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Theorem reps 14722
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 3485 . . 3 (𝑆𝑉𝑆 ∈ V)
21adantr 480 . 2 ((𝑆𝑉𝑁 ∈ ℕ0) → 𝑆 ∈ V)
3 simpr 484 . 2 ((𝑆𝑉𝑁 ∈ ℕ0) → 𝑁 ∈ ℕ0)
4 ovex 7435 . . 3 (0..^𝑁) ∈ V
5 mptexg 7215 . . 3 ((0..^𝑁) ∈ V → (𝑥 ∈ (0..^𝑁) ↦ 𝑆) ∈ V)
64, 5mp1i 13 . 2 ((𝑆𝑉𝑁 ∈ ℕ0) → (𝑥 ∈ (0..^𝑁) ↦ 𝑆) ∈ V)
7 oveq2 7410 . . . . 5 (𝑛 = 𝑁 → (0..^𝑛) = (0..^𝑁))
87adantl 481 . . . 4 ((𝑠 = 𝑆𝑛 = 𝑁) → (0..^𝑛) = (0..^𝑁))
9 simpll 764 . . . 4 (((𝑠 = 𝑆𝑛 = 𝑁) ∧ 𝑥 ∈ (0..^𝑛)) → 𝑠 = 𝑆)
108, 9mpteq12dva 5228 . . 3 ((𝑠 = 𝑆𝑛 = 𝑁) → (𝑥 ∈ (0..^𝑛) ↦ 𝑠) = (𝑥 ∈ (0..^𝑁) ↦ 𝑆))
11 df-reps 14721 . . 3 repeatS = (𝑠 ∈ V, 𝑛 ∈ ℕ0 ↦ (𝑥 ∈ (0..^𝑛) ↦ 𝑠))
1210, 11ovmpoga 7555 . 2 ((𝑆 ∈ V ∧ 𝑁 ∈ ℕ0 ∧ (𝑥 ∈ (0..^𝑁) ↦ 𝑆) ∈ V) → (𝑆 repeatS 𝑁) = (𝑥 ∈ (0..^𝑁) ↦ 𝑆))
132, 3, 6, 12syl3anc 1368 1 ((𝑆𝑉𝑁 ∈ ℕ0) → (𝑆 repeatS 𝑁) = (𝑥 ∈ (0..^𝑁) ↦ 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  Vcvv 3466  cmpt 5222  (class class class)co 7402  0cc0 11107  0cn0 12471  ..^cfzo 13628   repeatS creps 14720
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 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-reps 14721
This theorem is referenced by:  repsconst  14724  repsf  14725  repswsymb  14726  repswswrd  14736  repswccat  14738  repswrevw  14739  repsco  14793
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