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Theorem nlmnrg 18206
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 2296 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
2 eqid 2296 . . . 4  |-  ( norm `  W )  =  (
norm `  W )
3 eqid 2296 . . . 4  |-  ( .s
`  W )  =  ( .s `  W
)
4 nlmnrg.1 . . . 4  |-  F  =  (Scalar `  W )
5 eqid 2296 . . . 4  |-  ( Base `  F )  =  (
Base `  F )
6 eqid 2296 . . . 4  |-  ( norm `  F )  =  (
norm `  F )
71, 2, 3, 4, 5, 6isnlm 18202 . . 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 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:  nlmngp2  18207  nlmtlm  18220  nvctvc  18226  lssnlm  18227
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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