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Theorem rlmval 17271
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 5690 . . . 4  |-  ( a  =  W  ->  (subringAlg  `  a )  =  (subringAlg  `  W ) )
2 fveq2 5690 . . . 4  |-  ( a  =  W  ->  ( Base `  a )  =  ( Base `  W
) )
31, 2fveq12d 5696 . . 3  |-  ( a  =  W  ->  (
(subringAlg  `  a ) `  ( Base `  a )
)  =  ( (subringAlg  `  W ) `  ( Base `  W ) ) )
4 df-rgmod 17253 . . 3  |- ringLMod  =  ( a  e.  _V  |->  ( (subringAlg  `  a ) `  ( Base `  a )
) )
5 fvex 5700 . . 3  |-  ( (subringAlg  `  W ) `  ( Base `  W ) )  e.  _V
63, 4, 5fvmpt 5773 . 2  |-  ( W  e.  _V  ->  (ringLMod `  W )  =  ( (subringAlg  `  W ) `  ( Base `  W )
) )
7 0fv 5722 . . . 4  |-  ( (/) `  ( Base `  W
) )  =  (/)
87eqcomi 2446 . . 3  |-  (/)  =  (
(/) `  ( Base `  W ) )
9 fvprc 5684 . . 3  |-  ( -.  W  e.  _V  ->  (ringLMod `  W )  =  (/) )
10 fvprc 5684 . . . 4  |-  ( -.  W  e.  _V  ->  (subringAlg  `  W )  =  (/) )
1110fveq1d 5692 . . 3  |-  ( -.  W  e.  _V  ->  ( (subringAlg  `  W ) `  ( Base `  W )
)  =  ( (/) `  ( Base `  W
) ) )
128, 9, 113eqtr4a 2500 . 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 1369    e. wcel 1756   _Vcvv 2971   (/)c0 3636   ` cfv 5417   Basecbs 14173  subringAlg csra 17248  ringLModcrglmod 17249
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-sbc 3186  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-iota 5380  df-fun 5419  df-fv 5425  df-rgmod 17253
This theorem is referenced by:  rlmval2  17274  rlmbas  17275  rlmplusg  17276  rlm0  17277  rlmmulr  17279  rlmsca  17280  rlmsca2  17281  rlmvsca  17282  rlmtopn  17283  rlmds  17284  rlmlmod  17285  rlmassa  17396  frlmip  18202  rlmnlm  20268  rlmbn  20872  rrxprds  20892
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