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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 2610 | . . 3 ⊢ (norm‘𝐺) = (norm‘𝐺) | |
2 | eqid 2610 | . . 3 ⊢ (-g‘𝐺) = (-g‘𝐺) | |
3 | eqid 2610 | . . 3 ⊢ (dist‘𝐺) = (dist‘𝐺) | |
4 | 1, 2, 3 | isngp 22210 | . 2 ⊢ (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ ((norm‘𝐺) ∘ (-g‘𝐺)) ⊆ (dist‘𝐺))) |
5 | 4 | simp1bi 1069 | 1 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 ⊆ wss 3540 ∘ ccom 5042 ‘cfv 5804 distcds 15777 Grpcgrp 17245 -gcsg 17247 MetSpcmt 21933 normcnm 22191 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-co 5047 df-iota 5768 df-fv 5812 df-ngp 22198 |
This theorem is referenced by: ngpds 22218 ngpds2 22220 ngpds3 22222 ngprcan 22224 isngp4 22226 ngpinvds 22227 ngpsubcan 22228 nmf 22229 nmge0 22231 nmeq0 22232 nminv 22235 nmmtri 22236 nmsub 22237 nmrtri 22238 nm2dif 22239 nmtri 22240 nmtri2 22241 ngpi 22242 nm0 22243 ngptgp 22250 tngngp2 22266 tnggrpr 22269 nrmtngnrm 22272 nlmdsdi 22295 nlmdsdir 22296 nrginvrcnlem 22305 ngpocelbl 22318 nmo0 22349 nmotri 22353 0nghm 22355 nmoid 22356 idnghm 22357 nmods 22358 nmcn 22455 nmoleub2lem2 22724 nmhmcn 22728 cphipval2 22848 4cphipval2 22849 cphipval 22850 ipcnlem2 22851 nglmle 22908 qqhcn 29363 |
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