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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 24613 | . 2 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ MetSp) | |
| 2 | mstps 24465 | . 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 2108 TopSpctps 22938 MetSpcms 24328 NrmGrpcngp 24590 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-xp 5691 df-co 5694 df-res 5697 df-iota 6514 df-fv 6569 df-xms 24330 df-ms 24331 df-ngp 24596 | 
| This theorem is referenced by: nmcn 24866 cnmpt1ip 25281 cnmpt2ip 25282 csscld 25283 clsocv 25284 rrxtps 46301 | 
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