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Mirrors > Home > MPE Home > Th. List > Mathboxes > k0004ss1 | Structured version Visualization version GIF version |
Description: The topological simplex of dimension 𝑁 is a subset of the real vectors of dimension (𝑁 + 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 |
---|---|
k0004ss1 | ⊢ (𝑁 ∈ ℕ0 → (𝐴‘𝑁) ⊆ (ℝ ↑m (1...(𝑁 + 1)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | k0004.a | . . . 4 ⊢ 𝐴 = (𝑛 ∈ ℕ0 ↦ {𝑡 ∈ ((0[,]1) ↑m (1...(𝑛 + 1))) ∣ Σ𝑘 ∈ (1...(𝑛 + 1))(𝑡‘𝑘) = 1}) | |
2 | 1 | k0004val 40493 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (𝐴‘𝑁) = {𝑡 ∈ ((0[,]1) ↑m (1...(𝑁 + 1))) ∣ Σ𝑘 ∈ (1...(𝑁 + 1))(𝑡‘𝑘) = 1}) |
3 | simp2 1133 | . . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑡 ∈ ((0[,]1) ↑m (1...(𝑁 + 1))) ∧ Σ𝑘 ∈ (1...(𝑁 + 1))(𝑡‘𝑘) = 1) → 𝑡 ∈ ((0[,]1) ↑m (1...(𝑁 + 1)))) | |
4 | 3 | rabssdv 4050 | . . 3 ⊢ (𝑁 ∈ ℕ0 → {𝑡 ∈ ((0[,]1) ↑m (1...(𝑁 + 1))) ∣ Σ𝑘 ∈ (1...(𝑁 + 1))(𝑡‘𝑘) = 1} ⊆ ((0[,]1) ↑m (1...(𝑁 + 1)))) |
5 | 2, 4 | eqsstrd 4004 | . 2 ⊢ (𝑁 ∈ ℕ0 → (𝐴‘𝑁) ⊆ ((0[,]1) ↑m (1...(𝑁 + 1)))) |
6 | reex 10622 | . . 3 ⊢ ℝ ∈ V | |
7 | unitssre 12879 | . . 3 ⊢ (0[,]1) ⊆ ℝ | |
8 | mapss 8447 | . . 3 ⊢ ((ℝ ∈ V ∧ (0[,]1) ⊆ ℝ) → ((0[,]1) ↑m (1...(𝑁 + 1))) ⊆ (ℝ ↑m (1...(𝑁 + 1)))) | |
9 | 6, 7, 8 | mp2an 690 | . 2 ⊢ ((0[,]1) ↑m (1...(𝑁 + 1))) ⊆ (ℝ ↑m (1...(𝑁 + 1))) |
10 | 5, 9 | sstrdi 3978 | 1 ⊢ (𝑁 ∈ ℕ0 → (𝐴‘𝑁) ⊆ (ℝ ↑m (1...(𝑁 + 1)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 {crab 3142 Vcvv 3494 ⊆ wss 3935 ↦ cmpt 5138 ‘cfv 6349 (class class class)co 7150 ↑m cmap 8400 ℝcr 10530 0cc0 10531 1c1 10532 + caddc 10534 ℕ0cn0 11891 [,]cicc 12735 ...cfz 12886 Σcsu 15036 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-i2m1 10599 ax-1ne0 10600 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-po 5468 df-so 5469 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-icc 12739 df-seq 13364 df-sum 15037 |
This theorem is referenced by: k0004ss2 40495 k0004ss3 40496 |
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