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Theorem lmodvsdi2OLD 15653
Description: Obsolete version of lmodvsdir 15652 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 15652 . . . 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 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208  Scalarcsca 13211   .scvsca 13212   LModclmod 15627
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-nul 4149
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-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-iota 5219  df-fv 5263  df-ov 5861  df-lmod 15629
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