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Mirrors > Home > MPE Home > Th. List > ngpgrp | Structured version Visualization version GIF version |
Description: A normed group is a group. (Contributed by Mario Carneiro, 2-Oct-2015.) |
Ref | Expression |
---|---|
ngpgrp | ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ Grp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2778 | . . 3 ⊢ (norm‘𝐺) = (norm‘𝐺) | |
2 | eqid 2778 | . . 3 ⊢ (-g‘𝐺) = (-g‘𝐺) | |
3 | eqid 2778 | . . 3 ⊢ (dist‘𝐺) = (dist‘𝐺) | |
4 | 1, 2, 3 | isngp 22812 | . 2 ⊢ (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ ((norm‘𝐺) ∘ (-g‘𝐺)) ⊆ (dist‘𝐺))) |
5 | 4 | simp1bi 1136 | 1 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 ⊆ wss 3792 ∘ ccom 5361 ‘cfv 6137 distcds 16351 Grpcgrp 17813 -gcsg 17815 MetSpcms 22535 normcnm 22793 NrmGrpcngp 22794 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-rex 3096 df-rab 3099 df-v 3400 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-br 4889 df-opab 4951 df-co 5366 df-iota 6101 df-fv 6145 df-ngp 22800 |
This theorem is referenced by: ngpds 22820 ngpds2 22822 ngpds3 22824 ngprcan 22826 isngp4 22828 ngpinvds 22829 ngpsubcan 22830 nmf 22831 nmge0 22833 nmeq0 22834 nminv 22837 nmmtri 22838 nmsub 22839 nmrtri 22840 nm2dif 22841 nmtri 22842 nmtri2 22843 ngpi 22844 nm0 22845 ngptgp 22852 tngngp2 22868 tnggrpr 22871 nrmtngnrm 22874 nlmdsdi 22897 nlmdsdir 22898 nrginvrcnlem 22907 ngpocelbl 22920 nmo0 22951 nmotri 22955 0nghm 22957 nmoid 22958 idnghm 22959 nmods 22960 nmcn 23059 nmoleub2lem2 23327 nmhmcn 23331 cphipval2 23451 4cphipval2 23452 cphipval 23453 ipcnlem2 23454 nglmle 23512 qqhcn 30637 |
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