Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > distgp | Structured version Visualization version GIF version |
Description: Any group equipped with the discrete 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 |
---|---|
distgp | ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → 𝐺 ∈ TopGrp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 485 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → 𝐺 ∈ Grp) | |
2 | simpr 487 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → 𝐽 = 𝒫 𝐵) | |
3 | distgp.1 | . . . . . 6 ⊢ 𝐵 = (Base‘𝐺) | |
4 | 3 | fvexi 6684 | . . . . 5 ⊢ 𝐵 ∈ V |
5 | distopon 21605 | . . . . 5 ⊢ (𝐵 ∈ V → 𝒫 𝐵 ∈ (TopOn‘𝐵)) | |
6 | 4, 5 | ax-mp 5 | . . . 4 ⊢ 𝒫 𝐵 ∈ (TopOn‘𝐵) |
7 | 2, 6 | eqeltrdi 2921 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → 𝐽 ∈ (TopOn‘𝐵)) |
8 | distgp.2 | . . . 4 ⊢ 𝐽 = (TopOpen‘𝐺) | |
9 | 3, 8 | istps 21542 | . . 3 ⊢ (𝐺 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝐵)) |
10 | 7, 9 | sylibr 236 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → 𝐺 ∈ TopSp) |
11 | eqid 2821 | . . . . . 6 ⊢ (-g‘𝐺) = (-g‘𝐺) | |
12 | 3, 11 | grpsubf 18178 | . . . . 5 ⊢ (𝐺 ∈ Grp → (-g‘𝐺):(𝐵 × 𝐵)⟶𝐵) |
13 | 12 | adantr 483 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → (-g‘𝐺):(𝐵 × 𝐵)⟶𝐵) |
14 | 4, 4 | xpex 7476 | . . . . 5 ⊢ (𝐵 × 𝐵) ∈ V |
15 | 4, 14 | elmap 8435 | . . . 4 ⊢ ((-g‘𝐺) ∈ (𝐵 ↑m (𝐵 × 𝐵)) ↔ (-g‘𝐺):(𝐵 × 𝐵)⟶𝐵) |
16 | 13, 15 | sylibr 236 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → (-g‘𝐺) ∈ (𝐵 ↑m (𝐵 × 𝐵))) |
17 | 2, 2 | oveq12d 7174 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → (𝐽 ×t 𝐽) = (𝒫 𝐵 ×t 𝒫 𝐵)) |
18 | txdis 22240 | . . . . . . 7 ⊢ ((𝐵 ∈ V ∧ 𝐵 ∈ V) → (𝒫 𝐵 ×t 𝒫 𝐵) = 𝒫 (𝐵 × 𝐵)) | |
19 | 4, 4, 18 | mp2an 690 | . . . . . 6 ⊢ (𝒫 𝐵 ×t 𝒫 𝐵) = 𝒫 (𝐵 × 𝐵) |
20 | 17, 19 | syl6eq 2872 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → (𝐽 ×t 𝐽) = 𝒫 (𝐵 × 𝐵)) |
21 | 20 | oveq1d 7171 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → ((𝐽 ×t 𝐽) Cn 𝐽) = (𝒫 (𝐵 × 𝐵) Cn 𝐽)) |
22 | cndis 21899 | . . . . 5 ⊢ (((𝐵 × 𝐵) ∈ V ∧ 𝐽 ∈ (TopOn‘𝐵)) → (𝒫 (𝐵 × 𝐵) Cn 𝐽) = (𝐵 ↑m (𝐵 × 𝐵))) | |
23 | 14, 7, 22 | sylancr 589 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → (𝒫 (𝐵 × 𝐵) Cn 𝐽) = (𝐵 ↑m (𝐵 × 𝐵))) |
24 | 21, 23 | eqtrd 2856 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → ((𝐽 ×t 𝐽) Cn 𝐽) = (𝐵 ↑m (𝐵 × 𝐵))) |
25 | 16, 24 | eleqtrrd 2916 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → (-g‘𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) |
26 | 8, 11 | istgp2 22699 | . 2 ⊢ (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ (-g‘𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))) |
27 | 1, 10, 25, 26 | syl3anbrc 1339 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → 𝐺 ∈ TopGrp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3494 𝒫 cpw 4539 × cxp 5553 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 ↑m cmap 8406 Basecbs 16483 TopOpenctopn 16695 Grpcgrp 18103 -gcsg 18105 TopOnctopon 21518 TopSpctps 21540 Cn ccn 21832 ×t ctx 22168 TopGrpctgp 22679 |
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 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-map 8408 df-0g 16715 df-topgen 16717 df-plusf 17851 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-minusg 18107 df-sbg 18108 df-top 21502 df-topon 21519 df-topsp 21541 df-bases 21554 df-cn 21835 df-cnp 21836 df-tx 22170 df-tmd 22680 df-tgp 22681 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |