Theorem reps 13737

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 3352	. . 3 ⊢ (𝑆 ∈ 𝑉 → 𝑆 ∈ V)
2	1	adantr 472	. 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → 𝑆 ∈ V)
3		simpr 479	. 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → 𝑁 ∈ ℕ0)
4		ovex 6842	. . 3 ⊢ (0..^𝑁) ∈ V
5		mptexg 6649	. . 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 6822	. . . . 5 ⊢ (𝑛 = 𝑁 → (0..^𝑛) = (0..^𝑁))
8	7	adantl 473	. . . 4 ⊢ ((𝑠 = 𝑆 ∧ 𝑛 = 𝑁) → (0..^𝑛) = (0..^𝑁))
9		simpll 807	. . . 4 ⊢ (((𝑠 = 𝑆 ∧ 𝑛 = 𝑁) ∧ 𝑥 ∈ (0..^𝑛)) → 𝑠 = 𝑆)
10	8, 9	mpteq12dva 4884	. . 3 ⊢ ((𝑠 = 𝑆 ∧ 𝑛 = 𝑁) → (𝑥 ∈ (0..^𝑛) ↦ 𝑠) = (𝑥 ∈ (0..^𝑁) ↦ 𝑆))
11		df-reps 13512	. . 3 ⊢ repeatS = (𝑠 ∈ V, 𝑛 ∈ ℕ0 ↦ (𝑥 ∈ (0..^𝑛) ↦ 𝑠))
12	10, 11	ovmpt2ga 6956	. 2 ⊢ ((𝑆 ∈ V ∧ 𝑁 ∈ ℕ0 ∧ (𝑥 ∈ (0..^𝑁) ↦ 𝑆) ∈ V) → (𝑆 repeatS 𝑁) = (𝑥 ∈ (0..^𝑁) ↦ 𝑆))
13	2, 3, 6, 12	syl3anc 1477	1 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑆 repeatS 𝑁) = (𝑥 ∈ (0..^𝑁) ↦ 𝑆))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1632   ∈ wcel 2139  Vcvv 3340   ↦ cmpt 4881  (class class class)co 6814  0cc0 10148  ℕ₀cn0 11504  ..^cfzo 12679   repeatS creps 13504

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pr 5055

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-reps 13512

This theorem is referenced by:  repsconst  13739  repsf  13740  repswsymb  13741  repswswrd  13751  repswccat  13752  repswrevw  13753  repsco  13805