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Mirrors > Home > MPE Home > Th. List > tgpsubcn | Structured version Visualization version GIF version |
Description: In a topological group, the "subtraction" (or "division") is continuous. Axiom GT' of [BourbakiTop1] p. III.1. (Contributed by FL, 21-Jun-2010.) (Revised by Mario Carneiro, 19-Mar-2015.) |
Ref | Expression |
---|---|
tgpsubcn.2 | ⊢ 𝐽 = (TopOpen‘𝐺) |
tgpsubcn.3 | ⊢ − = (-g‘𝐺) |
Ref | Expression |
---|---|
tgpsubcn | ⊢ (𝐺 ∈ TopGrp → − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2821 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | eqid 2821 | . . 3 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
3 | eqid 2821 | . . 3 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
4 | tgpsubcn.3 | . . 3 ⊢ − = (-g‘𝐺) | |
5 | 1, 2, 3, 4 | grpsubfval 18087 | . 2 ⊢ − = (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g‘𝐺)((invg‘𝐺)‘𝑦))) |
6 | tgpsubcn.2 | . . 3 ⊢ 𝐽 = (TopOpen‘𝐺) | |
7 | tgptmd 22617 | . . 3 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd) | |
8 | 6, 1 | tgptopon 22620 | . . 3 ⊢ (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘(Base‘𝐺))) |
9 | 8, 8 | cnmpt1st 22206 | . . 3 ⊢ (𝐺 ∈ TopGrp → (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ 𝑥) ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) |
10 | 8, 8 | cnmpt2nd 22207 | . . . 4 ⊢ (𝐺 ∈ TopGrp → (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ 𝑦) ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) |
11 | 6, 3 | tgpinv 22623 | . . . 4 ⊢ (𝐺 ∈ TopGrp → (invg‘𝐺) ∈ (𝐽 Cn 𝐽)) |
12 | 8, 8, 10, 11 | cnmpt21f 22210 | . . 3 ⊢ (𝐺 ∈ TopGrp → (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ ((invg‘𝐺)‘𝑦)) ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) |
13 | 6, 2, 7, 8, 8, 9, 12 | cnmpt2plusg 22626 | . 2 ⊢ (𝐺 ∈ TopGrp → (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g‘𝐺)((invg‘𝐺)‘𝑦))) ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) |
14 | 5, 13 | eqeltrid 2917 | 1 ⊢ (𝐺 ∈ TopGrp → − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ‘cfv 6349 (class class class)co 7145 ∈ cmpo 7147 Basecbs 16473 +gcplusg 16555 TopOpenctopn 16685 invgcminusg 18044 -gcsg 18045 Cn ccn 21762 ×t ctx 22098 TopGrpctgp 22609 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4833 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-fo 6355 df-fv 6357 df-ov 7148 df-oprab 7149 df-mpo 7150 df-1st 7680 df-2nd 7681 df-map 8398 df-topgen 16707 df-plusf 17841 df-sbg 18048 df-top 21432 df-topon 21449 df-topsp 21471 df-bases 21484 df-cn 21765 df-tx 22100 df-tmd 22610 df-tgp 22611 |
This theorem is referenced by: istgp2 22629 clssubg 22646 clsnsg 22647 tgphaus 22654 tgpt0 22656 qustgplem 22658 |
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