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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 2622 | . . 3 ⊢ (norm‘𝐺) = (norm‘𝐺) | |
2 | eqid 2622 | . . 3 ⊢ (-g‘𝐺) = (-g‘𝐺) | |
3 | eqid 2622 | . . 3 ⊢ (dist‘𝐺) = (dist‘𝐺) | |
4 | 1, 2, 3 | isngp 22400 | . 2 ⊢ (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ ((norm‘𝐺) ∘ (-g‘𝐺)) ⊆ (dist‘𝐺))) |
5 | 4 | simp2bi 1077 | 1 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ MetSp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1990 ⊆ wss 3574 ∘ ccom 5118 ‘cfv 5888 distcds 15950 Grpcgrp 17422 -gcsg 17424 MetSpcmt 22123 normcnm 22381 NrmGrpcngp 22382 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-co 5123 df-iota 5851 df-fv 5896 df-ngp 22388 |
This theorem is referenced by: ngpxms 22405 ngptps 22406 ngpmet 22407 isngp4 22416 nmf 22419 nmmtri 22426 nmrtri 22428 subgngp 22439 ngptgp 22440 tngngp2 22456 nlmvscnlem2 22489 nlmvscnlem1 22490 nlmvscn 22491 nrginvrcn 22496 nghmcn 22549 nmcn 22647 nmhmcn 22920 ipcnlem2 23043 ipcnlem1 23044 ipcn 23045 nglmle 23100 minveclem2 23197 minveclem3b 23199 minveclem3 23200 minveclem4 23203 minveclem7 23206 |
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