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| Description: The symbols of a "repeated symbol word". (Contributed by AV, 4-Nov-2018.) | 
| Ref | Expression | 
|---|---|
| repswsymb | ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ∧ 𝐼 ∈ (0..^𝑁)) → ((𝑆 repeatS 𝑁)‘𝐼) = 𝑆) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | reps 14809 | . . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑆 repeatS 𝑁) = (𝑥 ∈ (0..^𝑁) ↦ 𝑆)) | |
| 2 | 1 | 3adant3 1132 | . 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ∧ 𝐼 ∈ (0..^𝑁)) → (𝑆 repeatS 𝑁) = (𝑥 ∈ (0..^𝑁) ↦ 𝑆)) | 
| 3 | eqidd 2737 | . 2 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ∧ 𝐼 ∈ (0..^𝑁)) ∧ 𝑥 = 𝐼) → 𝑆 = 𝑆) | |
| 4 | simp3 1138 | . 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ∧ 𝐼 ∈ (0..^𝑁)) → 𝐼 ∈ (0..^𝑁)) | |
| 5 | simp1 1136 | . 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ∧ 𝐼 ∈ (0..^𝑁)) → 𝑆 ∈ 𝑉) | |
| 6 | 2, 3, 4, 5 | fvmptd 7022 | 1 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ∧ 𝐼 ∈ (0..^𝑁)) → ((𝑆 repeatS 𝑁)‘𝐼) = 𝑆) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ↦ cmpt 5224 ‘cfv 6560 (class class class)co 7432 0cc0 11156 ℕ0cn0 12528 ..^cfzo 13695 repeatS creps 14807 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-reps 14808 | 
| This theorem is referenced by: repswfsts 14820 repswlsw 14821 repswswrd 14823 repswpfx 14824 repswccat 14825 repswrevw 14826 repsco 14880 | 
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