Theorem repsco 14370

Description: Mapping of words commutes with the "repeated symbol" operation. (Contributed by AV, 11-Nov-2018.)

Assertion

Ref	Expression
repsco	⊢ ((𝑆 ∈ 𝐴 ∧ 𝑁 ∈ ℕ0 ∧ 𝐹:𝐴⟶𝐵) → (𝐹 ∘ (𝑆 repeatS 𝑁)) = ((𝐹‘𝑆) repeatS 𝑁))

Proof of Theorem repsco

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl1 1193	. . . . 5 ⊢ (((𝑆 ∈ 𝐴 ∧ 𝑁 ∈ ℕ0 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^𝑁)) → 𝑆 ∈ 𝐴)
2		simpl2 1194	. . . . 5 ⊢ (((𝑆 ∈ 𝐴 ∧ 𝑁 ∈ ℕ0 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^𝑁)) → 𝑁 ∈ ℕ0)
3		simpr 488	. . . . 5 ⊢ (((𝑆 ∈ 𝐴 ∧ 𝑁 ∈ ℕ0 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^𝑁)) → 𝑥 ∈ (0..^𝑁))
4		repswsymb 14304	. . . . 5 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝑁 ∈ ℕ0 ∧ 𝑥 ∈ (0..^𝑁)) → ((𝑆 repeatS 𝑁)‘𝑥) = 𝑆)
5	1, 2, 3, 4	syl3anc 1373	. . . 4 ⊢ (((𝑆 ∈ 𝐴 ∧ 𝑁 ∈ ℕ0 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^𝑁)) → ((𝑆 repeatS 𝑁)‘𝑥) = 𝑆)
6	5	fveq2d 6699	. . 3 ⊢ (((𝑆 ∈ 𝐴 ∧ 𝑁 ∈ ℕ0 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^𝑁)) → (𝐹‘((𝑆 repeatS 𝑁)‘𝑥)) = (𝐹‘𝑆))
7	6	mpteq2dva 5135	. 2 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝑁 ∈ ℕ0 ∧ 𝐹:𝐴⟶𝐵) → (𝑥 ∈ (0..^𝑁) ↦ (𝐹‘((𝑆 repeatS 𝑁)‘𝑥))) = (𝑥 ∈ (0..^𝑁) ↦ (𝐹‘𝑆)))
8		simp3 1140	. . 3 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝑁 ∈ ℕ0 ∧ 𝐹:𝐴⟶𝐵) → 𝐹:𝐴⟶𝐵)
9		repsf 14303	. . . 4 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝑁 ∈ ℕ0) → (𝑆 repeatS 𝑁):(0..^𝑁)⟶𝐴)
10	9	3adant3 1134	. . 3 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝑁 ∈ ℕ0 ∧ 𝐹:𝐴⟶𝐵) → (𝑆 repeatS 𝑁):(0..^𝑁)⟶𝐴)
11		fcompt 6926	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑆 repeatS 𝑁):(0..^𝑁)⟶𝐴) → (𝐹 ∘ (𝑆 repeatS 𝑁)) = (𝑥 ∈ (0..^𝑁) ↦ (𝐹‘((𝑆 repeatS 𝑁)‘𝑥))))
12	8, 10, 11	syl2anc 587	. 2 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝑁 ∈ ℕ0 ∧ 𝐹:𝐴⟶𝐵) → (𝐹 ∘ (𝑆 repeatS 𝑁)) = (𝑥 ∈ (0..^𝑁) ↦ (𝐹‘((𝑆 repeatS 𝑁)‘𝑥))))
13		fvexd 6710	. . . . 5 ⊢ (𝑆 ∈ 𝐴 → (𝐹‘𝑆) ∈ V)
14	13	anim1i 618	. . . 4 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝑁 ∈ ℕ0) → ((𝐹‘𝑆) ∈ V ∧ 𝑁 ∈ ℕ0))
15	14	3adant3 1134	. . 3 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝑁 ∈ ℕ0 ∧ 𝐹:𝐴⟶𝐵) → ((𝐹‘𝑆) ∈ V ∧ 𝑁 ∈ ℕ0))
16		reps 14300	. . 3 ⊢ (((𝐹‘𝑆) ∈ V ∧ 𝑁 ∈ ℕ0) → ((𝐹‘𝑆) repeatS 𝑁) = (𝑥 ∈ (0..^𝑁) ↦ (𝐹‘𝑆)))
17	15, 16	syl 17	. 2 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝑁 ∈ ℕ0 ∧ 𝐹:𝐴⟶𝐵) → ((𝐹‘𝑆) repeatS 𝑁) = (𝑥 ∈ (0..^𝑁) ↦ (𝐹‘𝑆)))
18	7, 12, 17	3eqtr4d 2781	1 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝑁 ∈ ℕ0 ∧ 𝐹:𝐴⟶𝐵) → (𝐹 ∘ (𝑆 repeatS 𝑁)) = ((𝐹‘𝑆) repeatS 𝑁))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2112  Vcvv 3398   ↦ cmpt 5120   ∘ ccom 5540  ⟶wf 6354  ‘cfv 6358  (class class class)co 7191  0cc0 10694  ℕ₀cn0 12055  ..^cfzo 13203   repeatS creps 14298

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pr 5307

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3400  df-sbc 3684  df-csb 3799  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-iun 4892  df-br 5040  df-opab 5102  df-mpt 5121  df-id 5440  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6316  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-ov 7194  df-oprab 7195  df-mpo 7196  df-reps 14299

This theorem is referenced by: (None)