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Theorem clmrng 18960
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 18957 . 2  |-  ( W  e. CMod  ->  W  e.  LMod )
2 clm0.f . . 3  |-  F  =  (Scalar `  W )
32lmodrng 15879 . 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 1649    e. wcel 1717   ` cfv 5388  Scalarcsca 13453   Ringcrg 15581   LModclmod 15871  CModcclm 18952
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 2362  ax-nul 4273
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 2236  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2506  df-ne 2546  df-ral 2648  df-rex 2649  df-rab 2652  df-v 2895  df-sbc 3099  df-dif 3260  df-un 3262  df-in 3264  df-ss 3271  df-nul 3566  df-if 3677  df-sn 3757  df-pr 3758  df-op 3760  df-uni 3952  df-br 4148  df-iota 5352  df-fv 5396  df-ov 6017  df-lmod 15873  df-clm 18953
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