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Mirrors > Home > MPE Home > Th. List > reps | Structured version Visualization version GIF version |
Description: Construct a function mapping a half-open range of nonnegative integers to a constant. (Contributed by AV, 4-Nov-2018.) |
Ref | Expression |
---|---|
reps | ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑆 repeatS 𝑁) = (𝑥 ∈ (0..^𝑁) ↦ 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3499 | . . 3 ⊢ (𝑆 ∈ 𝑉 → 𝑆 ∈ V) | |
2 | 1 | adantr 480 | . 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → 𝑆 ∈ V) |
3 | simpr 484 | . 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → 𝑁 ∈ ℕ0) | |
4 | ovex 7464 | . . 3 ⊢ (0..^𝑁) ∈ V | |
5 | mptexg 7241 | . . 3 ⊢ ((0..^𝑁) ∈ V → (𝑥 ∈ (0..^𝑁) ↦ 𝑆) ∈ V) | |
6 | 4, 5 | mp1i 13 | . 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑥 ∈ (0..^𝑁) ↦ 𝑆) ∈ V) |
7 | oveq2 7439 | . . . . 5 ⊢ (𝑛 = 𝑁 → (0..^𝑛) = (0..^𝑁)) | |
8 | 7 | adantl 481 | . . . 4 ⊢ ((𝑠 = 𝑆 ∧ 𝑛 = 𝑁) → (0..^𝑛) = (0..^𝑁)) |
9 | simpll 767 | . . . 4 ⊢ (((𝑠 = 𝑆 ∧ 𝑛 = 𝑁) ∧ 𝑥 ∈ (0..^𝑛)) → 𝑠 = 𝑆) | |
10 | 8, 9 | mpteq12dva 5237 | . . 3 ⊢ ((𝑠 = 𝑆 ∧ 𝑛 = 𝑁) → (𝑥 ∈ (0..^𝑛) ↦ 𝑠) = (𝑥 ∈ (0..^𝑁) ↦ 𝑆)) |
11 | df-reps 14804 | . . 3 ⊢ repeatS = (𝑠 ∈ V, 𝑛 ∈ ℕ0 ↦ (𝑥 ∈ (0..^𝑛) ↦ 𝑠)) | |
12 | 10, 11 | ovmpoga 7587 | . 2 ⊢ ((𝑆 ∈ V ∧ 𝑁 ∈ ℕ0 ∧ (𝑥 ∈ (0..^𝑁) ↦ 𝑆) ∈ V) → (𝑆 repeatS 𝑁) = (𝑥 ∈ (0..^𝑁) ↦ 𝑆)) |
13 | 2, 3, 6, 12 | syl3anc 1370 | 1 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑆 repeatS 𝑁) = (𝑥 ∈ (0..^𝑁) ↦ 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ↦ cmpt 5231 (class class class)co 7431 0cc0 11153 ℕ0cn0 12524 ..^cfzo 13691 repeatS creps 14803 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-reps 14804 |
This theorem is referenced by: repsconst 14807 repsf 14808 repswsymb 14809 repswswrd 14819 repswccat 14821 repswrevw 14822 repsco 14876 |
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