Theorem sseqval 34565

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 34561	. . 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 6846	. . . . 5 ⊢ ((𝜑 ∧ (𝑚 = 𝑀 ∧ 𝑓 = 𝐹)) → (♯‘𝑚) = (♯‘𝑀))
5		simp1rr 1241	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑚 = 𝑀 ∧ 𝑓 = 𝐹)) ∧ 𝑥 ∈ V ∧ 𝑦 ∈ V) → 𝑓 = 𝐹)
6	5	fveq1d 6844	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑚 = 𝑀 ∧ 𝑓 = 𝐹)) ∧ 𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑓‘𝑥) = (𝐹‘𝑥))
7	6	s1eqd 14537	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑚 = 𝑀 ∧ 𝑓 = 𝐹)) ∧ 𝑥 ∈ V ∧ 𝑦 ∈ V) → ⟨“(𝑓‘𝑥)”⟩ = ⟨“(𝐹‘𝑥)”⟩)
8	7	oveq2d 7384	. . . . . 6 ⊢ (((𝜑 ∧ (𝑚 = 𝑀 ∧ 𝑓 = 𝐹)) ∧ 𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ++ ⟨“(𝑓‘𝑥)”⟩) = (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩))
9	8	mpoeq3dva 7445	. . . . 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 6849	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 = 𝑀 ∧ 𝑓 = 𝐹)) → (𝑓‘𝑚) = (𝐹‘𝑀))
12	11	s1eqd 14537	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑚 = 𝑀 ∧ 𝑓 = 𝐹)) → ⟨“(𝑓‘𝑚)”⟩ = ⟨“(𝐹‘𝑀)”⟩)
13	3, 12	oveq12d 7386	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 = 𝑀 ∧ 𝑓 = 𝐹)) → (𝑚 ++ ⟨“(𝑓‘𝑚)”⟩) = (𝑀 ++ ⟨“(𝐹‘𝑀)”⟩))
14	13	sneqd 4594	. . . . . 6 ⊢ ((𝜑 ∧ (𝑚 = 𝑀 ∧ 𝑓 = 𝐹)) → {(𝑚 ++ ⟨“(𝑓‘𝑚)”⟩)} = {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)})
15	14	xpeq2d 5662	. . . . 5 ⊢ ((𝜑 ∧ (𝑚 = 𝑀 ∧ 𝑓 = 𝐹)) → (ℕ0 × {(𝑚 ++ ⟨“(𝑓‘𝑚)”⟩)}) = (ℕ0 × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)}))
16	4, 9, 15	seqeq123d 13945	. . . 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 5819	. . 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 4123	. 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 3463	. . 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 14459	. . . . 5 ⊢ (𝑆 ∈ V → Word 𝑆 ∈ V)
26		inex1g 5266	. . . . 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 2841	. . 3 ⊢ (𝜑 → 𝑊 ∈ V)
29	22, 28	fexd 7183	. 2 ⊢ (𝜑 → 𝐹 ∈ V)
30		df-lsw 14498	. . . . . 6 ⊢ lastS = (𝑥 ∈ V ↦ (𝑥‘((♯‘𝑥) − 1)))
31	30	funmpt2 6539	. . . . 5 ⊢ Fun lastS
32	31	a1i 11	. . . 4 ⊢ (𝜑 → Fun lastS)
33		seqex 13938	. . . . 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 7903	. . . 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 585	. . 3 ⊢ (𝜑 → (lastS ∘ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ0 × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)}))) ∈ V)
37		unexg 7698	. . 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 585	. 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 7520	1 ⊢ (𝜑 → (𝑀seqstr𝐹) = (𝑀 ∪ (lastS ∘ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ0 × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)})))))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  Vcvv 3442   ∪ cun 3901   ∩ cin 3902  {csn 4582   × cxp 5630  ^◡ccnv 5631   “ cima 5635   ∘ ccom 5636  Fun wfun 6494  ⟶wf 6496  ‘cfv 6500  (class class class)co 7368   ∈ cmpo 7370  1c1 11039   − cmin 11376  ℕ₀cn0 12413  ℤ_≥cuz 12763  seqcseq 13936  ♯chash 14265  Word cword 14448  lastSclsw 14497   ++ cconcat 14505  ⟨“cs1 14531  seq_strcsseq 34560

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-inf2 9562  ax-cnex 11094  ax-1cn 11096  ax-addcl 11098

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-map 8777  df-nn 12158  df-n0 12414  df-seq 13937  df-word 14449  df-lsw 14498  df-s1 14532  df-sseq 34561

This theorem is referenced by:  sseqfv1  34566  sseqfn  34567  sseqf  34569  sseqfv2  34571