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Theorem repsco 14197
 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 488 . . . . 5 (((𝑆𝐴𝑁 ∈ ℕ0𝐹:𝐴𝐵) ∧ 𝑥 ∈ (0..^𝑁)) → 𝑥 ∈ (0..^𝑁))
4 repswsymb 14131 . . . . 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 6653 . . 3 (((𝑆𝐴𝑁 ∈ ℕ0𝐹:𝐴𝐵) ∧ 𝑥 ∈ (0..^𝑁)) → (𝐹‘((𝑆 repeatS 𝑁)‘𝑥)) = (𝐹𝑆))
76mpteq2dva 5128 . 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 14130 . . . 4 ((𝑆𝐴𝑁 ∈ ℕ0) → (𝑆 repeatS 𝑁):(0..^𝑁)⟶𝐴)
1093adant3 1129 . . 3 ((𝑆𝐴𝑁 ∈ ℕ0𝐹:𝐴𝐵) → (𝑆 repeatS 𝑁):(0..^𝑁)⟶𝐴)
11 fcompt 6876 . . 3 ((𝐹:𝐴𝐵 ∧ (𝑆 repeatS 𝑁):(0..^𝑁)⟶𝐴) → (𝐹 ∘ (𝑆 repeatS 𝑁)) = (𝑥 ∈ (0..^𝑁) ↦ (𝐹‘((𝑆 repeatS 𝑁)‘𝑥))))
128, 10, 11syl2anc 587 . 2 ((𝑆𝐴𝑁 ∈ ℕ0𝐹:𝐴𝐵) → (𝐹 ∘ (𝑆 repeatS 𝑁)) = (𝑥 ∈ (0..^𝑁) ↦ (𝐹‘((𝑆 repeatS 𝑁)‘𝑥))))
13 fvexd 6664 . . . . 5 (𝑆𝐴 → (𝐹𝑆) ∈ V)
1413anim1i 617 . . . 4 ((𝑆𝐴𝑁 ∈ ℕ0) → ((𝐹𝑆) ∈ V ∧ 𝑁 ∈ ℕ0))
15143adant3 1129 . . 3 ((𝑆𝐴𝑁 ∈ ℕ0𝐹:𝐴𝐵) → ((𝐹𝑆) ∈ V ∧ 𝑁 ∈ ℕ0))
16 reps 14127 . . 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 2846 1 ((𝑆𝐴𝑁 ∈ ℕ0𝐹:𝐴𝐵) → (𝐹 ∘ (𝑆 repeatS 𝑁)) = ((𝐹𝑆) repeatS 𝑁))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2112  Vcvv 3444   ↦ cmpt 5113   ∘ ccom 5527  ⟶wf 6324  ‘cfv 6328  (class class class)co 7139  0cc0 10530  ℕ0cn0 11889  ..^cfzo 13032   repeatS creps 14125 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-reps 14126 This theorem is referenced by: (None)
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