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Theorem nlmnrg 18707
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 2435 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
2 eqid 2435 . . . 4  |-  ( norm `  W )  =  (
norm `  W )
3 eqid 2435 . . . 4  |-  ( .s
`  W )  =  ( .s `  W
)
4 nlmnrg.1 . . . 4  |-  F  =  (Scalar `  W )
5 eqid 2435 . . . 4  |-  ( Base `  F )  =  (
Base `  F )
6 eqid 2435 . . . 4  |-  ( norm `  F )  =  (
norm `  F )
71, 2, 3, 4, 5, 6isnlm 18703 . . 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 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:  nlmngp2  18708  nlmtlm  18721  nvctvc  18727  lssnlm  18728  sitgclbn  24649
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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