Theorem k0004val 45259

Description: The topological simplex of dimension 𝑁 is the set of real vectors where the components are nonnegative and sum to 1. (Contributed by RP, 29-Mar-2021.)

Hypothesis

Ref	Expression
k0004.a	⊢ 𝐴 = (𝑛 ∈ ℕ0 ↦ {𝑡 ∈ ((0[,]1) ↑m (0...𝑛)) ∣ Σ𝑘 ∈ (0...𝑛)(𝑡‘𝑘) = 1})

Assertion

Ref	Expression
k0004val	⊢ (𝑁 ∈ ℕ0 → (𝐴‘𝑁) = {𝑡 ∈ ((0[,]1) ↑m (0...𝑁)) ∣ Σ𝑘 ∈ (0...𝑁)(𝑡‘𝑘) = 1})

Distinct variable groups:   𝑘,𝑛   𝑡,𝑛   𝑘,𝑁   𝑡,𝑁,𝑛

Allowed substitution hints:   𝐴(𝑡, 𝑘, 𝑛)

Proof of Theorem k0004val

Step	Hyp	Ref	Expression
1		oveq2 7432	. . . 4 ⊢ (𝑛 = 𝑁 → (0...𝑛) = (0...𝑁))
2	1	oveq2d 7440	. . 3 ⊢ (𝑛 = 𝑁 → ((0[,]1) ↑m (0...𝑛)) = ((0[,]1) ↑m (0...𝑁)))
3	1	sumeq1d 15876	. . . 4 ⊢ (𝑛 = 𝑁 → Σ𝑘 ∈ (0...𝑛)(𝑡‘𝑘) = Σ𝑘 ∈ (0...𝑁)(𝑡‘𝑘))
4	3	eqeq1d 2762	. . 3 ⊢ (𝑛 = 𝑁 → (Σ𝑘 ∈ (0...𝑛)(𝑡‘𝑘) = 1 ↔ Σ𝑘 ∈ (0...𝑁)(𝑡‘𝑘) = 1))
5	2, 4	rabeqbidv 3429	. 2 ⊢ (𝑛 = 𝑁 → {𝑡 ∈ ((0[,]1) ↑m (0...𝑛)) ∣ Σ𝑘 ∈ (0...𝑛)(𝑡‘𝑘) = 1} = {𝑡 ∈ ((0[,]1) ↑m (0...𝑁)) ∣ Σ𝑘 ∈ (0...𝑁)(𝑡‘𝑘) = 1})
6		k0004.a	. 2 ⊢ 𝐴 = (𝑛 ∈ ℕ0 ↦ {𝑡 ∈ ((0[,]1) ↑m (0...𝑛)) ∣ Σ𝑘 ∈ (0...𝑛)(𝑡‘𝑘) = 1})
7		ovex 7457	. . 3 ⊢ ((0[,]1) ↑m (0...𝑁)) ∈ V
8	7	rabex 5300	. 2 ⊢ {𝑡 ∈ ((0[,]1) ↑m (0...𝑁)) ∣ Σ𝑘 ∈ (0...𝑁)(𝑡‘𝑘) = 1} ∈ V
9	5, 6, 8	fvmpt 6998	1 ⊢ (𝑁 ∈ ℕ0 → (𝐴‘𝑁) = {𝑡 ∈ ((0[,]1) ↑m (0...𝑁)) ∣ Σ𝑘 ∈ (0...𝑁)(𝑡‘𝑘) = 1})

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144  {crab 3412   ↦ cmpt 5185  ‘cfv 6543  (class class class)co 7424   ↑_m cmap 8854  0cc0 11207  1c1 11208  ℕ₀cn0 12613  [,]cicc 13486  ...cfz 13646  Σcsu 15862

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6307  df-iota 6497  df-fun 6545  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7427  df-oprab 7428  df-mpo 7429  df-frecs 8306  df-wrecs 8337  df-recs 8386  df-rdg 8425  df-seq 14152  df-sum 15863

This theorem is used by:  k0004ss1  45260  k0004val0  45263