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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 2734 | . . 3 ⊢ (norm‘𝐺) = (norm‘𝐺) | |
2 | eqid 2734 | . . 3 ⊢ (-g‘𝐺) = (-g‘𝐺) | |
3 | eqid 2734 | . . 3 ⊢ (dist‘𝐺) = (dist‘𝐺) | |
4 | 1, 2, 3 | isngp 24624 | . 2 ⊢ (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ ((norm‘𝐺) ∘ (-g‘𝐺)) ⊆ (dist‘𝐺))) |
5 | 4 | simp1bi 1144 | 1 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 ⊆ wss 3962 ∘ ccom 5692 ‘cfv 6562 distcds 17306 Grpcgrp 18963 -gcsg 18965 MetSpcms 24343 normcnm 24604 NrmGrpcngp 24605 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-co 5697 df-iota 6515 df-fv 6570 df-ngp 24611 |
This theorem is referenced by: ngpds 24632 ngpds2 24634 ngpds3 24636 ngprcan 24638 isngp4 24640 ngpinvds 24641 ngpsubcan 24642 nmf 24643 nmge0 24645 nmeq0 24646 nminv 24649 nmmtri 24650 nmsub 24651 nmrtri 24652 nm2dif 24653 nmtri 24654 nmtri2 24655 ngpi 24656 nm0 24657 ngptgp 24664 tngngp2 24688 tnggrpr 24691 nrmtngnrm 24694 nlmdsdi 24717 nlmdsdir 24718 nrginvrcnlem 24727 ngpocelbl 24740 nmo0 24771 nmotri 24775 0nghm 24777 nmoid 24778 idnghm 24779 nmods 24780 nmcn 24879 nmoleub2lem2 25162 nmhmcn 25166 cphpyth 25263 cphipval2 25288 4cphipval2 25289 cphipval 25290 ipcnlem2 25291 nglmle 25349 qqhcn 33953 |
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