥 ∈ V ∧ 𝑦 ∈ V) → 𝑓 = 𝐹)
6	5	fveq1d 6498	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑚 = 𝑀 ∧ 𝑓 = 𝐹)) ∧ 𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑓‘𝑥) = (𝐹‘𝑥))
7	6	s1eqd 13762	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑚 = 𝑀 ∧ 𝑓 = 𝐹)) ∧ 𝑥 ∈ V ∧ 𝑦 ∈ V) → ⟨“(𝑓‘𝑥)”⟩ = ⟨“(𝐹‘𝑥)”⟩)
8	7	oveq2d 6990	. . . . . 6 ⊢ (((𝜑 ∧ (𝑚 = 𝑀 ∧ 𝑓 = 𝐹)) ∧ 𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ++ ⟨“(𝑓‘𝑥)”⟩) = (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩))
9	8	mpoeq3dva 7047	. . . . 5 ⊢ ((𝜑 ∧ (𝑚 = 𝑀 ∧ 𝑓 = 𝐹)) → (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝑓‘𝑥)”⟩)) = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)))
10		simprr 761	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 = 𝑀 ∧ 𝑓 = 𝐹)) → 𝑓 = 𝐹)
11	10, 3	fveq12d 6503	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 = 𝑀 ∧ 𝑓 = 𝐹)) → (𝑓‘𝑚) = (𝐹‘𝑀))
12	11	s1eqd 13762	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑚 = 𝑀 ∧ 𝑓 = 𝐹)) → ⟨“(𝑓‘𝑚)”⟩ = ⟨“(𝐹‘𝑀)”⟩)
13	3, 12	oveq12d 6992	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 = 𝑀 ∧ 𝑓 = 𝐹)) → (𝑚 ++ ⟨“(𝑓‘𝑚)”⟩) = (𝑀 ++ ⟨“(𝐹‘𝑀)”⟩))
14	13	sneqd 4447	. . . . . 6 ⊢ ((𝜑 ∧ (𝑚 = 𝑀 ∧ 𝑓 = 𝐹)) → {(𝑚 ++ ⟨“(𝑓‘𝑚)”⟩)} = {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)})
15	14	xpeq2d 5433	. . . . 5 ⊢ ((𝜑 ∧ (𝑚 = 𝑀 ∧ 𝑓 = 𝐹)) → (ℕ0 × {(𝑚 ++ ⟨“(𝑓‘𝑚)”⟩)}) = (ℕ0 × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)}))
16	4, 9, 15	seqeq123d 13191	. . . 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 5579	. . 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 4022	. 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 3426	. . 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 13680	. . . . 5 ⊢ (𝑆 ∈ V → Word 𝑆 ∈ V)
26		inex1g 5076	. . . . 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	syl5eqel 2863	. . 3 ⊢ (𝜑 → 𝑊 ∈ V)
29		fex 6813	. . 3 ⊢ ((𝐹:𝑊⟶𝑆 ∧ 𝑊 ∈ V) → 𝐹 ∈ V)
30	22, 28, 29	syl2anc 576	. 2 ⊢ (𝜑 → 𝐹 ∈ V)
31		df-lsw 13724	. . . . . 6 ⊢ lastS = (𝑥 ∈ V ↦ (𝑥‘((♯‘𝑥) − 1)))
32	31	funmpt2 6224	. . . . 5 ⊢ Fun lastS
33	32	a1i 11	. . . 4 ⊢ (𝜑 → Fun lastS)
34		seqex 13184	. . . . 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 7460	. . . 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 576	. . 3 ⊢ (𝜑 → (lastS ∘ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ0 × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)}))) ∈ V)
38		unexg 7287	. . 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 576	. 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 7116	1 ⊢ (𝜑 → (𝑀seqstr𝐹) = (𝑀 ∪ (lastS ∘ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ0 × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)})))))

Syntax hints:   → wi 4   ∧ wa 387   ∧ w3a 1069   = wceq 1508   ∈ wcel 2051  Vcvv 3408   ∪ cun 3820   ∩ cin 3821  {csn 4435   × cxp 5401  ^◡ccnv 5402   “ cima 5406   ∘ ccom 5407  Fun wfun 6179  ⟶wf 6181  ‘cfv 6185  (class class class)co 6974   ∈ cmpo 6976  1c1 10334   − cmin 10668  ℕ₀cn0 11705  ℤ_≥cuz 12056  seqcseq 13182  ♯chash 13503  Word cword 13670  lastSclsw 13723   ++ cconcat 13731  ⟨“cs1 13756  seq_strcsseq 31318

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-rep 5045  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277  ax-inf2 8896  ax-cnex 10389  ax-1cn 10391  ax-addcl 10393

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3or 1070  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ne 2961  df-ral 3086  df-rex 3087  df-reu 3088  df-rab 3090  df-v 3410  df-sbc 3675  df-csb 3780  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-pss 3838  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4709  df-iun 4790  df-br 4926  df-opab 4988  df-mpt 5005  df-tr 5027  df-id 5308  df-eprel 5313  df-po 5322  df-so 5323  df-fr 5362  df-we 5364  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-pred 5983  df-ord 6029  df-on 6030  df-lim 6031  df-suc 6032  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-f1 6190  df-fo 6191  df-f1o 6192  df-fv 6193  df-ov 6977  df-oprab 6978  df-mpo 6979  df-om 7395  df-wrecs 7748  df-recs 7810  df-rdg 7848  df-map 8206  df-nn 11438  df-n0 11706  df-seq 13183  df-word 13671  df-lsw 13724  df-s1 13757  df-sseq 31319

This theorem is referenced by:  sseqfv1  31325  sseqfn  31326  sseqf  31328  sseqfv2  31330