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Theorem rlmval 15961
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 5541 . . . 4  |-  ( a  =  W  ->  ( subringAlg  `  a )  =  ( subringAlg  `  W ) )
2 fveq2 5541 . . . 4  |-  ( a  =  W  ->  ( Base `  a )  =  ( Base `  W
) )
31, 2fveq12d 5547 . . 3  |-  ( a  =  W  ->  (
( subringAlg  `  a ) `  ( Base `  a )
)  =  ( ( subringAlg  `  W ) `  ( Base `  W ) ) )
4 df-rgmod 15942 . . 3  |- ringLMod  =  ( a  e.  _V  |->  ( ( subringAlg  `  a ) `  ( Base `  a )
) )
5 fvex 5555 . . 3  |-  ( ( subringAlg  `  W ) `  ( Base `  W ) )  e.  _V
63, 4, 5fvmpt 5618 . 2  |-  ( W  e.  _V  ->  (ringLMod `  W )  =  ( ( subringAlg  `  W ) `  ( Base `  W )
) )
7 fv01 5575 . . . 4  |-  ( (/) `  ( Base `  W
) )  =  (/)
87eqcomi 2300 . . 3  |-  (/)  =  (
(/) `  ( Base `  W ) )
9 fvprc 5535 . . 3  |-  ( -.  W  e.  _V  ->  (ringLMod `  W )  =  (/) )
10 fvprc 5535 . . . 4  |-  ( -.  W  e.  _V  ->  ( subringAlg  `  W )  =  (/) )
1110fveq1d 5543 . . 3  |-  ( -.  W  e.  _V  ->  ( ( subringAlg  `  W ) `  ( Base `  W )
)  =  ( (/) `  ( Base `  W
) ) )
128, 9, 113eqtr4a 2354 . 2  |-  ( -.  W  e.  _V  ->  (ringLMod `  W )  =  ( ( subringAlg  `  W ) `  ( Base `  W )
) )
136, 12pm2.61i 156 1  |-  (ringLMod `  W
)  =  ( ( subringAlg  `  W ) `  ( Base `  W ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1632    e. wcel 1696   _Vcvv 2801   (/)c0 3468   ` cfv 5271   Basecbs 13164   subringAlg csra 15937  ringLModcrglmod 15938
This theorem is referenced by:  rlmbas  15964  rlmplusg  15965  rlm0  15966  rlmmulr  15967  rlmsca  15968  rlmsca2  15969  rlmvsca  15970  rlmtopn  15971  rlmds  15972  rlmlmod  15973  rlmassa  16082  rlmnlm  18215  rlmbn  18794
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-rgmod 15942
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