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Mirrors > Home > MPE Home > Th. List > indistgp | Structured version Visualization version GIF version |
Description: Any group equipped with the indiscrete topology is a topological group. (Contributed by Mario Carneiro, 14-Aug-2015.) |
Ref | Expression |
---|---|
distgp.1 | ⊢ 𝐵 = (Base‘𝐺) |
distgp.2 | ⊢ 𝐽 = (TopOpen‘𝐺) |
Ref | Expression |
---|---|
indistgp | ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → 𝐺 ∈ TopGrp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 485 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → 𝐺 ∈ Grp) | |
2 | simpr 487 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → 𝐽 = {∅, 𝐵}) | |
3 | distgp.1 | . . . . . 6 ⊢ 𝐵 = (Base‘𝐺) | |
4 | 3 | fvexi 6686 | . . . . 5 ⊢ 𝐵 ∈ V |
5 | indistopon 21611 | . . . . 5 ⊢ (𝐵 ∈ V → {∅, 𝐵} ∈ (TopOn‘𝐵)) | |
6 | 4, 5 | ax-mp 5 | . . . 4 ⊢ {∅, 𝐵} ∈ (TopOn‘𝐵) |
7 | 2, 6 | eqeltrdi 2923 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → 𝐽 ∈ (TopOn‘𝐵)) |
8 | distgp.2 | . . . 4 ⊢ 𝐽 = (TopOpen‘𝐺) | |
9 | 3, 8 | istps 21544 | . . 3 ⊢ (𝐺 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝐵)) |
10 | 7, 9 | sylibr 236 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → 𝐺 ∈ TopSp) |
11 | eqid 2823 | . . . . . 6 ⊢ (-g‘𝐺) = (-g‘𝐺) | |
12 | 3, 11 | grpsubf 18180 | . . . . 5 ⊢ (𝐺 ∈ Grp → (-g‘𝐺):(𝐵 × 𝐵)⟶𝐵) |
13 | 12 | adantr 483 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → (-g‘𝐺):(𝐵 × 𝐵)⟶𝐵) |
14 | 4, 4 | xpex 7478 | . . . . 5 ⊢ (𝐵 × 𝐵) ∈ V |
15 | 4, 14 | elmap 8437 | . . . 4 ⊢ ((-g‘𝐺) ∈ (𝐵 ↑m (𝐵 × 𝐵)) ↔ (-g‘𝐺):(𝐵 × 𝐵)⟶𝐵) |
16 | 13, 15 | sylibr 236 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → (-g‘𝐺) ∈ (𝐵 ↑m (𝐵 × 𝐵))) |
17 | 2 | oveq2d 7174 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → ((𝐽 ×t 𝐽) Cn 𝐽) = ((𝐽 ×t 𝐽) Cn {∅, 𝐵})) |
18 | txtopon 22201 | . . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝐵) ∧ 𝐽 ∈ (TopOn‘𝐵)) → (𝐽 ×t 𝐽) ∈ (TopOn‘(𝐵 × 𝐵))) | |
19 | 7, 7, 18 | syl2anc 586 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → (𝐽 ×t 𝐽) ∈ (TopOn‘(𝐵 × 𝐵))) |
20 | cnindis 21902 | . . . . 5 ⊢ (((𝐽 ×t 𝐽) ∈ (TopOn‘(𝐵 × 𝐵)) ∧ 𝐵 ∈ V) → ((𝐽 ×t 𝐽) Cn {∅, 𝐵}) = (𝐵 ↑m (𝐵 × 𝐵))) | |
21 | 19, 4, 20 | sylancl 588 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → ((𝐽 ×t 𝐽) Cn {∅, 𝐵}) = (𝐵 ↑m (𝐵 × 𝐵))) |
22 | 17, 21 | eqtrd 2858 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → ((𝐽 ×t 𝐽) Cn 𝐽) = (𝐵 ↑m (𝐵 × 𝐵))) |
23 | 16, 22 | eleqtrrd 2918 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → (-g‘𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) |
24 | 8, 11 | istgp2 22701 | . 2 ⊢ (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ (-g‘𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))) |
25 | 1, 10, 23, 24 | syl3anbrc 1339 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → 𝐺 ∈ TopGrp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ∅c0 4293 {cpr 4571 × cxp 5555 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ↑m cmap 8408 Basecbs 16485 TopOpenctopn 16697 Grpcgrp 18105 -gcsg 18107 TopOnctopon 21520 TopSpctps 21542 Cn ccn 21834 ×t ctx 22170 TopGrpctgp 22681 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-map 8410 df-0g 16717 df-topgen 16719 df-plusf 17853 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-grp 18108 df-minusg 18109 df-sbg 18110 df-top 21504 df-topon 21521 df-topsp 21543 df-bases 21556 df-cn 21837 df-cnp 21838 df-tx 22172 df-tmd 22682 df-tgp 22683 |
This theorem is referenced by: (None) |
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