MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  reps Structured version   Visualization version   GIF version

Theorem reps 14808
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 3501 . . 3 (𝑆𝑉𝑆 ∈ V)
21adantr 480 . 2 ((𝑆𝑉𝑁 ∈ ℕ0) → 𝑆 ∈ V)
3 simpr 484 . 2 ((𝑆𝑉𝑁 ∈ ℕ0) → 𝑁 ∈ ℕ0)
4 ovex 7464 . . 3 (0..^𝑁) ∈ V
5 mptexg 7241 . . 3 ((0..^𝑁) ∈ V → (𝑥 ∈ (0..^𝑁) ↦ 𝑆) ∈ V)
64, 5mp1i 13 . 2 ((𝑆𝑉𝑁 ∈ ℕ0) → (𝑥 ∈ (0..^𝑁) ↦ 𝑆) ∈ V)
7 oveq2 7439 . . . . 5 (𝑛 = 𝑁 → (0..^𝑛) = (0..^𝑁))
87adantl 481 . . . 4 ((𝑠 = 𝑆𝑛 = 𝑁) → (0..^𝑛) = (0..^𝑁))
9 simpll 767 . . . 4 (((𝑠 = 𝑆𝑛 = 𝑁) ∧ 𝑥 ∈ (0..^𝑛)) → 𝑠 = 𝑆)
108, 9mpteq12dva 5231 . . 3 ((𝑠 = 𝑆𝑛 = 𝑁) → (𝑥 ∈ (0..^𝑛) ↦ 𝑠) = (𝑥 ∈ (0..^𝑁) ↦ 𝑆))
11 df-reps 14807 . . 3 repeatS = (𝑠 ∈ V, 𝑛 ∈ ℕ0 ↦ (𝑥 ∈ (0..^𝑛) ↦ 𝑠))
1210, 11ovmpoga 7587 . 2 ((𝑆 ∈ V ∧ 𝑁 ∈ ℕ0 ∧ (𝑥 ∈ (0..^𝑁) ↦ 𝑆) ∈ V) → (𝑆 repeatS 𝑁) = (𝑥 ∈ (0..^𝑁) ↦ 𝑆))
132, 3, 6, 12syl3anc 1373 1 ((𝑆𝑉𝑁 ∈ ℕ0) → (𝑆 repeatS 𝑁) = (𝑥 ∈ (0..^𝑁) ↦ 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  cmpt 5225  (class class class)co 7431  0cc0 11155  0cn0 12526  ..^cfzo 13694   repeatS creps 14806
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-reps 14807
This theorem is referenced by:  repsconst  14810  repsf  14811  repswsymb  14812  repswswrd  14822  repswccat  14824  repswrevw  14825  repsco  14879
  Copyright terms: Public domain W3C validator