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Theorem nlmlmod 18706
Description: A normed module is a left module. (Contributed by Mario Carneiro, 4-Oct-2015.)
Assertion
Ref Expression
nlmlmod  |-  ( W  e. NrmMod  ->  W  e.  LMod )

Proof of Theorem nlmlmod
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2435 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
2 eqid 2435 . . . 4  |-  ( norm `  W )  =  (
norm `  W )
3 eqid 2435 . . . 4  |-  ( .s
`  W )  =  ( .s `  W
)
4 eqid 2435 . . . 4  |-  (Scalar `  W )  =  (Scalar `  W )
5 eqid 2435 . . . 4  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
6 eqid 2435 . . . 4  |-  ( norm `  (Scalar `  W )
)  =  ( norm `  (Scalar `  W )
)
71, 2, 3, 4, 5, 6isnlm 18703 . . 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 ) )
98simp2d 970 1  |-  ( W  e. NrmMod  ->  W  e.  LMod )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2697   ` cfv 5446  (class class class)co 6073    x. cmul 8987   Basecbs 13461  Scalarcsca 13524   .scvsca 13525   LModclmod 15942   normcnm 18616  NrmGrpcngp 18617  NrmRingcnrg 18619  NrmModcnlm 18620
This theorem is referenced by:  nlmdsdi  18709  nlmdsdir  18710  nlmmul0or  18711  nlmvscnlem2  18713  nlmvscn  18715  nlmtlm  18721  nvclmod  18725  isnvc2  18726  lssnlm  18728  idnmhm  18780  0nmhm  18781  nmhmplusg  18783  nmoleub2lem3  19115  nmoleub3  19119  nmhmcn  19120  cphlmod  19129  bnlmod  19288  nmmulg  24344
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 2416  ax-nul 4330
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 2284  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454  df-ov 6076  df-nlm 18626
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