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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 22775 | . 2 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ MetSp) | |
2 | msxms 22630 | . 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 2166 ∞MetSpcxms 22493 MetSpcms 22494 NrmGrpcngp 22753 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-rex 3124 df-rab 3127 df-v 3417 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4660 df-br 4875 df-opab 4937 df-xp 5349 df-co 5352 df-res 5355 df-iota 6087 df-fv 6132 df-ms 22497 df-ngp 22759 |
This theorem is referenced by: ngpdsr 22780 ngpds2r 22782 ngpds3 22783 ngpds3r 22784 nmge0 22792 nmeq0 22793 minveclem4a 23599 minveclem4 23601 qqhcn 30581 qqhucn 30582 rrhcn 30587 rrhf 30588 rrexttps 30596 rrexthaus 30597 |
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