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Mirrors > Home > MPE Home > Th. List > Mathboxes > k0004val | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
k0004.a | ⊢ 𝐴 = (𝑛 ∈ ℕ0 ↦ {𝑡 ∈ ((0[,]1) ↑𝑚 (1...(𝑛 + 1))) ∣ Σ𝑘 ∈ (1...(𝑛 + 1))(𝑡‘𝑘) = 1}) |
Ref | Expression |
---|---|
k0004val | ⊢ (𝑁 ∈ ℕ0 → (𝐴‘𝑁) = {𝑡 ∈ ((0[,]1) ↑𝑚 (1...(𝑁 + 1))) ∣ Σ𝑘 ∈ (1...(𝑁 + 1))(𝑡‘𝑘) = 1}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 6556 | . . . . 5 ⊢ (𝑛 = 𝑁 → (𝑛 + 1) = (𝑁 + 1)) | |
2 | 1 | oveq2d 6565 | . . . 4 ⊢ (𝑛 = 𝑁 → (1...(𝑛 + 1)) = (1...(𝑁 + 1))) |
3 | 2 | oveq2d 6565 | . . 3 ⊢ (𝑛 = 𝑁 → ((0[,]1) ↑𝑚 (1...(𝑛 + 1))) = ((0[,]1) ↑𝑚 (1...(𝑁 + 1)))) |
4 | 2 | sumeq1d 14279 | . . . 4 ⊢ (𝑛 = 𝑁 → Σ𝑘 ∈ (1...(𝑛 + 1))(𝑡‘𝑘) = Σ𝑘 ∈ (1...(𝑁 + 1))(𝑡‘𝑘)) |
5 | 4 | eqeq1d 2612 | . . 3 ⊢ (𝑛 = 𝑁 → (Σ𝑘 ∈ (1...(𝑛 + 1))(𝑡‘𝑘) = 1 ↔ Σ𝑘 ∈ (1...(𝑁 + 1))(𝑡‘𝑘) = 1)) |
6 | 3, 5 | rabeqbidv 3168 | . 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 6577 | . . 3 ⊢ ((0[,]1) ↑𝑚 (1...(𝑁 + 1))) ∈ V | |
9 | 8 | rabex 4740 | . 2 ⊢ {𝑡 ∈ ((0[,]1) ↑𝑚 (1...(𝑁 + 1))) ∣ Σ𝑘 ∈ (1...(𝑁 + 1))(𝑡‘𝑘) = 1} ∈ V |
10 | 6, 7, 9 | fvmpt 6191 | 1 ⊢ (𝑁 ∈ ℕ0 → (𝐴‘𝑁) = {𝑡 ∈ ((0[,]1) ↑𝑚 (1...(𝑁 + 1))) ∣ Σ𝑘 ∈ (1...(𝑁 + 1))(𝑡‘𝑘) = 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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