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Theorem repsco 14541
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 1190 . . . . 5 (((𝑆𝐴𝑁 ∈ ℕ0𝐹:𝐴𝐵) ∧ 𝑥 ∈ (0..^𝑁)) → 𝑆𝐴)
2 simpl2 1191 . . . . 5 (((𝑆𝐴𝑁 ∈ ℕ0𝐹:𝐴𝐵) ∧ 𝑥 ∈ (0..^𝑁)) → 𝑁 ∈ ℕ0)
3 simpr 485 . . . . 5 (((𝑆𝐴𝑁 ∈ ℕ0𝐹:𝐴𝐵) ∧ 𝑥 ∈ (0..^𝑁)) → 𝑥 ∈ (0..^𝑁))
4 repswsymb 14475 . . . . 5 ((𝑆𝐴𝑁 ∈ ℕ0𝑥 ∈ (0..^𝑁)) → ((𝑆 repeatS 𝑁)‘𝑥) = 𝑆)
51, 2, 3, 4syl3anc 1370 . . . 4 (((𝑆𝐴𝑁 ∈ ℕ0𝐹:𝐴𝐵) ∧ 𝑥 ∈ (0..^𝑁)) → ((𝑆 repeatS 𝑁)‘𝑥) = 𝑆)
65fveq2d 6771 . . 3 (((𝑆𝐴𝑁 ∈ ℕ0𝐹:𝐴𝐵) ∧ 𝑥 ∈ (0..^𝑁)) → (𝐹‘((𝑆 repeatS 𝑁)‘𝑥)) = (𝐹𝑆))
76mpteq2dva 5174 . 2 ((𝑆𝐴𝑁 ∈ ℕ0𝐹:𝐴𝐵) → (𝑥 ∈ (0..^𝑁) ↦ (𝐹‘((𝑆 repeatS 𝑁)‘𝑥))) = (𝑥 ∈ (0..^𝑁) ↦ (𝐹𝑆)))
8 simp3 1137 . . 3 ((𝑆𝐴𝑁 ∈ ℕ0𝐹:𝐴𝐵) → 𝐹:𝐴𝐵)
9 repsf 14474 . . . 4 ((𝑆𝐴𝑁 ∈ ℕ0) → (𝑆 repeatS 𝑁):(0..^𝑁)⟶𝐴)
1093adant3 1131 . . 3 ((𝑆𝐴𝑁 ∈ ℕ0𝐹:𝐴𝐵) → (𝑆 repeatS 𝑁):(0..^𝑁)⟶𝐴)
11 fcompt 6998 . . 3 ((𝐹:𝐴𝐵 ∧ (𝑆 repeatS 𝑁):(0..^𝑁)⟶𝐴) → (𝐹 ∘ (𝑆 repeatS 𝑁)) = (𝑥 ∈ (0..^𝑁) ↦ (𝐹‘((𝑆 repeatS 𝑁)‘𝑥))))
128, 10, 11syl2anc 584 . 2 ((𝑆𝐴𝑁 ∈ ℕ0𝐹:𝐴𝐵) → (𝐹 ∘ (𝑆 repeatS 𝑁)) = (𝑥 ∈ (0..^𝑁) ↦ (𝐹‘((𝑆 repeatS 𝑁)‘𝑥))))
13 fvexd 6782 . . . . 5 (𝑆𝐴 → (𝐹𝑆) ∈ V)
1413anim1i 615 . . . 4 ((𝑆𝐴𝑁 ∈ ℕ0) → ((𝐹𝑆) ∈ V ∧ 𝑁 ∈ ℕ0))
15143adant3 1131 . . 3 ((𝑆𝐴𝑁 ∈ ℕ0𝐹:𝐴𝐵) → ((𝐹𝑆) ∈ V ∧ 𝑁 ∈ ℕ0))
16 reps 14471 . . 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 2788 1 ((𝑆𝐴𝑁 ∈ ℕ0𝐹:𝐴𝐵) → (𝐹 ∘ (𝑆 repeatS 𝑁)) = ((𝐹𝑆) repeatS 𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1086   = wceq 1539  wcel 2106  Vcvv 3430  cmpt 5157  ccom 5589  wf 6423  cfv 6427  (class class class)co 7268  0cc0 10859  0cn0 12221  ..^cfzo 13370   repeatS creps 14469
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5222  ax-nul 5229  ax-pr 5351
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3071  df-rab 3073  df-v 3432  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4258  df-if 4461  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4841  df-iun 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5485  df-xp 5591  df-rel 5592  df-cnv 5593  df-co 5594  df-dm 5595  df-rn 5596  df-res 5597  df-ima 5598  df-iota 6385  df-fun 6429  df-fn 6430  df-f 6431  df-f1 6432  df-fo 6433  df-f1o 6434  df-fv 6435  df-ov 7271  df-oprab 7272  df-mpo 7273  df-reps 14470
This theorem is referenced by: (None)
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