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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 (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 31319 . . 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 759 . . 3 ((𝜑 ∧ (𝑚 = 𝑀𝑓 = 𝐹)) → 𝑚 = 𝑀)
43fveq2d 6500 . . . . 5 ((𝜑 ∧ (𝑚 = 𝑀𝑓 = 𝐹)) → (♯‘𝑚) = (♯‘𝑀))
5 simp1rr 1220 . . . . . . . . 9 (((𝜑 ∧ (𝑚 = 𝑀𝑓 = 𝐹)) ∧ 𝑥 ∈ V ∧ 𝑦 ∈ V) → 𝑓 = 𝐹)
65fveq1d 6498 . . . . . . . 8 (((𝜑 ∧ (𝑚 = 𝑀𝑓 = 𝐹)) ∧ 𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑓𝑥) = (𝐹𝑥))
76s1eqd 13762 . . . . . . 7 (((𝜑 ∧ (𝑚 = 𝑀𝑓 = 𝐹)) ∧ 𝑥 ∈ V ∧ 𝑦 ∈ V) → ⟨“(𝑓𝑥)”⟩ = ⟨“(𝐹𝑥)”⟩)
87oveq2d 6990 . . . . . 6 (((𝜑 ∧ (𝑚 = 𝑀𝑓 = 𝐹)) ∧ 𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ++ ⟨“(𝑓𝑥)”⟩) = (𝑥 ++ ⟨“(𝐹𝑥)”⟩))
98mpoeq3dva 7047 . . . . 5 ((𝜑 ∧ (𝑚 = 𝑀𝑓 = 𝐹)) → (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝑓𝑥)”⟩)) = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹𝑥)”⟩)))
10 simprr 761 . . . . . . . . . 10 ((𝜑 ∧ (𝑚 = 𝑀𝑓 = 𝐹)) → 𝑓 = 𝐹)
1110, 3fveq12d 6503 . . . . . . . . 9 ((𝜑 ∧ (𝑚 = 𝑀𝑓 = 𝐹)) → (𝑓𝑚) = (𝐹𝑀))
1211s1eqd 13762 . . . . . . . 8 ((𝜑 ∧ (𝑚 = 𝑀𝑓 = 𝐹)) → ⟨“(𝑓𝑚)”⟩ = ⟨“(𝐹𝑀)”⟩)
133, 12oveq12d 6992 . . . . . . 7 ((𝜑 ∧ (𝑚 = 𝑀𝑓 = 𝐹)) → (𝑚 ++ ⟨“(𝑓𝑚)”⟩) = (𝑀 ++ ⟨“(𝐹𝑀)”⟩))
1413sneqd 4447 . . . . . 6 ((𝜑 ∧ (𝑚 = 𝑀𝑓 = 𝐹)) → {(𝑚 ++ ⟨“(𝑓𝑚)”⟩)} = {(𝑀 ++ ⟨“(𝐹𝑀)”⟩)})
1514xpeq2d 5433 . . . . 5 ((𝜑 ∧ (𝑚 = 𝑀𝑓 = 𝐹)) → (ℕ0 × {(𝑚 ++ ⟨“(𝑓𝑚)”⟩)}) = (ℕ0 × {(𝑀 ++ ⟨“(𝐹𝑀)”⟩)}))
164, 9, 15seqeq123d 13191 . . . 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 5579 . . 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 4022 . 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 3426 . . 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 13680 . . . . 5 (𝑆 ∈ V → Word 𝑆 ∈ V)
26 inex1g 5076 . . . . 5 (Word 𝑆 ∈ V → (Word 𝑆 ∩ (♯ “ (ℤ‘(♯‘𝑀)))) ∈ V)
2724, 25, 263syl 18 . . . 4 (𝜑 → (Word 𝑆 ∩ (♯ “ (ℤ‘(♯‘𝑀)))) ∈ V)
2823, 27syl5eqel 2863 . . 3 (𝜑𝑊 ∈ V)
29 fex 6813 . . 3 ((𝐹:𝑊𝑆𝑊 ∈ V) → 𝐹 ∈ V)
3022, 28, 29syl2anc 576 . 2 (𝜑𝐹 ∈ V)
31 df-lsw 13724 . . . . . 6 lastS = (𝑥 ∈ V ↦ (𝑥‘((♯‘𝑥) − 1)))
3231funmpt2 6224 . . . . 5 Fun lastS
3332a1i 11 . . . 4 (𝜑 → Fun lastS)
34 seqex 13184 . . . . 5 seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹𝑥)”⟩)), (ℕ0 × {(𝑀 ++ ⟨“(𝐹𝑀)”⟩)})) ∈ V
3534a1i 11 . . . 4 (𝜑 → seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹𝑥)”⟩)), (ℕ0 × {(𝑀 ++ ⟨“(𝐹𝑀)”⟩)})) ∈ V)
36 cofunexg 7460 . . . 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)
3733, 35, 36syl2anc 576 . . 3 (𝜑 → (lastS ∘ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹𝑥)”⟩)), (ℕ0 × {(𝑀 ++ ⟨“(𝐹𝑀)”⟩)}))) ∈ V)
38 unexg 7287 . . 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)
3921, 37, 38syl2anc 576 . 2 (𝜑 → (𝑀 ∪ (lastS ∘ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹𝑥)”⟩)), (ℕ0 × {(𝑀 ++ ⟨“(𝐹𝑀)”⟩)})))) ∈ V)
402, 18, 21, 30, 39ovmpod 7116 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 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  0cn0 11705  cuz 12056  seqcseq 13182  chash 13503  Word cword 13670  lastSclsw 13723   ++ cconcat 13731  ⟨“cs1 13756  seqstrcsseq 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
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