Theorem repsco 14205

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 1187	. . . . 5 ⊢ (((𝑆 ∈ 𝐴 ∧ 𝑁 ∈ ℕ0 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^𝑁)) → 𝑆 ∈ 𝐴)
2		simpl2 1188	. . . . 5 ⊢ (((𝑆 ∈ 𝐴 ∧ 𝑁 ∈ ℕ0 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^𝑁)) → 𝑁 ∈ ℕ0)
3		simpr 487	. . . . 5 ⊢ (((𝑆 ∈ 𝐴 ∧ 𝑁 ∈ ℕ0 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^𝑁)) → 𝑥 ∈ (0..^𝑁))
4		repswsymb 14139	. . . . 5 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝑁 ∈ ℕ0 ∧ 𝑥 ∈ (0..^𝑁)) → ((𝑆 repeatS 𝑁)‘𝑥) = 𝑆)
5	1, 2, 3, 4	syl3anc 1367	. . . 4 ⊢ (((𝑆 ∈ 𝐴 ∧ 𝑁 ∈ ℕ0 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^𝑁)) → ((𝑆 repeatS 𝑁)‘𝑥) = 𝑆)
6	5	fveq2d 6677	. . 3 ⊢ (((𝑆 ∈ 𝐴 ∧ 𝑁 ∈ ℕ0 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^𝑁)) → (𝐹‘((𝑆 repeatS 𝑁)‘𝑥)) = (𝐹‘𝑆))
7	6	mpteq2dva 5164	. 2 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝑁 ∈ ℕ0 ∧ 𝐹:𝐴⟶𝐵) → (𝑥 ∈ (0..^𝑁) ↦ (𝐹‘((𝑆 repeatS 𝑁)‘𝑥))) = (𝑥 ∈ (0..^𝑁) ↦ (𝐹‘𝑆)))
8		simp3 1134	. . 3 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝑁 ∈ ℕ0 ∧ 𝐹:𝐴⟶𝐵) → 𝐹:𝐴⟶𝐵)
9		repsf 14138	. . . 4 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝑁 ∈ ℕ0) → (𝑆 repeatS 𝑁):(0..^𝑁)⟶𝐴)
10	9	3adant3 1128	. . 3 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝑁 ∈ ℕ0 ∧ 𝐹:𝐴⟶𝐵) → (𝑆 repeatS 𝑁):(0..^𝑁)⟶𝐴)
11		fcompt 6898	. . 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 586	. 2 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝑁 ∈ ℕ0 ∧ 𝐹:𝐴⟶𝐵) → (𝐹 ∘ (𝑆 repeatS 𝑁)) = (𝑥 ∈ (0..^𝑁) ↦ (𝐹‘((𝑆 repeatS 𝑁)‘𝑥))))
13		fvexd 6688	. . . . 5 ⊢ (𝑆 ∈ 𝐴 → (𝐹‘𝑆) ∈ V)
14	13	anim1i 616	. . . 4 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝑁 ∈ ℕ0) → ((𝐹‘𝑆) ∈ V ∧ 𝑁 ∈ ℕ0))
15	14	3adant3 1128	. . 3 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝑁 ∈ ℕ0 ∧ 𝐹:𝐴⟶𝐵) → ((𝐹‘𝑆) ∈ V ∧ 𝑁 ∈ ℕ0))
16		reps 14135	. . 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 2869	1 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝑁 ∈ ℕ0 ∧ 𝐹:𝐴⟶𝐵) → (𝐹 ∘ (𝑆 repeatS 𝑁)) = ((𝐹‘𝑆) repeatS 𝑁))

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1536   ∈ wcel 2113  Vcvv 3497   ↦ cmpt 5149   ∘ ccom 5562  ⟶wf 6354  ‘cfv 6358  (class class class)co 7159  0cc0 10540  ℕ₀cn0 11900  ..^cfzo 13036   repeatS creps 14133

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-reu 3148  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-iun 4924  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-ov 7162  df-oprab 7163  df-mpo 7164  df-reps 14134

This theorem is referenced by: (None)