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Theorem rlmval 16266
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 5730 . . . 4  |-  ( a  =  W  ->  ( subringAlg  `  a )  =  ( subringAlg  `  W ) )
2 fveq2 5730 . . . 4  |-  ( a  =  W  ->  ( Base `  a )  =  ( Base `  W
) )
31, 2fveq12d 5736 . . 3  |-  ( a  =  W  ->  (
( subringAlg  `  a ) `  ( Base `  a )
)  =  ( ( subringAlg  `  W ) `  ( Base `  W ) ) )
4 df-rgmod 16247 . . 3  |- ringLMod  =  ( a  e.  _V  |->  ( ( subringAlg  `  a ) `  ( Base `  a )
) )
5 fvex 5744 . . 3  |-  ( ( subringAlg  `  W ) `  ( Base `  W ) )  e.  _V
63, 4, 5fvmpt 5808 . 2  |-  ( W  e.  _V  ->  (ringLMod `  W )  =  ( ( subringAlg  `  W ) `  ( Base `  W )
) )
7 fv01 5765 . . . 4  |-  ( (/) `  ( Base `  W
) )  =  (/)
87eqcomi 2442 . . 3  |-  (/)  =  (
(/) `  ( Base `  W ) )
9 fvprc 5724 . . 3  |-  ( -.  W  e.  _V  ->  (ringLMod `  W )  =  (/) )
10 fvprc 5724 . . . 4  |-  ( -.  W  e.  _V  ->  ( subringAlg  `  W )  =  (/) )
1110fveq1d 5732 . . 3  |-  ( -.  W  e.  _V  ->  ( ( subringAlg  `  W ) `  ( Base `  W )
)  =  ( (/) `  ( Base `  W
) ) )
128, 9, 113eqtr4a 2496 . 2  |-  ( -.  W  e.  _V  ->  (ringLMod `  W )  =  ( ( subringAlg  `  W ) `  ( Base `  W )
) )
136, 12pm2.61i 159 1  |-  (ringLMod `  W
)  =  ( ( subringAlg  `  W ) `  ( Base `  W ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1653    e. wcel 1726   _Vcvv 2958   (/)c0 3630   ` cfv 5456   Basecbs 13471   subringAlg csra 16242  ringLModcrglmod 16243
This theorem is referenced by:  rlmbas  16269  rlmplusg  16270  rlm0  16271  rlmmulr  16272  rlmsca  16273  rlmsca2  16274  rlmvsca  16275  rlmtopn  16276  rlmds  16277  rlmlmod  16278  rlmassa  16387  rlmnlm  18726  rlmbn  19317
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-iota 5420  df-fun 5458  df-fv 5464  df-rgmod 16247
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