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Theorem nlmngp 18713
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 2436 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
2 eqid 2436 . . . 4  |-  ( norm `  W )  =  (
norm `  W )
3 eqid 2436 . . . 4  |-  ( .s
`  W )  =  ( .s `  W
)
4 eqid 2436 . . . 4  |-  (Scalar `  W )  =  (Scalar `  W )
5 eqid 2436 . . . 4  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
6 eqid 2436 . . . 4  |-  ( norm `  (Scalar `  W )
)  =  ( norm `  (Scalar `  W )
)
71, 2, 3, 4, 5, 6isnlm 18711 . . 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 447 . 2  |-  ( W  e. NrmMod  ->  ( W  e. NrmGrp  /\  W  e.  LMod  /\  (Scalar `  W )  e. NrmRing ) )
98simp1d 969 1  |-  ( W  e. NrmMod  ->  W  e. NrmGrp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2705   ` cfv 5454  (class class class)co 6081    x. cmul 8995   Basecbs 13469  Scalarcsca 13532   .scvsca 13533   LModclmod 15950   normcnm 18624  NrmGrpcngp 18625  NrmRingcnrg 18627  NrmModcnlm 18628
This theorem is referenced by:  nlmdsdi  18717  nlmdsdir  18718  nlmmul0or  18719  nlmvscnlem2  18721  nlmvscnlem1  18722  nlmvscn  18723  nlmtlm  18729  lssnlm  18736  isnmhm2  18786  idnmhm  18788  0nmhm  18789  nmoleub2lem  19122  nmoleub2lem3  19123  nmoleub2lem2  19124  nmoleub3  19127  nmhmcn  19128  cphngp  19136  ipcnlem2  19198  ipcnlem1  19199  csscld  19203  bnngp  19295
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-nul 4338
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-iota 5418  df-fv 5462  df-ov 6084  df-nlm 18634
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