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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 2622 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | eqid 2622 | . . 3 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
3 | eqid 2622 | . . 3 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
4 | tgpsubcn.3 | . . 3 ⊢ − = (-g‘𝐺) | |
5 | 1, 2, 3, 4 | grpsubfval 17464 | . 2 ⊢ − = (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g‘𝐺)((invg‘𝐺)‘𝑦))) |
6 | tgpsubcn.2 | . . 3 ⊢ 𝐽 = (TopOpen‘𝐺) | |
7 | tgptmd 21883 | . . 3 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd) | |
8 | 6, 1 | tgptopon 21886 | . . 3 ⊢ (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘(Base‘𝐺))) |
9 | 8, 8 | cnmpt1st 21471 | . . 3 ⊢ (𝐺 ∈ TopGrp → (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ 𝑥) ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) |
10 | 8, 8 | cnmpt2nd 21472 | . . . 4 ⊢ (𝐺 ∈ TopGrp → (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ 𝑦) ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) |
11 | 6, 3 | tgpinv 21889 | . . . 4 ⊢ (𝐺 ∈ TopGrp → (invg‘𝐺) ∈ (𝐽 Cn 𝐽)) |
12 | 8, 8, 10, 11 | cnmpt21f 21475 | . . 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 21892 | . 2 ⊢ (𝐺 ∈ TopGrp → (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g‘𝐺)((invg‘𝐺)‘𝑦))) ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) |
14 | 5, 13 | syl5eqel 2705 | 1 ⊢ (𝐺 ∈ TopGrp → − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 Basecbs 15857 +gcplusg 15941 TopOpenctopn 16082 invgcminusg 17423 -gcsg 17424 Cn ccn 21028 ×t ctx 21363 TopGrpctgp 21875 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-map 7859 df-topgen 16104 df-plusf 17241 df-sbg 17427 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cn 21031 df-tx 21365 df-tmd 21876 df-tgp 21877 |
This theorem is referenced by: istgp2 21895 clssubg 21912 clsnsg 21913 tgphaus 21920 tgpt0 21922 qustgplem 21924 |
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