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Theorem nlmngp 21758
Description: A normed module is a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)
Assertion
Ref Expression
nlmngp  |-  ( W  e. NrmMod  ->  W  e. NrmGrp )

Proof of Theorem nlmngp
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2471 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
2 eqid 2471 . . . 4  |-  ( norm `  W )  =  (
norm `  W )
3 eqid 2471 . . . 4  |-  ( .s
`  W )  =  ( .s `  W
)
4 eqid 2471 . . . 4  |-  (Scalar `  W )  =  (Scalar `  W )
5 eqid 2471 . . . 4  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
6 eqid 2471 . . . 4  |-  ( norm `  (Scalar `  W )
)  =  ( norm `  (Scalar `  W )
)
71, 2, 3, 4, 5, 6isnlm 21756 . . 3  |-  ( W  e. NrmMod 
<->  ( ( W  e. NrmGrp  /\  W  e.  LMod  /\  (Scalar `  W )  e. NrmRing )  /\  A. x  e.  ( Base `  (Scalar `  W ) ) A. y  e.  ( Base `  W ) ( (
norm `  W ) `  ( x ( .s
`  W ) y ) )  =  ( ( ( norm `  (Scalar `  W ) ) `  x )  x.  (
( norm `  W ) `  y ) ) ) )
87simplbi 467 . 2  |-  ( W  e. NrmMod  ->  ( W  e. NrmGrp  /\  W  e.  LMod  /\  (Scalar `  W )  e. NrmRing ) )
98simp1d 1042 1  |-  ( W  e. NrmMod  ->  W  e. NrmGrp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 1007    = wceq 1452    e. wcel 1904   A.wral 2756   ` cfv 5589  (class class class)co 6308    x. cmul 9562   Basecbs 15199  Scalarcsca 15271   .scvsca 15272   LModclmod 18169   normcnm 21669  NrmGrpcngp 21670  NrmRingcnrg 21672  NrmModcnlm 21673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-nul 4527
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-iota 5553  df-fv 5597  df-ov 6311  df-nlm 21679
This theorem is referenced by:  nlmdsdi  21762  nlmdsdir  21763  nlmmul0or  21764  nlmvscnlem2  21766  nlmvscnlem1  21767  nlmvscn  21768  nlmtlm  21774  lssnlm  21781  isnmhm2  21851  idnmhm  21853  0nmhm  21854  nmoleub2lem  22206  nmoleub2lem3  22207  nmoleub2lem2  22208  nmoleub3  22211  nmhmcn  22212  cphngp  22229  ipcnlem2  22293  ipcnlem1  22294  csscld  22298  bnngp  22388
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