Theorem k0004val 44763

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 (1...(𝑛 + 1))) ∣ Σ𝑘 ∈ (1...(𝑛 + 1))(𝑡‘𝑘) = 1})

Assertion

Ref	Expression
k0004val	⊢ (𝑁 ∈ ℕ0 → (𝐴‘𝑁) = {𝑡 ∈ ((0[,]1) ↑m (1...(𝑁 + 1))) ∣ Σ𝑘 ∈ (1...(𝑁 + 1))(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		oveq1 7415	. . . . 5 ⊢ (𝑛 = 𝑁 → (𝑛 + 1) = (𝑁 + 1))
2	1	oveq2d 7424	. . . 4 ⊢ (𝑛 = 𝑁 → (1...(𝑛 + 1)) = (1...(𝑁 + 1)))
3	2	oveq2d 7424	. . 3 ⊢ (𝑛 = 𝑁 → ((0[,]1) ↑m (1...(𝑛 + 1))) = ((0[,]1) ↑m (1...(𝑁 + 1))))
4	2	sumeq1d 15747	. . . 4 ⊢ (𝑛 = 𝑁 → Σ𝑘 ∈ (1...(𝑛 + 1))(𝑡‘𝑘) = Σ𝑘 ∈ (1...(𝑁 + 1))(𝑡‘𝑘))
5	4	eqeq1d 2771	. . 3 ⊢ (𝑛 = 𝑁 → (Σ𝑘 ∈ (1...(𝑛 + 1))(𝑡‘𝑘) = 1 ↔ Σ𝑘 ∈ (1...(𝑁 + 1))(𝑡‘𝑘) = 1))
6	3, 5	rabeqbidv 3441	. 2 ⊢ (𝑛 = 𝑁 → {𝑡 ∈ ((0[,]1) ↑m (1...(𝑛 + 1))) ∣ Σ𝑘 ∈ (1...(𝑛 + 1))(𝑡‘𝑘) = 1} = {𝑡 ∈ ((0[,]1) ↑m (1...(𝑁 + 1))) ∣ Σ𝑘 ∈ (1...(𝑁 + 1))(𝑡‘𝑘) = 1})
7		k0004.a	. 2 ⊢ 𝐴 = (𝑛 ∈ ℕ0 ↦ {𝑡 ∈ ((0[,]1) ↑m (1...(𝑛 + 1))) ∣ Σ𝑘 ∈ (1...(𝑛 + 1))(𝑡‘𝑘) = 1})
8		ovex 7441	. . 3 ⊢ ((0[,]1) ↑m (1...(𝑁 + 1))) ∈ V
9	8	rabex 5307	. 2 ⊢ {𝑡 ∈ ((0[,]1) ↑m (1...(𝑁 + 1))) ∣ Σ𝑘 ∈ (1...(𝑁 + 1))(𝑡‘𝑘) = 1} ∈ V
10	6, 7, 9	fvmpt 6987	1 ⊢ (𝑁 ∈ ℕ0 → (𝐴‘𝑁) = {𝑡 ∈ ((0[,]1) ↑m (1...(𝑁 + 1))) ∣ Σ𝑘 ∈ (1...(𝑁 + 1))(𝑡‘𝑘) = 1})

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149  {crab 3423   ↦ cmpt 5193  ‘cfv 6534  (class class class)co 7408   ↑_m cmap 8820  0cc0 11096  1c1 11097   + caddc 11099  ℕ₀cn0 12500  [,]cicc 13371  ...cfz 13531  Σcsu 15733

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6300  df-iota 6490  df-fun 6536  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7411  df-oprab 7412  df-mpo 7413  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-seq 14034  df-sum 15734

This theorem is referenced by:  k0004ss1  44764  k0004val0  44767