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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 22812 | . 2 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ MetSp) | |
2 | mstps 22668 | . 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 2106 TopSpctps 21144 MetSpcms 22531 NrmGrpcngp 22790 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-rex 3095 df-rab 3098 df-v 3399 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-br 4887 df-opab 4949 df-xp 5361 df-co 5364 df-res 5367 df-iota 6099 df-fv 6143 df-xms 22533 df-ms 22534 df-ngp 22796 |
This theorem is referenced by: nmcn 23055 cnmpt1ip 23453 cnmpt2ip 23454 csscld 23455 clsocv 23456 rrxtps 41423 |
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