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Mirrors > Home > MPE Home > Th. List > ngptps | Structured version Visualization version GIF version |
Description: A normed group is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.) |
Ref | Expression |
---|---|
ngptps | ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ TopSp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ngpms 24600 | . 2 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ MetSp) | |
2 | mstps 24452 | . 2 ⊢ (𝐺 ∈ MetSp → 𝐺 ∈ TopSp) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ TopSp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2099 TopSpctps 22925 MetSpcms 24315 NrmGrpcngp 24577 |
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-ext 2697 |
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-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-rab 3420 df-v 3464 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-opab 5216 df-xp 5688 df-co 5691 df-res 5694 df-iota 6506 df-fv 6562 df-xms 24317 df-ms 24318 df-ngp 24583 |
This theorem is referenced by: nmcn 24851 cnmpt1ip 25266 cnmpt2ip 25267 csscld 25268 clsocv 25269 rrxtps 45907 |
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