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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 11915 | . . 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 40506 | . . 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 11762 | . . . . . . . 8 ⊢ (0 + 1) = 1 | |
6 | 5 | oveq2i 7170 | . . . . . . 7 ⊢ (1...(0 + 1)) = (1...1) |
7 | 1z 12015 | . . . . . . . 8 ⊢ 1 ∈ ℤ | |
8 | fzsn 12952 | . . . . . . . 8 ⊢ (1 ∈ ℤ → (1...1) = {1}) | |
9 | 7, 8 | ax-mp 5 | . . . . . . 7 ⊢ (1...1) = {1} |
10 | 6, 9 | eqtri 2847 | . . . . . 6 ⊢ (1...(0 + 1)) = {1} |
11 | 10 | oveq2i 7170 | . . . . 5 ⊢ ((0[,]1) ↑m (1...(0 + 1))) = ((0[,]1) ↑m {1}) |
12 | 11 | rabeqi 3485 | . . . 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 15058 | . . . . . . 7 ⊢ Σ𝑘 ∈ (1...(0 + 1))(𝑡‘𝑘) = Σ𝑘 ∈ {1} (𝑡‘𝑘) |
14 | elmapi 8431 | . . . . . . . . 9 ⊢ (𝑡 ∈ ((0[,]1) ↑m {1}) → 𝑡:{1}⟶(0[,]1)) | |
15 | fsn2g 6903 | . . . . . . . . . . 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 218 | . . . . . . . . 9 ⊢ (𝑡:{1}⟶(0[,]1) → ((𝑡‘1) ∈ (0[,]1) ∧ 𝑡 = {〈1, (𝑡‘1)〉})) |
18 | unitssre 12888 | . . . . . . . . . . . 12 ⊢ (0[,]1) ⊆ ℝ | |
19 | ax-resscn 10597 | . . . . . . . . . . . 12 ⊢ ℝ ⊆ ℂ | |
20 | 18, 19 | sstri 3979 | . . . . . . . . . . 11 ⊢ (0[,]1) ⊆ ℂ |
21 | 20 | sseli 3966 | . . . . . . . . . 10 ⊢ ((𝑡‘1) ∈ (0[,]1) → (𝑡‘1) ∈ ℂ) |
22 | 21 | adantr 483 | . . . . . . . . 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 6673 | . . . . . . . . 9 ⊢ (𝑘 = 1 → (𝑡‘𝑘) = (𝑡‘1)) | |
25 | 24 | sumsn 15104 | . . . . . . . 8 ⊢ ((1 ∈ ℤ ∧ (𝑡‘1) ∈ ℂ) → Σ𝑘 ∈ {1} (𝑡‘𝑘) = (𝑡‘1)) |
26 | 7, 23, 25 | sylancr 589 | . . . . . . 7 ⊢ (𝑡 ∈ ((0[,]1) ↑m {1}) → Σ𝑘 ∈ {1} (𝑡‘𝑘) = (𝑡‘1)) |
27 | 13, 26 | syl5eq 2871 | . . . . . 6 ⊢ (𝑡 ∈ ((0[,]1) ↑m {1}) → Σ𝑘 ∈ (1...(0 + 1))(𝑡‘𝑘) = (𝑡‘1)) |
28 | 27 | eqeq1d 2826 | . . . . 5 ⊢ (𝑡 ∈ ((0[,]1) ↑m {1}) → (Σ𝑘 ∈ (1...(0 + 1))(𝑡‘𝑘) = 1 ↔ (𝑡‘1) = 1)) |
29 | 28 | rabbiia 3475 | . . . 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 2847 | . . 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 4609 | . . . 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 7192 | . . . . 5 ⊢ (0[,]1) ∈ V | |
33 | 1elunit 12859 | . . . . 5 ⊢ 1 ∈ (0[,]1) | |
34 | k0004lem3 40505 | . . . . 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 1457 | . . . 4 ⊢ ((𝑡 ∈ ((0[,]1) ↑m {1}) ∧ (𝑡‘1) = 1) ↔ 𝑡 = {〈1, 1〉}) |
36 | 31, 35 | mpgbir 1799 | . . 3 ⊢ {𝑡 ∈ ((0[,]1) ↑m {1}) ∣ (𝑡‘1) = 1} = {{〈1, 1〉}} |
37 | 30, 36 | eqtri 2847 | . 2 ⊢ {𝑡 ∈ ((0[,]1) ↑m (1...(0 + 1))) ∣ Σ𝑘 ∈ (1...(0 + 1))(𝑡‘𝑘) = 1} = {{〈1, 1〉}} |
38 | 4, 37 | eqtri 2847 | 1 ⊢ (𝐴‘0) = {{〈1, 1〉}} |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 {crab 3145 Vcvv 3497 {csn 4570 〈cop 4576 ↦ cmpt 5149 ⟶wf 6354 ‘cfv 6358 (class class class)co 7159 ↑m cmap 8409 ℂcc 10538 ℝcr 10539 0cc0 10540 1c1 10541 + caddc 10543 ℕ0cn0 11900 ℤcz 11984 [,]cicc 12744 ...cfz 12895 Σcsu 15045 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-inf2 9107 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-se 5518 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-map 8411 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-sup 8909 df-oi 8977 df-card 9371 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-icc 12748 df-fz 12896 df-fzo 13037 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14461 df-re 14462 df-im 14463 df-sqrt 14597 df-abs 14598 df-clim 14848 df-sum 15046 |
This theorem is referenced by: (None) |
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