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Theorem k0004val 37468
 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))(𝑡𝑘) = 1})
Assertion
Ref Expression
k0004val (𝑁 ∈ ℕ0 → (𝐴𝑁) = {𝑡 ∈ ((0[,]1) ↑𝑚 (1...(𝑁 + 1))) ∣ Σ𝑘 ∈ (1...(𝑁 + 1))(𝑡𝑘) = 1})
Distinct variable groups:   𝑘,𝑛   𝑡,𝑛   𝑘,𝑁   𝑡,𝑁,𝑛
Allowed substitution hints:   𝐴(𝑡,𝑘,𝑛)

Proof of Theorem k0004val
StepHypRef Expression
1 oveq1 6556 . . . . 5 (𝑛 = 𝑁 → (𝑛 + 1) = (𝑁 + 1))
21oveq2d 6565 . . . 4 (𝑛 = 𝑁 → (1...(𝑛 + 1)) = (1...(𝑁 + 1)))
32oveq2d 6565 . . 3 (𝑛 = 𝑁 → ((0[,]1) ↑𝑚 (1...(𝑛 + 1))) = ((0[,]1) ↑𝑚 (1...(𝑁 + 1))))
42sumeq1d 14279 . . . 4 (𝑛 = 𝑁 → Σ𝑘 ∈ (1...(𝑛 + 1))(𝑡𝑘) = Σ𝑘 ∈ (1...(𝑁 + 1))(𝑡𝑘))
54eqeq1d 2612 . . 3 (𝑛 = 𝑁 → (Σ𝑘 ∈ (1...(𝑛 + 1))(𝑡𝑘) = 1 ↔ Σ𝑘 ∈ (1...(𝑁 + 1))(𝑡𝑘) = 1))
63, 5rabeqbidv 3168 . 2 (𝑛 = 𝑁 → {𝑡 ∈ ((0[,]1) ↑𝑚 (1...(𝑛 + 1))) ∣ Σ𝑘 ∈ (1...(𝑛 + 1))(𝑡𝑘) = 1} = {𝑡 ∈ ((0[,]1) ↑𝑚 (1...(𝑁 + 1))) ∣ Σ𝑘 ∈ (1...(𝑁 + 1))(𝑡𝑘) = 1})
7 k0004.a . 2 𝐴 = (𝑛 ∈ ℕ0 ↦ {𝑡 ∈ ((0[,]1) ↑𝑚 (1...(𝑛 + 1))) ∣ Σ𝑘 ∈ (1...(𝑛 + 1))(𝑡𝑘) = 1})
8 ovex 6577 . . 3 ((0[,]1) ↑𝑚 (1...(𝑁 + 1))) ∈ V
98rabex 4740 . 2 {𝑡 ∈ ((0[,]1) ↑𝑚 (1...(𝑁 + 1))) ∣ Σ𝑘 ∈ (1...(𝑁 + 1))(𝑡𝑘) = 1} ∈ V
106, 7, 9fvmpt 6191 1 (𝑁 ∈ ℕ0 → (𝐴𝑁) = {𝑡 ∈ ((0[,]1) ↑𝑚 (1...(𝑁 + 1))) ∣ Σ𝑘 ∈ (1...(𝑁 + 1))(𝑡𝑘) = 1})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  {crab 2900   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549   ↑𝑚 cmap 7744  0cc0 9815  1c1 9816   + caddc 9818  ℕ0cn0 11169  [,]cicc 12049  ...cfz 12197  Σcsu 14264 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-iota 5768  df-fun 5806  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-seq 12664  df-sum 14265 This theorem is referenced by:  k0004ss1  37469  k0004val0  37472
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