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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 2819 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | eqid 2819 | . . 3 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
3 | eqid 2819 | . . 3 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
4 | tgpsubcn.3 | . . 3 ⊢ − = (-g‘𝐺) | |
5 | 1, 2, 3, 4 | grpsubfval 18139 | . 2 ⊢ − = (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g‘𝐺)((invg‘𝐺)‘𝑦))) |
6 | tgpsubcn.2 | . . 3 ⊢ 𝐽 = (TopOpen‘𝐺) | |
7 | tgptmd 22679 | . . 3 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd) | |
8 | 6, 1 | tgptopon 22682 | . . 3 ⊢ (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘(Base‘𝐺))) |
9 | 8, 8 | cnmpt1st 22268 | . . 3 ⊢ (𝐺 ∈ TopGrp → (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ 𝑥) ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) |
10 | 8, 8 | cnmpt2nd 22269 | . . . 4 ⊢ (𝐺 ∈ TopGrp → (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ 𝑦) ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) |
11 | 6, 3 | tgpinv 22685 | . . . 4 ⊢ (𝐺 ∈ TopGrp → (invg‘𝐺) ∈ (𝐽 Cn 𝐽)) |
12 | 8, 8, 10, 11 | cnmpt21f 22272 | . . 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 22688 | . 2 ⊢ (𝐺 ∈ TopGrp → (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g‘𝐺)((invg‘𝐺)‘𝑦))) ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) |
14 | 5, 13 | eqeltrid 2915 | 1 ⊢ (𝐺 ∈ TopGrp → − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1531 ∈ wcel 2108 ‘cfv 6348 (class class class)co 7148 ∈ cmpo 7150 Basecbs 16475 +gcplusg 16557 TopOpenctopn 16687 invgcminusg 18096 -gcsg 18097 Cn ccn 21824 ×t ctx 22160 TopGrpctgp 22671 |
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 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-ral 3141 df-rex 3142 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fo 6354 df-fv 6356 df-ov 7151 df-oprab 7152 df-mpo 7153 df-1st 7681 df-2nd 7682 df-map 8400 df-topgen 16709 df-plusf 17843 df-sbg 18100 df-top 21494 df-topon 21511 df-topsp 21533 df-bases 21546 df-cn 21827 df-tx 22162 df-tmd 22672 df-tgp 22673 |
This theorem is referenced by: istgp2 22691 clssubg 22709 clsnsg 22710 tgphaus 22717 tgpt0 22719 qustgplem 22721 |
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