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Theorem repsco 13805
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 1228 . . . . 5 (((𝑆𝐴𝑁 ∈ ℕ0𝐹:𝐴𝐵) ∧ 𝑥 ∈ (0..^𝑁)) → 𝑆𝐴)
2 simpl2 1230 . . . . 5 (((𝑆𝐴𝑁 ∈ ℕ0𝐹:𝐴𝐵) ∧ 𝑥 ∈ (0..^𝑁)) → 𝑁 ∈ ℕ0)
3 simpr 479 . . . . 5 (((𝑆𝐴𝑁 ∈ ℕ0𝐹:𝐴𝐵) ∧ 𝑥 ∈ (0..^𝑁)) → 𝑥 ∈ (0..^𝑁))
4 repswsymb 13741 . . . . 5 ((𝑆𝐴𝑁 ∈ ℕ0𝑥 ∈ (0..^𝑁)) → ((𝑆 repeatS 𝑁)‘𝑥) = 𝑆)
51, 2, 3, 4syl3anc 1477 . . . 4 (((𝑆𝐴𝑁 ∈ ℕ0𝐹:𝐴𝐵) ∧ 𝑥 ∈ (0..^𝑁)) → ((𝑆 repeatS 𝑁)‘𝑥) = 𝑆)
65fveq2d 6357 . . 3 (((𝑆𝐴𝑁 ∈ ℕ0𝐹:𝐴𝐵) ∧ 𝑥 ∈ (0..^𝑁)) → (𝐹‘((𝑆 repeatS 𝑁)‘𝑥)) = (𝐹𝑆))
76mpteq2dva 4896 . 2 ((𝑆𝐴𝑁 ∈ ℕ0𝐹:𝐴𝐵) → (𝑥 ∈ (0..^𝑁) ↦ (𝐹‘((𝑆 repeatS 𝑁)‘𝑥))) = (𝑥 ∈ (0..^𝑁) ↦ (𝐹𝑆)))
8 simp3 1133 . . 3 ((𝑆𝐴𝑁 ∈ ℕ0𝐹:𝐴𝐵) → 𝐹:𝐴𝐵)
9 repsf 13740 . . . 4 ((𝑆𝐴𝑁 ∈ ℕ0) → (𝑆 repeatS 𝑁):(0..^𝑁)⟶𝐴)
1093adant3 1127 . . 3 ((𝑆𝐴𝑁 ∈ ℕ0𝐹:𝐴𝐵) → (𝑆 repeatS 𝑁):(0..^𝑁)⟶𝐴)
11 fcompt 6564 . . 3 ((𝐹:𝐴𝐵 ∧ (𝑆 repeatS 𝑁):(0..^𝑁)⟶𝐴) → (𝐹 ∘ (𝑆 repeatS 𝑁)) = (𝑥 ∈ (0..^𝑁) ↦ (𝐹‘((𝑆 repeatS 𝑁)‘𝑥))))
128, 10, 11syl2anc 696 . 2 ((𝑆𝐴𝑁 ∈ ℕ0𝐹:𝐴𝐵) → (𝐹 ∘ (𝑆 repeatS 𝑁)) = (𝑥 ∈ (0..^𝑁) ↦ (𝐹‘((𝑆 repeatS 𝑁)‘𝑥))))
13 fvexd 6365 . . . . 5 (𝑆𝐴 → (𝐹𝑆) ∈ V)
1413anim1i 593 . . . 4 ((𝑆𝐴𝑁 ∈ ℕ0) → ((𝐹𝑆) ∈ V ∧ 𝑁 ∈ ℕ0))
15143adant3 1127 . . 3 ((𝑆𝐴𝑁 ∈ ℕ0𝐹:𝐴𝐵) → ((𝐹𝑆) ∈ V ∧ 𝑁 ∈ ℕ0))
16 reps 13737 . . 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 2804 1 ((𝑆𝐴𝑁 ∈ ℕ0𝐹:𝐴𝐵) → (𝐹 ∘ (𝑆 repeatS 𝑁)) = ((𝐹𝑆) repeatS 𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1632  wcel 2139  Vcvv 3340  cmpt 4881  ccom 5270  wf 6045  cfv 6049  (class class class)co 6814  0cc0 10148  0cn0 11504  ..^cfzo 12679   repeatS creps 13504
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
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  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-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  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-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 6817  df-oprab 6818  df-mpt2 6819  df-reps 13512
This theorem is referenced by: (None)
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