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Theorem reps 13227
Description: Construct a function mapping a half-open range of nonnegative integers to a constant. (Contributed by AV, 4-Nov-2018.)
Assertion
Ref Expression
reps ((𝑆𝑉𝑁 ∈ ℕ0) → (𝑆 repeatS 𝑁) = (𝑥 ∈ (0..^𝑁) ↦ 𝑆))
Distinct variable groups:   𝑥,𝑁   𝑥,𝑆
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem reps
Dummy variables 𝑛 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3089 . . 3 (𝑆𝑉𝑆 ∈ V)
21adantr 479 . 2 ((𝑆𝑉𝑁 ∈ ℕ0) → 𝑆 ∈ V)
3 simpr 475 . 2 ((𝑆𝑉𝑁 ∈ ℕ0) → 𝑁 ∈ ℕ0)
4 ovex 6454 . . 3 (0..^𝑁) ∈ V
5 mptexg 6266 . . 3 ((0..^𝑁) ∈ V → (𝑥 ∈ (0..^𝑁) ↦ 𝑆) ∈ V)
64, 5mp1i 13 . 2 ((𝑆𝑉𝑁 ∈ ℕ0) → (𝑥 ∈ (0..^𝑁) ↦ 𝑆) ∈ V)
7 oveq2 6434 . . . . 5 (𝑛 = 𝑁 → (0..^𝑛) = (0..^𝑁))
87adantl 480 . . . 4 ((𝑠 = 𝑆𝑛 = 𝑁) → (0..^𝑛) = (0..^𝑁))
9 simpll 785 . . . 4 (((𝑠 = 𝑆𝑛 = 𝑁) ∧ 𝑥 ∈ (0..^𝑛)) → 𝑠 = 𝑆)
108, 9mpteq12dva 4560 . . 3 ((𝑠 = 𝑆𝑛 = 𝑁) → (𝑥 ∈ (0..^𝑛) ↦ 𝑠) = (𝑥 ∈ (0..^𝑁) ↦ 𝑆))
11 df-reps 13020 . . 3 repeatS = (𝑠 ∈ V, 𝑛 ∈ ℕ0 ↦ (𝑥 ∈ (0..^𝑛) ↦ 𝑠))
1210, 11ovmpt2ga 6565 . 2 ((𝑆 ∈ V ∧ 𝑁 ∈ ℕ0 ∧ (𝑥 ∈ (0..^𝑁) ↦ 𝑆) ∈ V) → (𝑆 repeatS 𝑁) = (𝑥 ∈ (0..^𝑁) ↦ 𝑆))
132, 3, 6, 12syl3anc 1317 1 ((𝑆𝑉𝑁 ∈ ℕ0) → (𝑆 repeatS 𝑁) = (𝑥 ∈ (0..^𝑁) ↦ 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1938  Vcvv 3077  cmpt 4541  (class class class)co 6426  0cc0 9691  0cn0 11047  ..^cfzo 12202   repeatS creps 13012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-rep 4597  ax-sep 4607  ax-nul 4616  ax-pr 4732
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-ral 2805  df-rex 2806  df-reu 2807  df-rab 2809  df-v 3079  df-sbc 3307  df-csb 3404  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-if 3940  df-sn 4029  df-pr 4031  df-op 4035  df-uni 4271  df-iun 4355  df-br 4482  df-opab 4542  df-mpt 4543  df-id 4847  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-iota 5653  df-fun 5691  df-fn 5692  df-f 5693  df-f1 5694  df-fo 5695  df-f1o 5696  df-fv 5697  df-ov 6429  df-oprab 6430  df-mpt2 6431  df-reps 13020
This theorem is referenced by:  repsconst  13229  repsf  13230  repswsymb  13231  repswswrd  13241  repswccat  13242  repswrevw  13243  repsco  13295
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