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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 12088 | . . 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 41389 | . . 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 11935 | . . . . . . . 8 ⊢ (0 + 1) = 1 | |
6 | 5 | oveq2i 7213 | . . . . . . 7 ⊢ (1...(0 + 1)) = (1...1) |
7 | 1z 12190 | . . . . . . . 8 ⊢ 1 ∈ ℤ | |
8 | fzsn 13137 | . . . . . . . 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 7213 | . . . . 5 ⊢ ((0[,]1) ↑m (1...(0 + 1))) = ((0[,]1) ↑m {1}) |
12 | 11 | rabeqi 3385 | . . . 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 15245 | . . . . . . 7 ⊢ Σ𝑘 ∈ (1...(0 + 1))(𝑡‘𝑘) = Σ𝑘 ∈ {1} (𝑡‘𝑘) |
14 | elmapi 8519 | . . . . . . . . 9 ⊢ (𝑡 ∈ ((0[,]1) ↑m {1}) → 𝑡:{1}⟶(0[,]1)) | |
15 | fsn2g 6942 | . . . . . . . . . . 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 219 | . . . . . . . . 9 ⊢ (𝑡:{1}⟶(0[,]1) → ((𝑡‘1) ∈ (0[,]1) ∧ 𝑡 = {〈1, (𝑡‘1)〉})) |
18 | unitssre 13070 | . . . . . . . . . . . 12 ⊢ (0[,]1) ⊆ ℝ | |
19 | ax-resscn 10769 | . . . . . . . . . . . 12 ⊢ ℝ ⊆ ℂ | |
20 | 18, 19 | sstri 3900 | . . . . . . . . . . 11 ⊢ (0[,]1) ⊆ ℂ |
21 | 20 | sseli 3887 | . . . . . . . . . 10 ⊢ ((𝑡‘1) ∈ (0[,]1) → (𝑡‘1) ∈ ℂ) |
22 | 21 | adantr 484 | . . . . . . . . 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 6706 | . . . . . . . . 9 ⊢ (𝑘 = 1 → (𝑡‘𝑘) = (𝑡‘1)) | |
25 | 24 | sumsn 15291 | . . . . . . . 8 ⊢ ((1 ∈ ℤ ∧ (𝑡‘1) ∈ ℂ) → Σ𝑘 ∈ {1} (𝑡‘𝑘) = (𝑡‘1)) |
26 | 7, 23, 25 | sylancr 590 | . . . . . . 7 ⊢ (𝑡 ∈ ((0[,]1) ↑m {1}) → Σ𝑘 ∈ {1} (𝑡‘𝑘) = (𝑡‘1)) |
27 | 13, 26 | syl5eq 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 3375 | . . . 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 4572 | . . . 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 7235 | . . . . 5 ⊢ (0[,]1) ∈ V | |
33 | 1elunit 13041 | . . . . 5 ⊢ 1 ∈ (0[,]1) | |
34 | k0004lem3 41388 | . . . . 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 1463 | . . . 4 ⊢ ((𝑡 ∈ ((0[,]1) ↑m {1}) ∧ (𝑡‘1) = 1) ↔ 𝑡 = {〈1, 1〉}) |
36 | 31, 35 | mpgbir 1807 | . . 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 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 {crab 3058 Vcvv 3401 {csn 4531 〈cop 4537 ↦ cmpt 5124 ⟶wf 6365 ‘cfv 6369 (class class class)co 7202 ↑m cmap 8497 ℂcc 10710 ℝcr 10711 0cc0 10712 1c1 10713 + caddc 10715 ℕ0cn0 12073 ℤcz 12159 [,]cicc 12921 ...cfz 13078 Σcsu 15232 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-inf2 9245 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 ax-pre-sup 10790 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-int 4850 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-se 5499 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-isom 6378 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-1st 7750 df-2nd 7751 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-er 8380 df-map 8499 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-sup 9047 df-oi 9115 df-card 9538 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-div 11473 df-nn 11814 df-2 11876 df-3 11877 df-n0 12074 df-z 12160 df-uz 12422 df-rp 12570 df-icc 12925 df-fz 13079 df-fzo 13222 df-seq 13558 df-exp 13619 df-hash 13880 df-cj 14645 df-re 14646 df-im 14647 df-sqrt 14781 df-abs 14782 df-clim 15032 df-sum 15233 |
This theorem is referenced by: (None) |
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