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Theorem lmodvsdi2OLD 15669
Description: Obsolete version of lmodvsdir 15668 as of 22-Sep-2015. (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
lmodvsdir.v  |-  V  =  ( Base `  W
)
lmodvsdir.a  |-  .+  =  ( +g  `  W )
lmodvsdir.f  |-  F  =  (Scalar `  W )
lmodvsdir.s  |-  .x.  =  ( .s `  W )
lmodvsdir.k  |-  K  =  ( Base `  F
)
lmodvsdir.p  |-  .+^  =  ( +g  `  F )
Assertion
Ref Expression
lmodvsdi2OLD  |-  ( ( W  e.  LMod  /\  ( Q  e.  K  /\  R  e.  K )  /\  X  e.  V
)  ->  ( ( Q  .+^  R )  .x.  X )  =  ( ( Q  .x.  X
)  .+  ( R  .x.  X ) ) )

Proof of Theorem lmodvsdi2OLD
StepHypRef Expression
1 lmodvsdir.v . . . . 5  |-  V  =  ( Base `  W
)
2 lmodvsdir.a . . . . 5  |-  .+  =  ( +g  `  W )
3 lmodvsdir.f . . . . 5  |-  F  =  (Scalar `  W )
4 lmodvsdir.s . . . . 5  |-  .x.  =  ( .s `  W )
5 lmodvsdir.k . . . . 5  |-  K  =  ( Base `  F
)
6 lmodvsdir.p . . . . 5  |-  .+^  =  ( +g  `  F )
71, 2, 3, 4, 5, 6lmodvsdir 15668 . . . 4  |-  ( ( W  e.  LMod  /\  ( Q  e.  K  /\  R  e.  K  /\  X  e.  V )
)  ->  ( ( Q  .+^  R )  .x.  X )  =  ( ( Q  .x.  X
)  .+  ( R  .x.  X ) ) )
873exp2 1169 . . 3  |-  ( W  e.  LMod  ->  ( Q  e.  K  ->  ( R  e.  K  ->  ( X  e.  V  -> 
( ( Q  .+^  R )  .x.  X )  =  ( ( Q 
.x.  X )  .+  ( R  .x.  X ) ) ) ) ) )
98imp3a 420 . 2  |-  ( W  e.  LMod  ->  ( ( Q  e.  K  /\  R  e.  K )  ->  ( X  e.  V  ->  ( ( Q  .+^  R )  .x.  X )  =  ( ( Q 
.x.  X )  .+  ( R  .x.  X ) ) ) ) )
1093imp 1145 1  |-  ( ( W  e.  LMod  /\  ( Q  e.  K  /\  R  e.  K )  /\  X  e.  V
)  ->  ( ( Q  .+^  R )  .x.  X )  =  ( ( Q  .x.  X
)  .+  ( R  .x.  X ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224  Scalarcsca 13227   .scvsca 13228   LModclmod 15643
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-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-nul 4165
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-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-iota 5235  df-fv 5279  df-ov 5877  df-lmod 15645
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