Theorem reps 14723

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	⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ 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.

Step	Hyp	Ref	Expression
1		elex 3451	. . 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 7393	. . 3 ⊢ (0..^𝑁) ∈ V
5		mptexg 7169	. . 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 7368	. . . . 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 5172	. . 3 ⊢ ((𝑠 = 𝑆 ∧ 𝑛 = 𝑁) → (𝑥 ∈ (0..^𝑛) ↦ 𝑠) = (𝑥 ∈ (0..^𝑁) ↦ 𝑆))
11		df-reps 14722	. . 3 ⊢ repeatS = (𝑠 ∈ V, 𝑛 ∈ ℕ0 ↦ (𝑥 ∈ (0..^𝑛) ↦ 𝑠))
12	10, 11	ovmpoga 7514	. 2 ⊢ ((𝑆 ∈ V ∧ 𝑁 ∈ ℕ0 ∧ (𝑥 ∈ (0..^𝑁) ↦ 𝑆) ∈ V) → (𝑆 repeatS 𝑁) = (𝑥 ∈ (0..^𝑁) ↦ 𝑆))
13	2, 3, 6, 12	syl3anc 1374	1 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑆 repeatS 𝑁) = (𝑥 ∈ (0..^𝑁) ↦ 𝑆))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3430   ↦ cmpt 5167  (class class class)co 7360  0cc0 11029  ℕ₀cn0 12428  ..^cfzo 13599   repeatS creps 14721

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-reps 14722

This theorem is referenced by:  repsconst  14725  repsf  14726  repswsymb  14727  repswswrd  14737  repswccat  14739  repswrevw  14740  repsco  14793