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Theorem sseqval 33028
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 33024 . . 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 6851 . . . . 5 ((𝜑 ∧ (𝑚 = 𝑀𝑓 = 𝐹)) → (♯‘𝑚) = (♯‘𝑀))
5 simp1rr 1240 . . . . . . . . 9 (((𝜑 ∧ (𝑚 = 𝑀𝑓 = 𝐹)) ∧ 𝑥 ∈ V ∧ 𝑦 ∈ V) → 𝑓 = 𝐹)
65fveq1d 6849 . . . . . . . 8 (((𝜑 ∧ (𝑚 = 𝑀𝑓 = 𝐹)) ∧ 𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑓𝑥) = (𝐹𝑥))
76s1eqd 14496 . . . . . . 7 (((𝜑 ∧ (𝑚 = 𝑀𝑓 = 𝐹)) ∧ 𝑥 ∈ V ∧ 𝑦 ∈ V) → ⟨“(𝑓𝑥)”⟩ = ⟨“(𝐹𝑥)”⟩)
87oveq2d 7378 . . . . . 6 (((𝜑 ∧ (𝑚 = 𝑀𝑓 = 𝐹)) ∧ 𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ++ ⟨“(𝑓𝑥)”⟩) = (𝑥 ++ ⟨“(𝐹𝑥)”⟩))
98mpoeq3dva 7439 . . . . 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 6854 . . . . . . . . 9 ((𝜑 ∧ (𝑚 = 𝑀𝑓 = 𝐹)) → (𝑓𝑚) = (𝐹𝑀))
1211s1eqd 14496 . . . . . . . 8 ((𝜑 ∧ (𝑚 = 𝑀𝑓 = 𝐹)) → ⟨“(𝑓𝑚)”⟩ = ⟨“(𝐹𝑀)”⟩)
133, 12oveq12d 7380 . . . . . . 7 ((𝜑 ∧ (𝑚 = 𝑀𝑓 = 𝐹)) → (𝑚 ++ ⟨“(𝑓𝑚)”⟩) = (𝑀 ++ ⟨“(𝐹𝑀)”⟩))
1413sneqd 4603 . . . . . 6 ((𝜑 ∧ (𝑚 = 𝑀𝑓 = 𝐹)) → {(𝑚 ++ ⟨“(𝑓𝑚)”⟩)} = {(𝑀 ++ ⟨“(𝐹𝑀)”⟩)})
1514xpeq2d 5668 . . . . 5 ((𝜑 ∧ (𝑚 = 𝑀𝑓 = 𝐹)) → (ℕ0 × {(𝑚 ++ ⟨“(𝑓𝑚)”⟩)}) = (ℕ0 × {(𝑀 ++ ⟨“(𝐹𝑀)”⟩)}))
164, 9, 15seqeq123d 13922 . . . 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 5823 . . 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 4129 . 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 3466 . . 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 14419 . . . . 5 (𝑆 ∈ V → Word 𝑆 ∈ V)
26 inex1g 5281 . . . . 5 (Word 𝑆 ∈ V → (Word 𝑆 ∩ (♯ “ (ℤ‘(♯‘𝑀)))) ∈ V)
2724, 25, 263syl 18 . . . 4 (𝜑 → (Word 𝑆 ∩ (♯ “ (ℤ‘(♯‘𝑀)))) ∈ V)
2823, 27eqeltrid 2842 . . 3 (𝜑𝑊 ∈ V)
2922, 28fexd 7182 . 2 (𝜑𝐹 ∈ V)
30 df-lsw 14458 . . . . . 6 lastS = (𝑥 ∈ V ↦ (𝑥‘((♯‘𝑥) − 1)))
3130funmpt2 6545 . . . . 5 Fun lastS
3231a1i 11 . . . 4 (𝜑 → Fun lastS)
33 seqex 13915 . . . . 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 7886 . . . 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 585 . . 3 (𝜑 → (lastS ∘ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹𝑥)”⟩)), (ℕ0 × {(𝑀 ++ ⟨“(𝐹𝑀)”⟩)}))) ∈ V)
37 unexg 7688 . . 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 585 . 2 (𝜑 → (𝑀 ∪ (lastS ∘ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹𝑥)”⟩)), (ℕ0 × {(𝑀 ++ ⟨“(𝐹𝑀)”⟩)})))) ∈ V)
392, 18, 21, 29, 38ovmpod 7512 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 397  w3a 1088   = wceq 1542  wcel 2107  Vcvv 3448  cun 3913  cin 3914  {csn 4591   × cxp 5636  ccnv 5637  cima 5641  ccom 5642  Fun wfun 6495  wf 6497  cfv 6501  (class class class)co 7362  cmpo 7364  1c1 11059  cmin 11392  0cn0 12420  cuz 12770  seqcseq 13913  chash 14237  Word cword 14409  lastSclsw 14457   ++ cconcat 14465  ⟨“cs1 14490  seqstrcsseq 33023
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-inf2 9584  ax-cnex 11114  ax-1cn 11116  ax-addcl 11118
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-map 8774  df-nn 12161  df-n0 12421  df-seq 13914  df-word 14410  df-lsw 14458  df-s1 14491  df-sseq 33024
This theorem is referenced by:  sseqfv1  33029  sseqfn  33030  sseqf  33032  sseqfv2  33034
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