Theorem k0004val 44595

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 7367	. . . . 5 ⊢ (𝑛 = 𝑁 → (𝑛 + 1) = (𝑁 + 1))
2	1	oveq2d 7376	. . . 4 ⊢ (𝑛 = 𝑁 → (1...(𝑛 + 1)) = (1...(𝑁 + 1)))
3	2	oveq2d 7376	. . 3 ⊢ (𝑛 = 𝑁 → ((0[,]1) ↑m (1...(𝑛 + 1))) = ((0[,]1) ↑m (1...(𝑁 + 1))))
4	2	sumeq1d 15653	. . . 4 ⊢ (𝑛 = 𝑁 → Σ𝑘 ∈ (1...(𝑛 + 1))(𝑡‘𝑘) = Σ𝑘 ∈ (1...(𝑁 + 1))(𝑡‘𝑘))
5	4	eqeq1d 2739	. . 3 ⊢ (𝑛 = 𝑁 → (Σ𝑘 ∈ (1...(𝑛 + 1))(𝑡‘𝑘) = 1 ↔ Σ𝑘 ∈ (1...(𝑁 + 1))(𝑡‘𝑘) = 1))
6	3, 5	rabeqbidv 3408	. 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 7393	. . 3 ⊢ ((0[,]1) ↑m (1...(𝑁 + 1))) ∈ V
9	8	rabex 5276	. 2 ⊢ {𝑡 ∈ ((0[,]1) ↑m (1...(𝑁 + 1))) ∣ Σ𝑘 ∈ (1...(𝑁 + 1))(𝑡‘𝑘) = 1} ∈ V
10	6, 7, 9	fvmpt 6941	1 ⊢ (𝑁 ∈ ℕ0 → (𝐴‘𝑁) = {𝑡 ∈ ((0[,]1) ↑m (1...(𝑁 + 1))) ∣ Σ𝑘 ∈ (1...(𝑁 + 1))(𝑡‘𝑘) = 1})

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  {crab 3390   ↦ cmpt 5167  ‘cfv 6492  (class class class)co 7360   ↑_m cmap 8766  0cc0 11029  1c1 11030   + caddc 11032  ℕ₀cn0 12428  [,]cicc 13292  ...cfz 13452  Σcsu 15639

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-iota 6448  df-fun 6494  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-seq 13955  df-sum 15640

This theorem is referenced by:  k0004ss1  44596  k0004val0  44599