Theorem k0004val 44687

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 7398	. . . . 5 ⊢ (𝑛 = 𝑁 → (𝑛 + 1) = (𝑁 + 1))
2	1	oveq2d 7407	. . . 4 ⊢ (𝑛 = 𝑁 → (1...(𝑛 + 1)) = (1...(𝑁 + 1)))
3	2	oveq2d 7407	. . 3 ⊢ (𝑛 = 𝑁 → ((0[,]1) ↑m (1...(𝑛 + 1))) = ((0[,]1) ↑m (1...(𝑁 + 1))))
4	2	sumeq1d 15718	. . . 4 ⊢ (𝑛 = 𝑁 → Σ𝑘 ∈ (1...(𝑛 + 1))(𝑡‘𝑘) = Σ𝑘 ∈ (1...(𝑁 + 1))(𝑡‘𝑘))
5	4	eqeq1d 2763	. . 3 ⊢ (𝑛 = 𝑁 → (Σ𝑘 ∈ (1...(𝑛 + 1))(𝑡‘𝑘) = 1 ↔ Σ𝑘 ∈ (1...(𝑁 + 1))(𝑡‘𝑘) = 1))
6	3, 5	rabeqbidv 3431	. 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 7424	. . 3 ⊢ ((0[,]1) ↑m (1...(𝑁 + 1))) ∈ V
9	8	rabex 5292	. 2 ⊢ {𝑡 ∈ ((0[,]1) ↑m (1...(𝑁 + 1))) ∣ Σ𝑘 ∈ (1...(𝑁 + 1))(𝑡‘𝑘) = 1} ∈ V
10	6, 7, 9	fvmpt 6970	1 ⊢ (𝑁 ∈ ℕ0 → (𝐴‘𝑁) = {𝑡 ∈ ((0[,]1) ↑m (1...(𝑁 + 1))) ∣ Σ𝑘 ∈ (1...(𝑁 + 1))(𝑡‘𝑘) = 1})

Syntax hints:   → wi 4   = wceq 1559   ∈ wcel 2141  {crab 3413   ↦ cmpt 5178  ‘cfv 6516  (class class class)co 7391   ↑_m cmap 8802  0cc0 11067  1c1 11068   + caddc 11070  ℕ₀cn0 12475  [,]cicc 13346  ...cfz 13506  Σcsu 15704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-pred 6283  df-iota 6472  df-fun 6518  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-ov 7394  df-oprab 7395  df-mpo 7396  df-frecs 8256  df-wrecs 8287  df-recs 8336  df-rdg 8375  df-seq 14009  df-sum 15705

This theorem is referenced by:  k0004ss1  44688  k0004val0  44691