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Theorem rlmval 17708
Description: Value of the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.)
Assertion
Ref Expression
rlmval  |-  (ringLMod `  W
)  =  ( (subringAlg  `  W ) `  ( Base `  W ) )

Proof of Theorem rlmval
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 fveq2 5872 . . . 4  |-  ( a  =  W  ->  (subringAlg  `  a )  =  (subringAlg  `  W ) )
2 fveq2 5872 . . . 4  |-  ( a  =  W  ->  ( Base `  a )  =  ( Base `  W
) )
31, 2fveq12d 5878 . . 3  |-  ( a  =  W  ->  (
(subringAlg  `  a ) `  ( Base `  a )
)  =  ( (subringAlg  `  W ) `  ( Base `  W ) ) )
4 df-rgmod 17690 . . 3  |- ringLMod  =  ( a  e.  _V  |->  ( (subringAlg  `  a ) `  ( Base `  a )
) )
5 fvex 5882 . . 3  |-  ( (subringAlg  `  W ) `  ( Base `  W ) )  e.  _V
63, 4, 5fvmpt 5957 . 2  |-  ( W  e.  _V  ->  (ringLMod `  W )  =  ( (subringAlg  `  W ) `  ( Base `  W )
) )
7 0fv 5905 . . . 4  |-  ( (/) `  ( Base `  W
) )  =  (/)
87eqcomi 2480 . . 3  |-  (/)  =  (
(/) `  ( Base `  W ) )
9 fvprc 5866 . . 3  |-  ( -.  W  e.  _V  ->  (ringLMod `  W )  =  (/) )
10 fvprc 5866 . . . 4  |-  ( -.  W  e.  _V  ->  (subringAlg  `  W )  =  (/) )
1110fveq1d 5874 . . 3  |-  ( -.  W  e.  _V  ->  ( (subringAlg  `  W ) `  ( Base `  W )
)  =  ( (/) `  ( Base `  W
) ) )
128, 9, 113eqtr4a 2534 . 2  |-  ( -.  W  e.  _V  ->  (ringLMod `  W )  =  ( (subringAlg  `  W ) `  ( Base `  W )
) )
136, 12pm2.61i 164 1  |-  (ringLMod `  W
)  =  ( (subringAlg  `  W ) `  ( Base `  W ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1379    e. wcel 1767   _Vcvv 3118   (/)c0 3790   ` cfv 5594   Basecbs 14507  subringAlg csra 17685  ringLModcrglmod 17686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-iota 5557  df-fun 5596  df-fv 5602  df-rgmod 17690
This theorem is referenced by:  rlmval2  17711  rlmbas  17712  rlmplusg  17713  rlm0  17714  rlmmulr  17716  rlmsca  17717  rlmsca2  17718  rlmvsca  17719  rlmtopn  17720  rlmds  17721  rlmlmod  17722  rlmassa  17845  frlmip  18678  rlmnlm  21065  rlmbn  21669  rrxprds  21689
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