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Mirrors > Home > MPE Home > Th. List > ngpxms | Structured version Visualization version GIF version |
Description: A normed group is a metric space. (Contributed by Mario Carneiro, 2-Oct-2015.) |
Ref | Expression |
---|---|
ngpxms | ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ ∞MetSp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ngpms 22214 | . 2 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ MetSp) | |
2 | msxms 22069 | . 2 ⊢ (𝐺 ∈ MetSp → 𝐺 ∈ ∞MetSp) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ ∞MetSp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 ∞MetSpcxme 21932 MetSpcmt 21933 NrmGrpcngp 22192 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-xp 5044 df-co 5047 df-res 5050 df-iota 5768 df-fv 5812 df-ms 21936 df-ngp 22198 |
This theorem is referenced by: ngpdsr 22219 ngpds2r 22221 ngpds3 22222 ngpds3r 22223 nmge0 22231 nmeq0 22232 minveclem4a 23009 minveclem4 23011 qqhcn 29363 qqhucn 29364 rrhcn 29369 rrhf 29370 rrexttps 29378 rrexthaus 29379 |
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