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Theorem repsco 14795
Description: Mapping of words commutes with the "repeated symbol" operation. (Contributed by AV, 11-Nov-2018.)
Assertion
Ref Expression
repsco ((𝑆𝐴𝑁 ∈ ℕ0𝐹:𝐴𝐵) → (𝐹 ∘ (𝑆 repeatS 𝑁)) = ((𝐹𝑆) repeatS 𝑁))

Proof of Theorem repsco
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 simpl1 1188 . . . . 5 (((𝑆𝐴𝑁 ∈ ℕ0𝐹:𝐴𝐵) ∧ 𝑥 ∈ (0..^𝑁)) → 𝑆𝐴)
2 simpl2 1189 . . . . 5 (((𝑆𝐴𝑁 ∈ ℕ0𝐹:𝐴𝐵) ∧ 𝑥 ∈ (0..^𝑁)) → 𝑁 ∈ ℕ0)
3 simpr 484 . . . . 5 (((𝑆𝐴𝑁 ∈ ℕ0𝐹:𝐴𝐵) ∧ 𝑥 ∈ (0..^𝑁)) → 𝑥 ∈ (0..^𝑁))
4 repswsymb 14728 . . . . 5 ((𝑆𝐴𝑁 ∈ ℕ0𝑥 ∈ (0..^𝑁)) → ((𝑆 repeatS 𝑁)‘𝑥) = 𝑆)
51, 2, 3, 4syl3anc 1368 . . . 4 (((𝑆𝐴𝑁 ∈ ℕ0𝐹:𝐴𝐵) ∧ 𝑥 ∈ (0..^𝑁)) → ((𝑆 repeatS 𝑁)‘𝑥) = 𝑆)
65fveq2d 6888 . . 3 (((𝑆𝐴𝑁 ∈ ℕ0𝐹:𝐴𝐵) ∧ 𝑥 ∈ (0..^𝑁)) → (𝐹‘((𝑆 repeatS 𝑁)‘𝑥)) = (𝐹𝑆))
76mpteq2dva 5241 . 2 ((𝑆𝐴𝑁 ∈ ℕ0𝐹:𝐴𝐵) → (𝑥 ∈ (0..^𝑁) ↦ (𝐹‘((𝑆 repeatS 𝑁)‘𝑥))) = (𝑥 ∈ (0..^𝑁) ↦ (𝐹𝑆)))
8 simp3 1135 . . 3 ((𝑆𝐴𝑁 ∈ ℕ0𝐹:𝐴𝐵) → 𝐹:𝐴𝐵)
9 repsf 14727 . . . 4 ((𝑆𝐴𝑁 ∈ ℕ0) → (𝑆 repeatS 𝑁):(0..^𝑁)⟶𝐴)
1093adant3 1129 . . 3 ((𝑆𝐴𝑁 ∈ ℕ0𝐹:𝐴𝐵) → (𝑆 repeatS 𝑁):(0..^𝑁)⟶𝐴)
11 fcompt 7126 . . 3 ((𝐹:𝐴𝐵 ∧ (𝑆 repeatS 𝑁):(0..^𝑁)⟶𝐴) → (𝐹 ∘ (𝑆 repeatS 𝑁)) = (𝑥 ∈ (0..^𝑁) ↦ (𝐹‘((𝑆 repeatS 𝑁)‘𝑥))))
128, 10, 11syl2anc 583 . 2 ((𝑆𝐴𝑁 ∈ ℕ0𝐹:𝐴𝐵) → (𝐹 ∘ (𝑆 repeatS 𝑁)) = (𝑥 ∈ (0..^𝑁) ↦ (𝐹‘((𝑆 repeatS 𝑁)‘𝑥))))
13 fvexd 6899 . . . . 5 (𝑆𝐴 → (𝐹𝑆) ∈ V)
1413anim1i 614 . . . 4 ((𝑆𝐴𝑁 ∈ ℕ0) → ((𝐹𝑆) ∈ V ∧ 𝑁 ∈ ℕ0))
15143adant3 1129 . . 3 ((𝑆𝐴𝑁 ∈ ℕ0𝐹:𝐴𝐵) → ((𝐹𝑆) ∈ V ∧ 𝑁 ∈ ℕ0))
16 reps 14724 . . 3 (((𝐹𝑆) ∈ V ∧ 𝑁 ∈ ℕ0) → ((𝐹𝑆) repeatS 𝑁) = (𝑥 ∈ (0..^𝑁) ↦ (𝐹𝑆)))
1715, 16syl 17 . 2 ((𝑆𝐴𝑁 ∈ ℕ0𝐹:𝐴𝐵) → ((𝐹𝑆) repeatS 𝑁) = (𝑥 ∈ (0..^𝑁) ↦ (𝐹𝑆)))
187, 12, 173eqtr4d 2776 1 ((𝑆𝐴𝑁 ∈ ℕ0𝐹:𝐴𝐵) → (𝐹 ∘ (𝑆 repeatS 𝑁)) = ((𝐹𝑆) repeatS 𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1084   = wceq 1533  wcel 2098  Vcvv 3468  cmpt 5224  ccom 5673  wf 6532  cfv 6536  (class class class)co 7404  0cc0 11109  0cn0 12473  ..^cfzo 13630   repeatS creps 14722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-reps 14723
This theorem is referenced by: (None)
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