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Theorem nlmlmod 18205
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 2296 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
2 eqid 2296 . . . 4  |-  ( norm `  W )  =  (
norm `  W )
3 eqid 2296 . . . 4  |-  ( .s
`  W )  =  ( .s `  W
)
4 eqid 2296 . . . 4  |-  (Scalar `  W )  =  (Scalar `  W )
5 eqid 2296 . . . 4  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
6 eqid 2296 . . . 4  |-  ( norm `  (Scalar `  W )
)  =  ( norm `  (Scalar `  W )
)
71, 2, 3, 4, 5, 6isnlm 18202 . . 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 ) )
98simp2d 968 1  |-  ( W  e. NrmMod  ->  W  e.  LMod )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   ` cfv 5271  (class class class)co 5874    x. cmul 8758   Basecbs 13164  Scalarcsca 13227   .scvsca 13228   LModclmod 15643   normcnm 18115  NrmGrpcngp 18116  NrmRingcnrg 18118  NrmModcnlm 18119
This theorem is referenced by:  nlmdsdi  18208  nlmdsdir  18209  nlmmul0or  18210  nlmvscnlem2  18212  nlmvscn  18214  nlmtlm  18220  nvclmod  18224  isnvc2  18225  lssnlm  18227  idnmhm  18279  0nmhm  18280  nmhmplusg  18282  nmoleub2lem3  18612  nmoleub3  18616  nmhmcn  18617  cphlmod  18626  bnlmod  18781
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-nul 4165
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-ov 5877  df-nlm 18125
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