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Theorem nlmngp 18188
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 2283 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
2 eqid 2283 . . . 4  |-  ( norm `  W )  =  (
norm `  W )
3 eqid 2283 . . . 4  |-  ( .s
`  W )  =  ( .s `  W
)
4 eqid 2283 . . . 4  |-  (Scalar `  W )  =  (Scalar `  W )
5 eqid 2283 . . . 4  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
6 eqid 2283 . . . 4  |-  ( norm `  (Scalar `  W )
)  =  ( norm `  (Scalar `  W )
)
71, 2, 3, 4, 5, 6isnlm 18186 . . 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 446 . 2  |-  ( W  e. NrmMod  ->  ( W  e. NrmGrp  /\  W  e.  LMod  /\  (Scalar `  W )  e. NrmRing ) )
98simp1d 967 1  |-  ( W  e. NrmMod  ->  W  e. NrmGrp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   ` cfv 5255  (class class class)co 5858    x. cmul 8742   Basecbs 13148  Scalarcsca 13211   .scvsca 13212   LModclmod 15627   normcnm 18099  NrmGrpcngp 18100  NrmRingcnrg 18102  NrmModcnlm 18103
This theorem is referenced by:  nlmdsdi  18192  nlmdsdir  18193  nlmmul0or  18194  nlmvscnlem2  18196  nlmvscnlem1  18197  nlmvscn  18198  nlmtlm  18204  lssnlm  18211  isnmhm2  18261  idnmhm  18263  0nmhm  18264  nmoleub2lem  18595  nmoleub2lem3  18596  nmoleub2lem2  18597  nmoleub3  18600  nmhmcn  18601  cphngp  18609  ipcnlem2  18671  ipcnlem1  18672  csscld  18676  bnngp  18764
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-nul 4149
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861  df-nlm 18109
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