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Theorem nlmnrg 18190
Description: The scalar component of a left module is a normed ring. (Contributed by Mario Carneiro, 4-Oct-2015.)
Hypothesis
Ref Expression
nlmnrg.1  |-  F  =  (Scalar `  W )
Assertion
Ref Expression
nlmnrg  |-  ( W  e. NrmMod  ->  F  e. NrmRing )

Proof of Theorem nlmnrg
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 nlmnrg.1 . . . 4  |-  F  =  (Scalar `  W )
5 eqid 2283 . . . 4  |-  ( Base `  F )  =  (
Base `  F )
6 eqid 2283 . . . 4  |-  ( norm `  F )  =  (
norm `  F )
71, 2, 3, 4, 5, 6isnlm 18186 . . 3  |-  ( W  e. NrmMod 
<->  ( ( W  e. NrmGrp  /\  W  e.  LMod  /\  F  e. NrmRing )  /\  A. x  e.  ( Base `  F ) A. y  e.  ( Base `  W
) ( ( norm `  W ) `  (
x ( .s `  W ) y ) )  =  ( ( ( norm `  F
) `  x )  x.  ( ( norm `  W
) `  y )
) ) )
87simplbi 446 . 2  |-  ( W  e. NrmMod  ->  ( W  e. NrmGrp  /\  W  e.  LMod  /\  F  e. NrmRing ) )
98simp3d 969 1  |-  ( W  e. NrmMod  ->  F  e. NrmRing )
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:  nlmngp2  18191  nlmtlm  18204  nvctvc  18210  lssnlm  18211
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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