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Theorem reps 13563
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 3243 . . 3 (𝑆𝑉𝑆 ∈ V)
21adantr 480 . 2 ((𝑆𝑉𝑁 ∈ ℕ0) → 𝑆 ∈ V)
3 simpr 476 . 2 ((𝑆𝑉𝑁 ∈ ℕ0) → 𝑁 ∈ ℕ0)
4 ovex 6718 . . 3 (0..^𝑁) ∈ V
5 mptexg 6525 . . 3 ((0..^𝑁) ∈ V → (𝑥 ∈ (0..^𝑁) ↦ 𝑆) ∈ V)
64, 5mp1i 13 . 2 ((𝑆𝑉𝑁 ∈ ℕ0) → (𝑥 ∈ (0..^𝑁) ↦ 𝑆) ∈ V)
7 oveq2 6698 . . . . 5 (𝑛 = 𝑁 → (0..^𝑛) = (0..^𝑁))
87adantl 481 . . . 4 ((𝑠 = 𝑆𝑛 = 𝑁) → (0..^𝑛) = (0..^𝑁))
9 simpll 805 . . . 4 (((𝑠 = 𝑆𝑛 = 𝑁) ∧ 𝑥 ∈ (0..^𝑛)) → 𝑠 = 𝑆)
108, 9mpteq12dva 4765 . . 3 ((𝑠 = 𝑆𝑛 = 𝑁) → (𝑥 ∈ (0..^𝑛) ↦ 𝑠) = (𝑥 ∈ (0..^𝑁) ↦ 𝑆))
11 df-reps 13338 . . 3 repeatS = (𝑠 ∈ V, 𝑛 ∈ ℕ0 ↦ (𝑥 ∈ (0..^𝑛) ↦ 𝑠))
1210, 11ovmpt2ga 6832 . 2 ((𝑆 ∈ V ∧ 𝑁 ∈ ℕ0 ∧ (𝑥 ∈ (0..^𝑁) ↦ 𝑆) ∈ V) → (𝑆 repeatS 𝑁) = (𝑥 ∈ (0..^𝑁) ↦ 𝑆))
132, 3, 6, 12syl3anc 1366 1 ((𝑆𝑉𝑁 ∈ ℕ0) → (𝑆 repeatS 𝑁) = (𝑥 ∈ (0..^𝑁) ↦ 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  Vcvv 3231  cmpt 4762  (class class class)co 6690  0cc0 9974  0cn0 11330  ..^cfzo 12504   repeatS creps 13330
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-reps 13338
This theorem is referenced by:  repsconst  13565  repsf  13566  repswsymb  13567  repswswrd  13577  repswccat  13578  repswrevw  13579  repsco  13631
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