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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 2726 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | eqid 2726 | . . 3 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
3 | eqid 2726 | . . 3 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
4 | tgpsubcn.3 | . . 3 ⊢ − = (-g‘𝐺) | |
5 | 1, 2, 3, 4 | grpsubfval 18973 | . 2 ⊢ − = (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g‘𝐺)((invg‘𝐺)‘𝑦))) |
6 | tgpsubcn.2 | . . 3 ⊢ 𝐽 = (TopOpen‘𝐺) | |
7 | tgptmd 24071 | . . 3 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd) | |
8 | 6, 1 | tgptopon 24074 | . . 3 ⊢ (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘(Base‘𝐺))) |
9 | 8, 8 | cnmpt1st 23660 | . . 3 ⊢ (𝐺 ∈ TopGrp → (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ 𝑥) ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) |
10 | 8, 8 | cnmpt2nd 23661 | . . . 4 ⊢ (𝐺 ∈ TopGrp → (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ 𝑦) ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) |
11 | 6, 3 | tgpinv 24077 | . . . 4 ⊢ (𝐺 ∈ TopGrp → (invg‘𝐺) ∈ (𝐽 Cn 𝐽)) |
12 | 8, 8, 10, 11 | cnmpt21f 23664 | . . 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 24080 | . 2 ⊢ (𝐺 ∈ TopGrp → (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g‘𝐺)((invg‘𝐺)‘𝑦))) ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) |
14 | 5, 13 | eqeltrid 2830 | 1 ⊢ (𝐺 ∈ TopGrp → − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ‘cfv 6546 (class class class)co 7416 ∈ cmpo 7418 Basecbs 17208 +gcplusg 17261 TopOpenctopn 17431 invgcminusg 18924 -gcsg 18925 Cn ccn 23216 ×t ctx 23552 TopGrpctgp 24063 |
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 2697 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-iun 4995 df-br 5146 df-opab 5208 df-mpt 5229 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-fo 6552 df-fv 6554 df-ov 7419 df-oprab 7420 df-mpo 7421 df-1st 7995 df-2nd 7996 df-map 8849 df-topgen 17453 df-plusf 18627 df-sbg 18928 df-top 22884 df-topon 22901 df-topsp 22923 df-bases 22937 df-cn 23219 df-tx 23554 df-tmd 24064 df-tgp 24065 |
This theorem is referenced by: istgp2 24083 clssubg 24101 clsnsg 24102 tgphaus 24109 tgpt0 24111 qustgplem 24113 |
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