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Theorem rlmval 15945
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 5525 . . . 4  |-  ( a  =  W  ->  ( subringAlg  `  a )  =  ( subringAlg  `  W ) )
2 fveq2 5525 . . . 4  |-  ( a  =  W  ->  ( Base `  a )  =  ( Base `  W
) )
31, 2fveq12d 5531 . . 3  |-  ( a  =  W  ->  (
( subringAlg  `  a ) `  ( Base `  a )
)  =  ( ( subringAlg  `  W ) `  ( Base `  W ) ) )
4 df-rgmod 15926 . . 3  |- ringLMod  =  ( a  e.  _V  |->  ( ( subringAlg  `  a ) `  ( Base `  a )
) )
5 fvex 5539 . . 3  |-  ( ( subringAlg  `  W ) `  ( Base `  W ) )  e.  _V
63, 4, 5fvmpt 5602 . 2  |-  ( W  e.  _V  ->  (ringLMod `  W )  =  ( ( subringAlg  `  W ) `  ( Base `  W )
) )
7 fv01 5559 . . . 4  |-  ( (/) `  ( Base `  W
) )  =  (/)
87eqcomi 2287 . . 3  |-  (/)  =  (
(/) `  ( Base `  W ) )
9 fvprc 5519 . . 3  |-  ( -.  W  e.  _V  ->  (ringLMod `  W )  =  (/) )
10 fvprc 5519 . . . 4  |-  ( -.  W  e.  _V  ->  ( subringAlg  `  W )  =  (/) )
1110fveq1d 5527 . . 3  |-  ( -.  W  e.  _V  ->  ( ( subringAlg  `  W ) `  ( Base `  W )
)  =  ( (/) `  ( Base `  W
) ) )
128, 9, 113eqtr4a 2341 . 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 1623    e. wcel 1684   _Vcvv 2788   (/)c0 3455   ` cfv 5255   Basecbs 13148   subringAlg csra 15921  ringLModcrglmod 15922
This theorem is referenced by:  rlmbas  15948  rlmplusg  15949  rlm0  15950  rlmmulr  15951  rlmsca  15952  rlmsca2  15953  rlmvsca  15954  rlmtopn  15955  rlmds  15956  rlmlmod  15957  rlmassa  16066  rlmnlm  18199  rlmbn  18778
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-rgmod 15926
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