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Theorem sseqval 30759
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 30755 . . 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 811 . . 3 ((𝜑 ∧ (𝑚 = 𝑀𝑓 = 𝐹)) → 𝑚 = 𝑀)
43fveq2d 6356 . . . . 5 ((𝜑 ∧ (𝑚 = 𝑀𝑓 = 𝐹)) → (♯‘𝑚) = (♯‘𝑀))
5 simp1rr 1306 . . . . . . . . 9 (((𝜑 ∧ (𝑚 = 𝑀𝑓 = 𝐹)) ∧ 𝑥 ∈ V ∧ 𝑦 ∈ V) → 𝑓 = 𝐹)
65fveq1d 6354 . . . . . . . 8 (((𝜑 ∧ (𝑚 = 𝑀𝑓 = 𝐹)) ∧ 𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑓𝑥) = (𝐹𝑥))
76s1eqd 13571 . . . . . . 7 (((𝜑 ∧ (𝑚 = 𝑀𝑓 = 𝐹)) ∧ 𝑥 ∈ V ∧ 𝑦 ∈ V) → ⟨“(𝑓𝑥)”⟩ = ⟨“(𝐹𝑥)”⟩)
87oveq2d 6829 . . . . . 6 (((𝜑 ∧ (𝑚 = 𝑀𝑓 = 𝐹)) ∧ 𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ++ ⟨“(𝑓𝑥)”⟩) = (𝑥 ++ ⟨“(𝐹𝑥)”⟩))
98mpt2eq3dva 6884 . . . . 5 ((𝜑 ∧ (𝑚 = 𝑀𝑓 = 𝐹)) → (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝑓𝑥)”⟩)) = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹𝑥)”⟩)))
10 simprr 813 . . . . . . . . . 10 ((𝜑 ∧ (𝑚 = 𝑀𝑓 = 𝐹)) → 𝑓 = 𝐹)
1110, 3fveq12d 6358 . . . . . . . . 9 ((𝜑 ∧ (𝑚 = 𝑀𝑓 = 𝐹)) → (𝑓𝑚) = (𝐹𝑀))
1211s1eqd 13571 . . . . . . . 8 ((𝜑 ∧ (𝑚 = 𝑀𝑓 = 𝐹)) → ⟨“(𝑓𝑚)”⟩ = ⟨“(𝐹𝑀)”⟩)
133, 12oveq12d 6831 . . . . . . 7 ((𝜑 ∧ (𝑚 = 𝑀𝑓 = 𝐹)) → (𝑚 ++ ⟨“(𝑓𝑚)”⟩) = (𝑀 ++ ⟨“(𝐹𝑀)”⟩))
1413sneqd 4333 . . . . . 6 ((𝜑 ∧ (𝑚 = 𝑀𝑓 = 𝐹)) → {(𝑚 ++ ⟨“(𝑓𝑚)”⟩)} = {(𝑀 ++ ⟨“(𝐹𝑀)”⟩)})
1514xpeq2d 5296 . . . . 5 ((𝜑 ∧ (𝑚 = 𝑀𝑓 = 𝐹)) → (ℕ0 × {(𝑚 ++ ⟨“(𝑓𝑚)”⟩)}) = (ℕ0 × {(𝑀 ++ ⟨“(𝐹𝑀)”⟩)}))
164, 9, 15seqeq123d 13004 . . . 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 5440 . . 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 3911 . 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 3352 . . 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 13501 . . . . 5 (𝑆 ∈ V → Word 𝑆 ∈ V)
26 inex1g 4953 . . . . 5 (Word 𝑆 ∈ V → (Word 𝑆 ∩ (♯ “ (ℤ‘(♯‘𝑀)))) ∈ V)
2724, 25, 263syl 18 . . . 4 (𝜑 → (Word 𝑆 ∩ (♯ “ (ℤ‘(♯‘𝑀)))) ∈ V)
2823, 27syl5eqel 2843 . . 3 (𝜑𝑊 ∈ V)
29 fex 6653 . . 3 ((𝐹:𝑊𝑆𝑊 ∈ V) → 𝐹 ∈ V)
3022, 28, 29syl2anc 696 . 2 (𝜑𝐹 ∈ V)
31 df-lsw 13486 . . . . . 6 lastS = (𝑥 ∈ V ↦ (𝑥‘((♯‘𝑥) − 1)))
3231funmpt2 6088 . . . . 5 Fun lastS
3332a1i 11 . . . 4 (𝜑 → Fun lastS)
34 seqex 12997 . . . . 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 7295 . . . 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 696 . . 3 (𝜑 → (lastS ∘ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹𝑥)”⟩)), (ℕ0 × {(𝑀 ++ ⟨“(𝐹𝑀)”⟩)}))) ∈ V)
38 unexg 7124 . . 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 696 . 2 (𝜑 → (𝑀 ∪ (lastS ∘ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹𝑥)”⟩)), (ℕ0 × {(𝑀 ++ ⟨“(𝐹𝑀)”⟩)})))) ∈ V)
402, 18, 21, 30, 39ovmpt2d 6953 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 383  w3a 1072   = wceq 1632  wcel 2139  Vcvv 3340  cun 3713  cin 3714  {csn 4321   × cxp 5264  ccnv 5265  cima 5269  ccom 5270  Fun wfun 6043  wf 6045  cfv 6049  (class class class)co 6813  cmpt2 6815  1c1 10129  cmin 10458  0cn0 11484  cuz 11879  seqcseq 12995  chash 13311  Word cword 13477  lastSclsw 13478   ++ cconcat 13479  ⟨“cs1 13480  seqstrcsseq 30754
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-inf2 8711  ax-cnex 10184  ax-resscn 10185
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-map 8025  df-pm 8026  df-neg 10461  df-z 11570  df-uz 11880  df-fz 12520  df-fzo 12660  df-seq 12996  df-word 13485  df-lsw 13486  df-s1 13488  df-sseq 30755
This theorem is referenced by:  sseqfv1  30760  sseqfn  30761  sseqf  30763  sseqfv2  30765
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