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Theorem lmodrng 15913
Description: The scalar component of a left module is a ring. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
Hypothesis
Ref Expression
lmodrng.1  |-  F  =  (Scalar `  W )
Assertion
Ref Expression
lmodrng  |-  ( W  e.  LMod  ->  F  e. 
Ring )

Proof of Theorem lmodrng
Dummy variables  r 
q  w  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2404 . . 3  |-  ( Base `  W )  =  (
Base `  W )
2 eqid 2404 . . 3  |-  ( +g  `  W )  =  ( +g  `  W )
3 eqid 2404 . . 3  |-  ( .s
`  W )  =  ( .s `  W
)
4 lmodrng.1 . . 3  |-  F  =  (Scalar `  W )
5 eqid 2404 . . 3  |-  ( Base `  F )  =  (
Base `  F )
6 eqid 2404 . . 3  |-  ( +g  `  F )  =  ( +g  `  F )
7 eqid 2404 . . 3  |-  ( .r
`  F )  =  ( .r `  F
)
8 eqid 2404 . . 3  |-  ( 1r
`  F )  =  ( 1r `  F
)
91, 2, 3, 4, 5, 6, 7, 8islmod 15909 . 2  |-  ( W  e.  LMod  <->  ( W  e. 
Grp  /\  F  e.  Ring  /\  A. q  e.  (
Base `  F ) A. r  e.  ( Base `  F ) A. x  e.  ( Base `  W ) A. w  e.  ( Base `  W
) ( ( ( r ( .s `  W ) w )  e.  ( Base `  W
)  /\  ( r
( .s `  W
) ( w ( +g  `  W ) x ) )  =  ( ( r ( .s `  W ) w ) ( +g  `  W ) ( r ( .s `  W
) x ) )  /\  ( ( q ( +g  `  F
) r ) ( .s `  W ) w )  =  ( ( q ( .s
`  W ) w ) ( +g  `  W
) ( r ( .s `  W ) w ) ) )  /\  ( ( ( q ( .r `  F ) r ) ( .s `  W
) w )  =  ( q ( .s
`  W ) ( r ( .s `  W ) w ) )  /\  ( ( 1r `  F ) ( .s `  W
) w )  =  w ) ) ) )
109simp2bi 973 1  |-  ( W  e.  LMod  ->  F  e. 
Ring )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2666   ` cfv 5413  (class class class)co 6040   Basecbs 13424   +g cplusg 13484   .rcmulr 13485  Scalarcsca 13487   .scvsca 13488   Grpcgrp 14640   Ringcrg 15615   1rcur 15617   LModclmod 15905
This theorem is referenced by:  lmodfgrp  15914  lmodmcl  15917  lmod0cl  15931  lmod1cl  15932  lmod0vs  15938  lmodvs0  15939  lmodvsneg  15943  lmodsubvs  15955  lmodsubdi  15956  lmodsubdir  15957  lssvnegcl  15987  islss3  15990  pwslmod  16001  lmodvsinv  16067  islmhm2  16069  lbsind2  16108  lspsneq  16149  lspexch  16156  asclghm  16352  ip2subdi  16830  isphld  16840  ocvlss  16854  tlmtgp  18178  clmrng  19048  frlmup1  27118  frlmup2  27119  frlmup3  27120  frlmup4  27121  islindf5  27177  lmisfree  27180  lfl0  29548  lfladd  29549  lflsub  29550  lfl0f  29552  lfladdcl  29554  lfladdcom  29555  lfladdass  29556  lfladd0l  29557  lflnegcl  29558  lflnegl  29559  lflvscl  29560  lflvsdi1  29561  lflvsdi2  29562  lflvsass  29564  lfl0sc  29565  lflsc0N  29566  lfl1sc  29567  lkrlss  29578  eqlkr  29582  eqlkr3  29584  lkrlsp  29585  ldualvsass  29624  lduallmodlem  29635  ldualvsubcl  29639  ldualvsubval  29640  lkrin  29647  dochfl1  31959  lcfl7lem  31982  lclkrlem2m  32002  lclkrlem2o  32004  lclkrlem2p  32005  lcfrlem1  32025  lcfrlem2  32026  lcfrlem3  32027  lcfrlem29  32054  lcfrlem33  32058  lcdvsubval  32101  mapdpglem30  32185  baerlem3lem1  32190  baerlem5alem1  32191  baerlem5blem1  32192  baerlem5blem2  32195  hgmapval1  32379  hdmapinvlem3  32406  hdmapinvlem4  32407  hdmapglem5  32408  hgmapvvlem1  32409  hdmapglem7b  32414  hdmapglem7  32415
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 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-nul 4298
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 2258  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-iota 5377  df-fv 5421  df-ov 6043  df-lmod 15907
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