Theorem repsco 14829

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 1188	. . . . 5 ⊢ (((𝑆 ∈ 𝐴 ∧ 𝑁 ∈ ℕ0 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^𝑁)) → 𝑆 ∈ 𝐴)
2		simpl2 1189	. . . . 5 ⊢ (((𝑆 ∈ 𝐴 ∧ 𝑁 ∈ ℕ0 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^𝑁)) → 𝑁 ∈ ℕ0)
3		simpr 483	. . . . 5 ⊢ (((𝑆 ∈ 𝐴 ∧ 𝑁 ∈ ℕ0 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^𝑁)) → 𝑥 ∈ (0..^𝑁))
4		repswsymb 14762	. . . . 5 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝑁 ∈ ℕ0 ∧ 𝑥 ∈ (0..^𝑁)) → ((𝑆 repeatS 𝑁)‘𝑥) = 𝑆)
5	1, 2, 3, 4	syl3anc 1368	. . . 4 ⊢ (((𝑆 ∈ 𝐴 ∧ 𝑁 ∈ ℕ0 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^𝑁)) → ((𝑆 repeatS 𝑁)‘𝑥) = 𝑆)
6	5	fveq2d 6904	. . 3 ⊢ (((𝑆 ∈ 𝐴 ∧ 𝑁 ∈ ℕ0 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^𝑁)) → (𝐹‘((𝑆 repeatS 𝑁)‘𝑥)) = (𝐹‘𝑆))
7	6	mpteq2dva 5250	. 2 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝑁 ∈ ℕ0 ∧ 𝐹:𝐴⟶𝐵) → (𝑥 ∈ (0..^𝑁) ↦ (𝐹‘((𝑆 repeatS 𝑁)‘𝑥))) = (𝑥 ∈ (0..^𝑁) ↦ (𝐹‘𝑆)))
8		simp3 1135	. . 3 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝑁 ∈ ℕ0 ∧ 𝐹:𝐴⟶𝐵) → 𝐹:𝐴⟶𝐵)
9		repsf 14761	. . . 4 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝑁 ∈ ℕ0) → (𝑆 repeatS 𝑁):(0..^𝑁)⟶𝐴)
10	9	3adant3 1129	. . 3 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝑁 ∈ ℕ0 ∧ 𝐹:𝐴⟶𝐵) → (𝑆 repeatS 𝑁):(0..^𝑁)⟶𝐴)
11		fcompt 7146	. . 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 582	. 2 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝑁 ∈ ℕ0 ∧ 𝐹:𝐴⟶𝐵) → (𝐹 ∘ (𝑆 repeatS 𝑁)) = (𝑥 ∈ (0..^𝑁) ↦ (𝐹‘((𝑆 repeatS 𝑁)‘𝑥))))
13		fvexd 6915	. . . . 5 ⊢ (𝑆 ∈ 𝐴 → (𝐹‘𝑆) ∈ V)
14	13	anim1i 613	. . . 4 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝑁 ∈ ℕ0) → ((𝐹‘𝑆) ∈ V ∧ 𝑁 ∈ ℕ0))
15	14	3adant3 1129	. . 3 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝑁 ∈ ℕ0 ∧ 𝐹:𝐴⟶𝐵) → ((𝐹‘𝑆) ∈ V ∧ 𝑁 ∈ ℕ0))
16		reps 14758	. . 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 2777	1 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝑁 ∈ ℕ0 ∧ 𝐹:𝐴⟶𝐵) → (𝐹 ∘ (𝑆 repeatS 𝑁)) = ((𝐹‘𝑆) repeatS 𝑁))

Syntax hints:   → wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  Vcvv 3471   ↦ cmpt 5233   ∘ ccom 5684  ⟶wf 6547  ‘cfv 6551  (class class class)co 7424  0cc0 11144  ℕ₀cn0 12508  ..^cfzo 13665   repeatS creps 14756

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-rep 5287  ax-sep 5301  ax-nul 5308  ax-pr 5431

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-ov 7427  df-oprab 7428  df-mpo 7429  df-reps 14757

This theorem is referenced by: (None)