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Theorem sseqval 33944
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 (𝜑𝑆 ∈ V)
sseqval.2 (𝜑𝑀 ∈ Word 𝑆)
sseqval.3 𝑊 = (Word 𝑆 ∩ (♯ “ (ℤ‘(♯‘𝑀))))
sseqval.4 (𝜑𝐹:𝑊𝑆)
Assertion
Ref Expression
sseqval (𝜑 → (𝑀seqstr𝐹) = (𝑀 ∪ (lastS ∘ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹𝑥)”⟩)), (ℕ0 × {(𝑀 ++ ⟨“(𝐹𝑀)”⟩)})))))
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.
StepHypRef Expression
1 df-sseq 33940 . . 3 seqstr = (𝑚 ∈ V, 𝑓 ∈ V ↦ (𝑚 ∪ (lastS ∘ seq(♯‘𝑚)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝑓𝑥)”⟩)), (ℕ0 × {(𝑚 ++ ⟨“(𝑓𝑚)”⟩)})))))
21a1i 11 . 2 (𝜑 → seqstr = (𝑚 ∈ V, 𝑓 ∈ V ↦ (𝑚 ∪ (lastS ∘ seq(♯‘𝑚)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝑓𝑥)”⟩)), (ℕ0 × {(𝑚 ++ ⟨“(𝑓𝑚)”⟩)}))))))
3 simprl 770 . . 3 ((𝜑 ∧ (𝑚 = 𝑀𝑓 = 𝐹)) → 𝑚 = 𝑀)
43fveq2d 6895 . . . . 5 ((𝜑 ∧ (𝑚 = 𝑀𝑓 = 𝐹)) → (♯‘𝑚) = (♯‘𝑀))
5 simp1rr 1237 . . . . . . . . 9 (((𝜑 ∧ (𝑚 = 𝑀𝑓 = 𝐹)) ∧ 𝑥 ∈ V ∧ 𝑦 ∈ V) → 𝑓 = 𝐹)
65fveq1d 6893 . . . . . . . 8 (((𝜑 ∧ (𝑚 = 𝑀𝑓 = 𝐹)) ∧ 𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑓𝑥) = (𝐹𝑥))
76s1eqd 14575 . . . . . . 7 (((𝜑 ∧ (𝑚 = 𝑀𝑓 = 𝐹)) ∧ 𝑥 ∈ V ∧ 𝑦 ∈ V) → ⟨“(𝑓𝑥)”⟩ = ⟨“(𝐹𝑥)”⟩)
87oveq2d 7430 . . . . . 6 (((𝜑 ∧ (𝑚 = 𝑀𝑓 = 𝐹)) ∧ 𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ++ ⟨“(𝑓𝑥)”⟩) = (𝑥 ++ ⟨“(𝐹𝑥)”⟩))
98mpoeq3dva 7491 . . . . 5 ((𝜑 ∧ (𝑚 = 𝑀𝑓 = 𝐹)) → (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝑓𝑥)”⟩)) = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹𝑥)”⟩)))
10 simprr 772 . . . . . . . . . 10 ((𝜑 ∧ (𝑚 = 𝑀𝑓 = 𝐹)) → 𝑓 = 𝐹)
1110, 3fveq12d 6898 . . . . . . . . 9 ((𝜑 ∧ (𝑚 = 𝑀𝑓 = 𝐹)) → (𝑓𝑚) = (𝐹𝑀))
1211s1eqd 14575 . . . . . . . 8 ((𝜑 ∧ (𝑚 = 𝑀𝑓 = 𝐹)) → ⟨“(𝑓𝑚)”⟩ = ⟨“(𝐹𝑀)”⟩)
133, 12oveq12d 7432 . . . . . . 7 ((𝜑 ∧ (𝑚 = 𝑀𝑓 = 𝐹)) → (𝑚 ++ ⟨“(𝑓𝑚)”⟩) = (𝑀 ++ ⟨“(𝐹𝑀)”⟩))
1413sneqd 4636 . . . . . 6 ((𝜑 ∧ (𝑚 = 𝑀𝑓 = 𝐹)) → {(𝑚 ++ ⟨“(𝑓𝑚)”⟩)} = {(𝑀 ++ ⟨“(𝐹𝑀)”⟩)})
1514xpeq2d 5702 . . . . 5 ((𝜑 ∧ (𝑚 = 𝑀𝑓 = 𝐹)) → (ℕ0 × {(𝑚 ++ ⟨“(𝑓𝑚)”⟩)}) = (ℕ0 × {(𝑀 ++ ⟨“(𝐹𝑀)”⟩)}))
164, 9, 15seqeq123d 13999 . . . 4 ((𝜑 ∧ (𝑚 = 𝑀𝑓 = 𝐹)) → seq(♯‘𝑚)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝑓𝑥)”⟩)), (ℕ0 × {(𝑚 ++ ⟨“(𝑓𝑚)”⟩)})) = seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹𝑥)”⟩)), (ℕ0 × {(𝑀 ++ ⟨“(𝐹𝑀)”⟩)})))
1716coeq2d 5859 . . 3 ((𝜑 ∧ (𝑚 = 𝑀𝑓 = 𝐹)) → (lastS ∘ seq(♯‘𝑚)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝑓𝑥)”⟩)), (ℕ0 × {(𝑚 ++ ⟨“(𝑓𝑚)”⟩)}))) = (lastS ∘ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹𝑥)”⟩)), (ℕ0 × {(𝑀 ++ ⟨“(𝐹𝑀)”⟩)}))))
183, 17uneq12d 4160 . 2 ((𝜑 ∧ (𝑚 = 𝑀𝑓 = 𝐹)) → (𝑚 ∪ (lastS ∘ seq(♯‘𝑚)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝑓𝑥)”⟩)), (ℕ0 × {(𝑚 ++ ⟨“(𝑓𝑚)”⟩)})))) = (𝑀 ∪ (lastS ∘ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹𝑥)”⟩)), (ℕ0 × {(𝑀 ++ ⟨“(𝐹𝑀)”⟩)})))))
19 sseqval.2 . . 3 (𝜑𝑀 ∈ Word 𝑆)
20 elex 3488 . . 3 (𝑀 ∈ Word 𝑆𝑀 ∈ V)
2119, 20syl 17 . 2 (𝜑𝑀 ∈ V)
22 sseqval.4 . . 3 (𝜑𝐹:𝑊𝑆)
23 sseqval.3 . . . 4 𝑊 = (Word 𝑆 ∩ (♯ “ (ℤ‘(♯‘𝑀))))
24 sseqval.1 . . . . 5 (𝜑𝑆 ∈ V)
25 wrdexg 14498 . . . . 5 (𝑆 ∈ V → Word 𝑆 ∈ V)
26 inex1g 5313 . . . . 5 (Word 𝑆 ∈ V → (Word 𝑆 ∩ (♯ “ (ℤ‘(♯‘𝑀)))) ∈ V)
2724, 25, 263syl 18 . . . 4 (𝜑 → (Word 𝑆 ∩ (♯ “ (ℤ‘(♯‘𝑀)))) ∈ V)
2823, 27eqeltrid 2832 . . 3 (𝜑𝑊 ∈ V)
2922, 28fexd 7233 . 2 (𝜑𝐹 ∈ V)
30 df-lsw 14537 . . . . . 6 lastS = (𝑥 ∈ V ↦ (𝑥‘((♯‘𝑥) − 1)))
3130funmpt2 6586 . . . . 5 Fun lastS
3231a1i 11 . . . 4 (𝜑 → Fun lastS)
33 seqex 13992 . . . . 5 seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹𝑥)”⟩)), (ℕ0 × {(𝑀 ++ ⟨“(𝐹𝑀)”⟩)})) ∈ V
3433a1i 11 . . . 4 (𝜑 → seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹𝑥)”⟩)), (ℕ0 × {(𝑀 ++ ⟨“(𝐹𝑀)”⟩)})) ∈ V)
35 cofunexg 7946 . . . 4 ((Fun lastS ∧ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹𝑥)”⟩)), (ℕ0 × {(𝑀 ++ ⟨“(𝐹𝑀)”⟩)})) ∈ V) → (lastS ∘ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹𝑥)”⟩)), (ℕ0 × {(𝑀 ++ ⟨“(𝐹𝑀)”⟩)}))) ∈ V)
3632, 34, 35syl2anc 583 . . 3 (𝜑 → (lastS ∘ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹𝑥)”⟩)), (ℕ0 × {(𝑀 ++ ⟨“(𝐹𝑀)”⟩)}))) ∈ V)
37 unexg 7745 . . 3 ((𝑀 ∈ V ∧ (lastS ∘ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹𝑥)”⟩)), (ℕ0 × {(𝑀 ++ ⟨“(𝐹𝑀)”⟩)}))) ∈ V) → (𝑀 ∪ (lastS ∘ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹𝑥)”⟩)), (ℕ0 × {(𝑀 ++ ⟨“(𝐹𝑀)”⟩)})))) ∈ V)
3821, 36, 37syl2anc 583 . 2 (𝜑 → (𝑀 ∪ (lastS ∘ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹𝑥)”⟩)), (ℕ0 × {(𝑀 ++ ⟨“(𝐹𝑀)”⟩)})))) ∈ V)
392, 18, 21, 29, 38ovmpod 7567 1 (𝜑 → (𝑀seqstr𝐹) = (𝑀 ∪ (lastS ∘ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹𝑥)”⟩)), (ℕ0 × {(𝑀 ++ ⟨“(𝐹𝑀)”⟩)})))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1085   = wceq 1534  wcel 2099  Vcvv 3469  cun 3942  cin 3943  {csn 4624   × cxp 5670  ccnv 5671  cima 5675  ccom 5676  Fun wfun 6536  wf 6538  cfv 6542  (class class class)co 7414  cmpo 7416  1c1 11131  cmin 11466  0cn0 12494  cuz 12844  seqcseq 13990  chash 14313  Word cword 14488  lastSclsw 14536   ++ cconcat 14544  ⟨“cs1 14569  seqstrcsseq 33939
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-inf2 9656  ax-cnex 11186  ax-1cn 11188  ax-addcl 11190
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-map 8838  df-nn 12235  df-n0 12495  df-seq 13991  df-word 14489  df-lsw 14537  df-s1 14570  df-sseq 33940
This theorem is referenced by:  sseqfv1  33945  sseqfn  33946  sseqf  33948  sseqfv2  33950
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