Theorem reps 14716

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 3492	. . 3 ⊢ (𝑆 ∈ 𝑉 → 𝑆 ∈ V)
2	1	adantr 481	. 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → 𝑆 ∈ V)
3		simpr 485	. 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → 𝑁 ∈ ℕ0)
4		ovex 7438	. . 3 ⊢ (0..^𝑁) ∈ V
5		mptexg 7219	. . 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 7413	. . . . 5 ⊢ (𝑛 = 𝑁 → (0..^𝑛) = (0..^𝑁))
8	7	adantl 482	. . . 4 ⊢ ((𝑠 = 𝑆 ∧ 𝑛 = 𝑁) → (0..^𝑛) = (0..^𝑁))
9		simpll 765	. . . 4 ⊢ (((𝑠 = 𝑆 ∧ 𝑛 = 𝑁) ∧ 𝑥 ∈ (0..^𝑛)) → 𝑠 = 𝑆)
10	8, 9	mpteq12dva 5236	. . 3 ⊢ ((𝑠 = 𝑆 ∧ 𝑛 = 𝑁) → (𝑥 ∈ (0..^𝑛) ↦ 𝑠) = (𝑥 ∈ (0..^𝑁) ↦ 𝑆))
11		df-reps 14715	. . 3 ⊢ repeatS = (𝑠 ∈ V, 𝑛 ∈ ℕ0 ↦ (𝑥 ∈ (0..^𝑛) ↦ 𝑠))
12	10, 11	ovmpoga 7558	. 2 ⊢ ((𝑆 ∈ V ∧ 𝑁 ∈ ℕ0 ∧ (𝑥 ∈ (0..^𝑁) ↦ 𝑆) ∈ V) → (𝑆 repeatS 𝑁) = (𝑥 ∈ (0..^𝑁) ↦ 𝑆))
13	2, 3, 6, 12	syl3anc 1371	1 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑆 repeatS 𝑁) = (𝑥 ∈ (0..^𝑁) ↦ 𝑆))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  Vcvv 3474   ↦ cmpt 5230  (class class class)co 7405  0cc0 11106  ℕ₀cn0 12468  ..^cfzo 13623   repeatS creps 14714

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-reps 14715

This theorem is referenced by:  repsconst  14718  repsf  14719  repswsymb  14720  repswswrd  14730  repswccat  14732  repswrevw  14733  repsco  14787