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Theorem clmrng 19087
Description: The scalar ring of a complex module is a ring. (Contributed by Mario Carneiro, 16-Oct-2015.)
Hypothesis
Ref Expression
clm0.f  |-  F  =  (Scalar `  W )
Assertion
Ref Expression
clmrng  |-  ( W  e. CMod  ->  F  e.  Ring )

Proof of Theorem clmrng
StepHypRef Expression
1 clmlmod 19084 . 2  |-  ( W  e. CMod  ->  W  e.  LMod )
2 clm0.f . . 3  |-  F  =  (Scalar `  W )
32lmodrng 15950 . 2  |-  ( W  e.  LMod  ->  F  e. 
Ring )
41, 3syl 16 1  |-  ( W  e. CMod  ->  F  e.  Ring )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   ` cfv 5446  Scalarcsca 13524   Ringcrg 15652   LModclmod 15942  CModcclm 19079
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-lmod 15944  df-clm 19080
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