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| Mirrors > Home > MPE Home > Th. List > ngpms | Structured version Visualization version GIF version | ||
| Description: A normed group is a metric space. (Contributed by Mario Carneiro, 2-Oct-2015.) |
| Ref | Expression |
|---|---|
| ngpms | ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ MetSp) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqid 2737 | . . 3 ⊢ (norm‘𝐺) = (norm‘𝐺) | |
| 2 | eqid 2737 | . . 3 ⊢ (-g‘𝐺) = (-g‘𝐺) | |
| 3 | eqid 2737 | . . 3 ⊢ (dist‘𝐺) = (dist‘𝐺) | |
| 4 | 1, 2, 3 | isngp 24609 | . 2 ⊢ (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ ((norm‘𝐺) ∘ (-g‘𝐺)) ⊆ (dist‘𝐺))) |
| 5 | 4 | simp2bi 1147 | 1 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ MetSp) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 ⊆ wss 3951 ∘ ccom 5689 ‘cfv 6561 distcds 17306 Grpcgrp 18951 -gcsg 18953 MetSpcms 24328 normcnm 24589 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-co 5694 df-iota 6514 df-fv 6569 df-ngp 24596 |
| This theorem is referenced by: ngpxms 24614 ngptps 24615 ngpmet 24616 isngp4 24625 nmmtri 24635 nmrtri 24637 subgngp 24648 ngptgp 24649 tngngp2 24673 nlmvscnlem2 24706 nlmvscnlem1 24707 nlmvscn 24708 nrginvrcn 24713 nghmcn 24766 nmcn 24866 nmhmcn 25153 ipcnlem2 25278 ipcnlem1 25279 ipcn 25280 nglmle 25336 cssbn 25409 minveclem2 25460 minveclem3b 25462 minveclem3 25463 minveclem4 25466 minveclem7 25469 |
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