![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 482 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → 𝐺 ∈ Grp) | |
2 | simpr 484 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → 𝐽 = {∅, 𝐵}) | |
3 | distgp.1 | . . . . . 6 ⊢ 𝐵 = (Base‘𝐺) | |
4 | 3 | fvexi 6911 | . . . . 5 ⊢ 𝐵 ∈ V |
5 | indistopon 22903 | . . . . 5 ⊢ (𝐵 ∈ V → {∅, 𝐵} ∈ (TopOn‘𝐵)) | |
6 | 4, 5 | ax-mp 5 | . . . 4 ⊢ {∅, 𝐵} ∈ (TopOn‘𝐵) |
7 | 2, 6 | eqeltrdi 2837 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → 𝐽 ∈ (TopOn‘𝐵)) |
8 | distgp.2 | . . . 4 ⊢ 𝐽 = (TopOpen‘𝐺) | |
9 | 3, 8 | istps 22835 | . . 3 ⊢ (𝐺 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝐵)) |
10 | 7, 9 | sylibr 233 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → 𝐺 ∈ TopSp) |
11 | eqid 2728 | . . . . . 6 ⊢ (-g‘𝐺) = (-g‘𝐺) | |
12 | 3, 11 | grpsubf 18974 | . . . . 5 ⊢ (𝐺 ∈ Grp → (-g‘𝐺):(𝐵 × 𝐵)⟶𝐵) |
13 | 12 | adantr 480 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → (-g‘𝐺):(𝐵 × 𝐵)⟶𝐵) |
14 | 4, 4 | xpex 7755 | . . . . 5 ⊢ (𝐵 × 𝐵) ∈ V |
15 | 4, 14 | elmap 8889 | . . . 4 ⊢ ((-g‘𝐺) ∈ (𝐵 ↑m (𝐵 × 𝐵)) ↔ (-g‘𝐺):(𝐵 × 𝐵)⟶𝐵) |
16 | 13, 15 | sylibr 233 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → (-g‘𝐺) ∈ (𝐵 ↑m (𝐵 × 𝐵))) |
17 | 2 | oveq2d 7436 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → ((𝐽 ×t 𝐽) Cn 𝐽) = ((𝐽 ×t 𝐽) Cn {∅, 𝐵})) |
18 | txtopon 23494 | . . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝐵) ∧ 𝐽 ∈ (TopOn‘𝐵)) → (𝐽 ×t 𝐽) ∈ (TopOn‘(𝐵 × 𝐵))) | |
19 | 7, 7, 18 | syl2anc 583 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → (𝐽 ×t 𝐽) ∈ (TopOn‘(𝐵 × 𝐵))) |
20 | cnindis 23195 | . . . . 5 ⊢ (((𝐽 ×t 𝐽) ∈ (TopOn‘(𝐵 × 𝐵)) ∧ 𝐵 ∈ V) → ((𝐽 ×t 𝐽) Cn {∅, 𝐵}) = (𝐵 ↑m (𝐵 × 𝐵))) | |
21 | 19, 4, 20 | sylancl 585 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → ((𝐽 ×t 𝐽) Cn {∅, 𝐵}) = (𝐵 ↑m (𝐵 × 𝐵))) |
22 | 17, 21 | eqtrd 2768 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → ((𝐽 ×t 𝐽) Cn 𝐽) = (𝐵 ↑m (𝐵 × 𝐵))) |
23 | 16, 22 | eleqtrrd 2832 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → (-g‘𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) |
24 | 8, 11 | istgp2 23994 | . 2 ⊢ (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ (-g‘𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))) |
25 | 1, 10, 23, 24 | syl3anbrc 1341 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → 𝐺 ∈ TopGrp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 Vcvv 3471 ∅c0 4323 {cpr 4631 × cxp 5676 ⟶wf 6544 ‘cfv 6548 (class class class)co 7420 ↑m cmap 8844 Basecbs 17179 TopOpenctopn 17402 Grpcgrp 18889 -gcsg 18891 TopOnctopon 22811 TopSpctps 22833 Cn ccn 23127 ×t ctx 23463 TopGrpctgp 23974 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-1st 7993 df-2nd 7994 df-map 8846 df-0g 17422 df-topgen 17424 df-plusf 18598 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-grp 18892 df-minusg 18893 df-sbg 18894 df-top 22795 df-topon 22812 df-topsp 22834 df-bases 22848 df-cn 23130 df-cnp 23131 df-tx 23465 df-tmd 23975 df-tgp 23976 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |