Theorem reps 14746

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 3489	. . 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 7447	. . 3 ⊢ (0..^𝑁) ∈ V
5		mptexg 7227	. . 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 7422	. . . . 5 ⊢ (𝑛 = 𝑁 → (0..^𝑛) = (0..^𝑁))
8	7	adantl 481	. . . 4 ⊢ ((𝑠 = 𝑆 ∧ 𝑛 = 𝑁) → (0..^𝑛) = (0..^𝑁))
9		simpll 766	. . . 4 ⊢ (((𝑠 = 𝑆 ∧ 𝑛 = 𝑁) ∧ 𝑥 ∈ (0..^𝑛)) → 𝑠 = 𝑆)
10	8, 9	mpteq12dva 5231	. . 3 ⊢ ((𝑠 = 𝑆 ∧ 𝑛 = 𝑁) → (𝑥 ∈ (0..^𝑛) ↦ 𝑠) = (𝑥 ∈ (0..^𝑁) ↦ 𝑆))
11		df-reps 14745	. . 3 ⊢ repeatS = (𝑠 ∈ V, 𝑛 ∈ ℕ0 ↦ (𝑥 ∈ (0..^𝑛) ↦ 𝑠))
12	10, 11	ovmpoga 7569	. 2 ⊢ ((𝑆 ∈ V ∧ 𝑁 ∈ ℕ0 ∧ (𝑥 ∈ (0..^𝑁) ↦ 𝑆) ∈ V) → (𝑆 repeatS 𝑁) = (𝑥 ∈ (0..^𝑁) ↦ 𝑆))
13	2, 3, 6, 12	syl3anc 1369	1 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑆 repeatS 𝑁) = (𝑥 ∈ (0..^𝑁) ↦ 𝑆))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099  Vcvv 3470   ↦ cmpt 5225  (class class class)co 7414  0cc0 11132  ℕ₀cn0 12496  ..^cfzo 13653   repeatS creps 14744

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-reps 14745

This theorem is referenced by:  repsconst  14748  repsf  14749  repswsymb  14750  repswswrd  14760  repswccat  14762  repswrevw  14763  repsco  14817