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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 2651 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | eqid 2651 | . . 3 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
3 | eqid 2651 | . . 3 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
4 | tgpsubcn.3 | . . 3 ⊢ − = (-g‘𝐺) | |
5 | 1, 2, 3, 4 | grpsubfval 17511 | . 2 ⊢ − = (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g‘𝐺)((invg‘𝐺)‘𝑦))) |
6 | tgpsubcn.2 | . . 3 ⊢ 𝐽 = (TopOpen‘𝐺) | |
7 | tgptmd 21930 | . . 3 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd) | |
8 | 6, 1 | tgptopon 21933 | . . 3 ⊢ (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘(Base‘𝐺))) |
9 | 8, 8 | cnmpt1st 21519 | . . 3 ⊢ (𝐺 ∈ TopGrp → (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ 𝑥) ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) |
10 | 8, 8 | cnmpt2nd 21520 | . . . 4 ⊢ (𝐺 ∈ TopGrp → (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ 𝑦) ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) |
11 | 6, 3 | tgpinv 21936 | . . . 4 ⊢ (𝐺 ∈ TopGrp → (invg‘𝐺) ∈ (𝐽 Cn 𝐽)) |
12 | 8, 8, 10, 11 | cnmpt21f 21523 | . . 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 21939 | . 2 ⊢ (𝐺 ∈ TopGrp → (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g‘𝐺)((invg‘𝐺)‘𝑦))) ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) |
14 | 5, 13 | syl5eqel 2734 | 1 ⊢ (𝐺 ∈ TopGrp → − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1523 ∈ wcel 2030 ‘cfv 5926 (class class class)co 6690 ↦ cmpt2 6692 Basecbs 15904 +gcplusg 15988 TopOpenctopn 16129 invgcminusg 17470 -gcsg 17471 Cn ccn 21076 ×t ctx 21411 TopGrpctgp 21922 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-1st 7210 df-2nd 7211 df-map 7901 df-topgen 16151 df-plusf 17288 df-sbg 17474 df-top 20747 df-topon 20764 df-topsp 20785 df-bases 20798 df-cn 21079 df-tx 21413 df-tmd 21923 df-tgp 21924 |
This theorem is referenced by: istgp2 21942 clssubg 21959 clsnsg 21960 tgphaus 21967 tgpt0 21969 qustgplem 21971 |
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