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Theorem repsco 13384
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 1056 . . . . 5 (((𝑆𝐴𝑁 ∈ ℕ0𝐹:𝐴𝐵) ∧ 𝑥 ∈ (0..^𝑁)) → 𝑆𝐴)
2 simpl2 1057 . . . . 5 (((𝑆𝐴𝑁 ∈ ℕ0𝐹:𝐴𝐵) ∧ 𝑥 ∈ (0..^𝑁)) → 𝑁 ∈ ℕ0)
3 simpr 475 . . . . 5 (((𝑆𝐴𝑁 ∈ ℕ0𝐹:𝐴𝐵) ∧ 𝑥 ∈ (0..^𝑁)) → 𝑥 ∈ (0..^𝑁))
4 repswsymb 13320 . . . . 5 ((𝑆𝐴𝑁 ∈ ℕ0𝑥 ∈ (0..^𝑁)) → ((𝑆 repeatS 𝑁)‘𝑥) = 𝑆)
51, 2, 3, 4syl3anc 1317 . . . 4 (((𝑆𝐴𝑁 ∈ ℕ0𝐹:𝐴𝐵) ∧ 𝑥 ∈ (0..^𝑁)) → ((𝑆 repeatS 𝑁)‘𝑥) = 𝑆)
65fveq2d 6091 . . 3 (((𝑆𝐴𝑁 ∈ ℕ0𝐹:𝐴𝐵) ∧ 𝑥 ∈ (0..^𝑁)) → (𝐹‘((𝑆 repeatS 𝑁)‘𝑥)) = (𝐹𝑆))
76mpteq2dva 4666 . 2 ((𝑆𝐴𝑁 ∈ ℕ0𝐹:𝐴𝐵) → (𝑥 ∈ (0..^𝑁) ↦ (𝐹‘((𝑆 repeatS 𝑁)‘𝑥))) = (𝑥 ∈ (0..^𝑁) ↦ (𝐹𝑆)))
8 simp3 1055 . . 3 ((𝑆𝐴𝑁 ∈ ℕ0𝐹:𝐴𝐵) → 𝐹:𝐴𝐵)
9 repsf 13319 . . . 4 ((𝑆𝐴𝑁 ∈ ℕ0) → (𝑆 repeatS 𝑁):(0..^𝑁)⟶𝐴)
1093adant3 1073 . . 3 ((𝑆𝐴𝑁 ∈ ℕ0𝐹:𝐴𝐵) → (𝑆 repeatS 𝑁):(0..^𝑁)⟶𝐴)
11 fcompt 6290 . . 3 ((𝐹:𝐴𝐵 ∧ (𝑆 repeatS 𝑁):(0..^𝑁)⟶𝐴) → (𝐹 ∘ (𝑆 repeatS 𝑁)) = (𝑥 ∈ (0..^𝑁) ↦ (𝐹‘((𝑆 repeatS 𝑁)‘𝑥))))
128, 10, 11syl2anc 690 . 2 ((𝑆𝐴𝑁 ∈ ℕ0𝐹:𝐴𝐵) → (𝐹 ∘ (𝑆 repeatS 𝑁)) = (𝑥 ∈ (0..^𝑁) ↦ (𝐹‘((𝑆 repeatS 𝑁)‘𝑥))))
13 fvex 6097 . . . . . 6 (𝐹𝑆) ∈ V
1413a1i 11 . . . . 5 (𝑆𝐴 → (𝐹𝑆) ∈ V)
1514anim1i 589 . . . 4 ((𝑆𝐴𝑁 ∈ ℕ0) → ((𝐹𝑆) ∈ V ∧ 𝑁 ∈ ℕ0))
16153adant3 1073 . . 3 ((𝑆𝐴𝑁 ∈ ℕ0𝐹:𝐴𝐵) → ((𝐹𝑆) ∈ V ∧ 𝑁 ∈ ℕ0))
17 reps 13316 . . 3 (((𝐹𝑆) ∈ V ∧ 𝑁 ∈ ℕ0) → ((𝐹𝑆) repeatS 𝑁) = (𝑥 ∈ (0..^𝑁) ↦ (𝐹𝑆)))
1816, 17syl 17 . 2 ((𝑆𝐴𝑁 ∈ ℕ0𝐹:𝐴𝐵) → ((𝐹𝑆) repeatS 𝑁) = (𝑥 ∈ (0..^𝑁) ↦ (𝐹𝑆)))
197, 12, 183eqtr4d 2653 1 ((𝑆𝐴𝑁 ∈ ℕ0𝐹:𝐴𝐵) → (𝐹 ∘ (𝑆 repeatS 𝑁)) = ((𝐹𝑆) repeatS 𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1030   = wceq 1474  wcel 1976  Vcvv 3172  cmpt 4637  ccom 5031  wf 5785  cfv 5789  (class class class)co 6526  0cc0 9792  0cn0 11141  ..^cfzo 12291   repeatS creps 13101
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4711  ax-pow 4763  ax-pr 4827
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4942  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040  df-iota 5753  df-fun 5791  df-fn 5792  df-f 5793  df-f1 5794  df-fo 5795  df-f1o 5796  df-fv 5797  df-ov 6529  df-oprab 6530  df-mpt2 6531  df-reps 13109
This theorem is referenced by: (None)
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