Theorem k0004val 44391

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 7365	. . . . 5 ⊢ (𝑛 = 𝑁 → (𝑛 + 1) = (𝑁 + 1))
2	1	oveq2d 7374	. . . 4 ⊢ (𝑛 = 𝑁 → (1...(𝑛 + 1)) = (1...(𝑁 + 1)))
3	2	oveq2d 7374	. . 3 ⊢ (𝑛 = 𝑁 → ((0[,]1) ↑m (1...(𝑛 + 1))) = ((0[,]1) ↑m (1...(𝑁 + 1))))
4	2	sumeq1d 15623	. . . 4 ⊢ (𝑛 = 𝑁 → Σ𝑘 ∈ (1...(𝑛 + 1))(𝑡‘𝑘) = Σ𝑘 ∈ (1...(𝑁 + 1))(𝑡‘𝑘))
5	4	eqeq1d 2738	. . 3 ⊢ (𝑛 = 𝑁 → (Σ𝑘 ∈ (1...(𝑛 + 1))(𝑡‘𝑘) = 1 ↔ Σ𝑘 ∈ (1...(𝑁 + 1))(𝑡‘𝑘) = 1))
6	3, 5	rabeqbidv 3417	. 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 7391	. . 3 ⊢ ((0[,]1) ↑m (1...(𝑁 + 1))) ∈ V
9	8	rabex 5284	. 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 1541   ∈ wcel 2113  {crab 3399   ↦ cmpt 5179  ‘cfv 6492  (class class class)co 7358   ↑_m cmap 8763  0cc0 11026  1c1 11027   + caddc 11029  ℕ₀cn0 12401  [,]cicc 13264  ...cfz 13423  Σcsu 15609

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  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 7361  df-oprab 7362  df-mpo 7363  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-seq 13925  df-sum 15610

This theorem is referenced by:  k0004ss1  44392  k0004val0  44395