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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 474 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → 𝐺 ∈ Grp) | |
2 | simpr 479 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → 𝐽 = {∅, 𝐵}) | |
3 | distgp.1 | . . . . . 6 ⊢ 𝐵 = (Base‘𝐺) | |
4 | fvex 6362 | . . . . . 6 ⊢ (Base‘𝐺) ∈ V | |
5 | 3, 4 | eqeltri 2835 | . . . . 5 ⊢ 𝐵 ∈ V |
6 | indistopon 21007 | . . . . 5 ⊢ (𝐵 ∈ V → {∅, 𝐵} ∈ (TopOn‘𝐵)) | |
7 | 5, 6 | ax-mp 5 | . . . 4 ⊢ {∅, 𝐵} ∈ (TopOn‘𝐵) |
8 | 2, 7 | syl6eqel 2847 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → 𝐽 ∈ (TopOn‘𝐵)) |
9 | distgp.2 | . . . 4 ⊢ 𝐽 = (TopOpen‘𝐺) | |
10 | 3, 9 | istps 20940 | . . 3 ⊢ (𝐺 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝐵)) |
11 | 8, 10 | sylibr 224 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → 𝐺 ∈ TopSp) |
12 | eqid 2760 | . . . . . 6 ⊢ (-g‘𝐺) = (-g‘𝐺) | |
13 | 3, 12 | grpsubf 17695 | . . . . 5 ⊢ (𝐺 ∈ Grp → (-g‘𝐺):(𝐵 × 𝐵)⟶𝐵) |
14 | 13 | adantr 472 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → (-g‘𝐺):(𝐵 × 𝐵)⟶𝐵) |
15 | 5, 5 | xpex 7127 | . . . . 5 ⊢ (𝐵 × 𝐵) ∈ V |
16 | 5, 15 | elmap 8052 | . . . 4 ⊢ ((-g‘𝐺) ∈ (𝐵 ↑𝑚 (𝐵 × 𝐵)) ↔ (-g‘𝐺):(𝐵 × 𝐵)⟶𝐵) |
17 | 14, 16 | sylibr 224 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → (-g‘𝐺) ∈ (𝐵 ↑𝑚 (𝐵 × 𝐵))) |
18 | 2 | oveq2d 6829 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → ((𝐽 ×t 𝐽) Cn 𝐽) = ((𝐽 ×t 𝐽) Cn {∅, 𝐵})) |
19 | txtopon 21596 | . . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝐵) ∧ 𝐽 ∈ (TopOn‘𝐵)) → (𝐽 ×t 𝐽) ∈ (TopOn‘(𝐵 × 𝐵))) | |
20 | 8, 8, 19 | syl2anc 696 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → (𝐽 ×t 𝐽) ∈ (TopOn‘(𝐵 × 𝐵))) |
21 | cnindis 21298 | . . . . 5 ⊢ (((𝐽 ×t 𝐽) ∈ (TopOn‘(𝐵 × 𝐵)) ∧ 𝐵 ∈ V) → ((𝐽 ×t 𝐽) Cn {∅, 𝐵}) = (𝐵 ↑𝑚 (𝐵 × 𝐵))) | |
22 | 20, 5, 21 | sylancl 697 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → ((𝐽 ×t 𝐽) Cn {∅, 𝐵}) = (𝐵 ↑𝑚 (𝐵 × 𝐵))) |
23 | 18, 22 | eqtrd 2794 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → ((𝐽 ×t 𝐽) Cn 𝐽) = (𝐵 ↑𝑚 (𝐵 × 𝐵))) |
24 | 17, 23 | eleqtrrd 2842 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → (-g‘𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) |
25 | 9, 12 | istgp2 22096 | . 2 ⊢ (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ (-g‘𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))) |
26 | 1, 11, 24, 25 | syl3anbrc 1429 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → 𝐺 ∈ TopGrp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1632 ∈ wcel 2139 Vcvv 3340 ∅c0 4058 {cpr 4323 × cxp 5264 ⟶wf 6045 ‘cfv 6049 (class class class)co 6813 ↑𝑚 cmap 8023 Basecbs 16059 TopOpenctopn 16284 Grpcgrp 17623 -gcsg 17625 TopOnctopon 20917 TopSpctps 20938 Cn ccn 21230 ×t ctx 21565 TopGrpctgp 22076 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-id 5174 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6774 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-1st 7333 df-2nd 7334 df-map 8025 df-0g 16304 df-topgen 16306 df-plusf 17442 df-mgm 17443 df-sgrp 17485 df-mnd 17496 df-grp 17626 df-minusg 17627 df-sbg 17628 df-top 20901 df-topon 20918 df-topsp 20939 df-bases 20952 df-cn 21233 df-cnp 21234 df-tx 21567 df-tmd 22077 df-tgp 22078 |
This theorem is referenced by: (None) |
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