Theorem k0004val 45411

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 7459	. . . 4 ⊢ (𝑛 = 𝑁 → (0...𝑛) = (0...𝑁))
2	1	oveq2d 7467	. . 3 ⊢ (𝑛 = 𝑁 → ((0[,]1) ↑m (0...𝑛)) = ((0[,]1) ↑m (0...𝑁)))
3	1	sumeq1d 15911	. . . 4 ⊢ (𝑛 = 𝑁 → Σ𝑘 ∈ (0...𝑛)(𝑡‘𝑘) = Σ𝑘 ∈ (0...𝑁)(𝑡‘𝑘))
4	3	eqeq1d 2765	. . 3 ⊢ (𝑛 = 𝑁 → (Σ𝑘 ∈ (0...𝑛)(𝑡‘𝑘) = 1 ↔ Σ𝑘 ∈ (0...𝑁)(𝑡‘𝑘) = 1))
5	2, 4	rabeqbidv 3433	. 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 7484	. . 3 ⊢ ((0[,]1) ↑m (0...𝑁)) ∈ V
8	7	rabex 5307	. 2 ⊢ {𝑡 ∈ ((0[,]1) ↑m (0...𝑁)) ∣ Σ𝑘 ∈ (0...𝑁)(𝑡‘𝑘) = 1} ∈ V
9	5, 6, 8	fvmpt 7024	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 1570   ∈ wcel 2147  {crab 3416   ↦ cmpt 5192  ‘cfv 6566  (class class class)co 7451   ↑_m cmap 8885  0cc0 11239  1c1 11240  ℕ₀cn0 12646  [,]cicc 13519  ...cfz 13679  Σcsu 15897

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-iota 6520  df-fun 6568  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-seq 14185  df-sum 15898

This theorem is used by:  k0004ss1  45412  k0004val0  45415