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Mirrors > Home > MPE Home > Th. List > Mathboxes > k0004val0 | Structured version Visualization version GIF version |
Description: The topological simplex of dimension 0 is a singleton. (Contributed by RP, 2-Apr-2021.) |
Ref | Expression |
---|---|
k0004.a | ⊢ 𝐴 = (𝑛 ∈ ℕ0 ↦ {𝑡 ∈ ((0[,]1) ↑m (1...(𝑛 + 1))) ∣ Σ𝑘 ∈ (1...(𝑛 + 1))(𝑡‘𝑘) = 1}) |
Ref | Expression |
---|---|
k0004val0 | ⊢ (𝐴‘0) = {{〈1, 1〉}} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0nn0 12538 | . . 3 ⊢ 0 ∈ ℕ0 | |
2 | k0004.a | . . . 4 ⊢ 𝐴 = (𝑛 ∈ ℕ0 ↦ {𝑡 ∈ ((0[,]1) ↑m (1...(𝑛 + 1))) ∣ Σ𝑘 ∈ (1...(𝑛 + 1))(𝑡‘𝑘) = 1}) | |
3 | 2 | k0004val 44139 | . . 3 ⊢ (0 ∈ ℕ0 → (𝐴‘0) = {𝑡 ∈ ((0[,]1) ↑m (1...(0 + 1))) ∣ Σ𝑘 ∈ (1...(0 + 1))(𝑡‘𝑘) = 1}) |
4 | 1, 3 | ax-mp 5 | . 2 ⊢ (𝐴‘0) = {𝑡 ∈ ((0[,]1) ↑m (1...(0 + 1))) ∣ Σ𝑘 ∈ (1...(0 + 1))(𝑡‘𝑘) = 1} |
5 | 0p1e1 12385 | . . . . . . . 8 ⊢ (0 + 1) = 1 | |
6 | 5 | oveq2i 7441 | . . . . . . 7 ⊢ (1...(0 + 1)) = (1...1) |
7 | 1z 12644 | . . . . . . . 8 ⊢ 1 ∈ ℤ | |
8 | fzsn 13602 | . . . . . . . 8 ⊢ (1 ∈ ℤ → (1...1) = {1}) | |
9 | 7, 8 | ax-mp 5 | . . . . . . 7 ⊢ (1...1) = {1} |
10 | 6, 9 | eqtri 2762 | . . . . . 6 ⊢ (1...(0 + 1)) = {1} |
11 | 10 | oveq2i 7441 | . . . . 5 ⊢ ((0[,]1) ↑m (1...(0 + 1))) = ((0[,]1) ↑m {1}) |
12 | 11 | rabeqi 3446 | . . . 4 ⊢ {𝑡 ∈ ((0[,]1) ↑m (1...(0 + 1))) ∣ Σ𝑘 ∈ (1...(0 + 1))(𝑡‘𝑘) = 1} = {𝑡 ∈ ((0[,]1) ↑m {1}) ∣ Σ𝑘 ∈ (1...(0 + 1))(𝑡‘𝑘) = 1} |
13 | 10 | sumeq1i 15729 | . . . . . . 7 ⊢ Σ𝑘 ∈ (1...(0 + 1))(𝑡‘𝑘) = Σ𝑘 ∈ {1} (𝑡‘𝑘) |
14 | elmapi 8887 | . . . . . . . . 9 ⊢ (𝑡 ∈ ((0[,]1) ↑m {1}) → 𝑡:{1}⟶(0[,]1)) | |
15 | fsn2g 7157 | . . . . . . . . . . 11 ⊢ (1 ∈ ℤ → (𝑡:{1}⟶(0[,]1) ↔ ((𝑡‘1) ∈ (0[,]1) ∧ 𝑡 = {〈1, (𝑡‘1)〉}))) | |
16 | 7, 15 | ax-mp 5 | . . . . . . . . . 10 ⊢ (𝑡:{1}⟶(0[,]1) ↔ ((𝑡‘1) ∈ (0[,]1) ∧ 𝑡 = {〈1, (𝑡‘1)〉})) |
17 | 16 | biimpi 216 | . . . . . . . . 9 ⊢ (𝑡:{1}⟶(0[,]1) → ((𝑡‘1) ∈ (0[,]1) ∧ 𝑡 = {〈1, (𝑡‘1)〉})) |
18 | unitssre 13535 | . . . . . . . . . . . 12 ⊢ (0[,]1) ⊆ ℝ | |
19 | ax-resscn 11209 | . . . . . . . . . . . 12 ⊢ ℝ ⊆ ℂ | |
20 | 18, 19 | sstri 4004 | . . . . . . . . . . 11 ⊢ (0[,]1) ⊆ ℂ |
21 | 20 | sseli 3990 | . . . . . . . . . 10 ⊢ ((𝑡‘1) ∈ (0[,]1) → (𝑡‘1) ∈ ℂ) |
22 | 21 | adantr 480 | . . . . . . . . 9 ⊢ (((𝑡‘1) ∈ (0[,]1) ∧ 𝑡 = {〈1, (𝑡‘1)〉}) → (𝑡‘1) ∈ ℂ) |
23 | 14, 17, 22 | 3syl 18 | . . . . . . . 8 ⊢ (𝑡 ∈ ((0[,]1) ↑m {1}) → (𝑡‘1) ∈ ℂ) |
24 | fveq2 6906 | . . . . . . . . 9 ⊢ (𝑘 = 1 → (𝑡‘𝑘) = (𝑡‘1)) | |
25 | 24 | sumsn 15778 | . . . . . . . 8 ⊢ ((1 ∈ ℤ ∧ (𝑡‘1) ∈ ℂ) → Σ𝑘 ∈ {1} (𝑡‘𝑘) = (𝑡‘1)) |
26 | 7, 23, 25 | sylancr 587 | . . . . . . 7 ⊢ (𝑡 ∈ ((0[,]1) ↑m {1}) → Σ𝑘 ∈ {1} (𝑡‘𝑘) = (𝑡‘1)) |
27 | 13, 26 | eqtrid 2786 | . . . . . 6 ⊢ (𝑡 ∈ ((0[,]1) ↑m {1}) → Σ𝑘 ∈ (1...(0 + 1))(𝑡‘𝑘) = (𝑡‘1)) |
28 | 27 | eqeq1d 2736 | . . . . 5 ⊢ (𝑡 ∈ ((0[,]1) ↑m {1}) → (Σ𝑘 ∈ (1...(0 + 1))(𝑡‘𝑘) = 1 ↔ (𝑡‘1) = 1)) |
29 | 28 | rabbiia 3436 | . . . 4 ⊢ {𝑡 ∈ ((0[,]1) ↑m {1}) ∣ Σ𝑘 ∈ (1...(0 + 1))(𝑡‘𝑘) = 1} = {𝑡 ∈ ((0[,]1) ↑m {1}) ∣ (𝑡‘1) = 1} |
30 | 12, 29 | eqtri 2762 | . . 3 ⊢ {𝑡 ∈ ((0[,]1) ↑m (1...(0 + 1))) ∣ Σ𝑘 ∈ (1...(0 + 1))(𝑡‘𝑘) = 1} = {𝑡 ∈ ((0[,]1) ↑m {1}) ∣ (𝑡‘1) = 1} |
31 | rabeqsn 4671 | . . . 4 ⊢ ({𝑡 ∈ ((0[,]1) ↑m {1}) ∣ (𝑡‘1) = 1} = {{〈1, 1〉}} ↔ ∀𝑡((𝑡 ∈ ((0[,]1) ↑m {1}) ∧ (𝑡‘1) = 1) ↔ 𝑡 = {〈1, 1〉})) | |
32 | ovex 7463 | . . . . 5 ⊢ (0[,]1) ∈ V | |
33 | 1elunit 13506 | . . . . 5 ⊢ 1 ∈ (0[,]1) | |
34 | k0004lem3 44138 | . . . . 5 ⊢ ((1 ∈ ℤ ∧ (0[,]1) ∈ V ∧ 1 ∈ (0[,]1)) → ((𝑡 ∈ ((0[,]1) ↑m {1}) ∧ (𝑡‘1) = 1) ↔ 𝑡 = {〈1, 1〉})) | |
35 | 7, 32, 33, 34 | mp3an 1460 | . . . 4 ⊢ ((𝑡 ∈ ((0[,]1) ↑m {1}) ∧ (𝑡‘1) = 1) ↔ 𝑡 = {〈1, 1〉}) |
36 | 31, 35 | mpgbir 1795 | . . 3 ⊢ {𝑡 ∈ ((0[,]1) ↑m {1}) ∣ (𝑡‘1) = 1} = {{〈1, 1〉}} |
37 | 30, 36 | eqtri 2762 | . 2 ⊢ {𝑡 ∈ ((0[,]1) ↑m (1...(0 + 1))) ∣ Σ𝑘 ∈ (1...(0 + 1))(𝑡‘𝑘) = 1} = {{〈1, 1〉}} |
38 | 4, 37 | eqtri 2762 | 1 ⊢ (𝐴‘0) = {{〈1, 1〉}} |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1536 ∈ wcel 2105 {crab 3432 Vcvv 3477 {csn 4630 〈cop 4636 ↦ cmpt 5230 ⟶wf 6558 ‘cfv 6562 (class class class)co 7430 ↑m cmap 8864 ℂcc 11150 ℝcr 11151 0cc0 11152 1c1 11153 + caddc 11155 ℕ0cn0 12523 ℤcz 12610 [,]cicc 13386 ...cfz 13543 Σcsu 15718 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-inf2 9678 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-er 8743 df-map 8866 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-sup 9479 df-oi 9547 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-n0 12524 df-z 12611 df-uz 12876 df-rp 13032 df-icc 13390 df-fz 13544 df-fzo 13691 df-seq 14039 df-exp 14099 df-hash 14366 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-clim 15520 df-sum 15719 |
This theorem is referenced by: (None) |
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