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Theorem nlmnrg 18587
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 2388 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
2 eqid 2388 . . . 4  |-  ( norm `  W )  =  (
norm `  W )
3 eqid 2388 . . . 4  |-  ( .s
`  W )  =  ( .s `  W
)
4 nlmnrg.1 . . . 4  |-  F  =  (Scalar `  W )
5 eqid 2388 . . . 4  |-  ( Base `  F )  =  (
Base `  F )
6 eqid 2388 . . . 4  |-  ( norm `  F )  =  (
norm `  F )
71, 2, 3, 4, 5, 6isnlm 18583 . . 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 447 . 2  |-  ( W  e. NrmMod  ->  ( W  e. NrmGrp  /\  W  e.  LMod  /\  F  e. NrmRing ) )
98simp3d 971 1  |-  ( W  e. NrmMod  ->  F  e. NrmRing )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1649    e. wcel 1717   A.wral 2650   ` cfv 5395  (class class class)co 6021    x. cmul 8929   Basecbs 13397  Scalarcsca 13460   .scvsca 13461   LModclmod 15878   normcnm 18496  NrmGrpcngp 18497  NrmRingcnrg 18499  NrmModcnlm 18500
This theorem is referenced by:  nlmngp2  18588  nlmtlm  18601  nvctvc  18607  lssnlm  18608
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-nul 4280
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-iota 5359  df-fv 5403  df-ov 6024  df-nlm 18506
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