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Theorem lmodass 15642
Description: Left module vector sum is associative. (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
Hypotheses
Ref Expression
lmodvacl.v  |-  V  =  ( Base `  W
)
lmodvacl.a  |-  .+  =  ( +g  `  W )
Assertion
Ref Expression
lmodass  |-  ( ( W  e.  LMod  /\  ( X  e.  V  /\  Y  e.  V  /\  Z  e.  V )
)  ->  ( ( X  .+  Y )  .+  Z )  =  ( X  .+  ( Y 
.+  Z ) ) )

Proof of Theorem lmodass
StepHypRef Expression
1 lmodgrp 15634 . 2  |-  ( W  e.  LMod  ->  W  e. 
Grp )
2 lmodvacl.v . . 3  |-  V  =  ( Base `  W
)
3 lmodvacl.a . . 3  |-  .+  =  ( +g  `  W )
42, 3grpass 14496 . 2  |-  ( ( W  e.  Grp  /\  ( X  e.  V  /\  Y  e.  V  /\  Z  e.  V
) )  ->  (
( X  .+  Y
)  .+  Z )  =  ( X  .+  ( Y  .+  Z ) ) )
51, 4sylan 457 1  |-  ( ( W  e.  LMod  /\  ( X  e.  V  /\  Y  e.  V  /\  Z  e.  V )
)  ->  ( ( X  .+  Y )  .+  Z )  =  ( X  .+  ( Y 
.+  Z ) ) )
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   Grpcgrp 14362   LModclmod 15627
This theorem is referenced by:  lmodvneg1  15667  lmodcom  15671  baerlem5alem1  31898  mapdh6gN  31932  mapdh6hN  31933  hdmap1l6g  32007  hdmap1l6h  32008
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-nul 4149  ax-pow 4188
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-iota 5219  df-fv 5263  df-ov 5861  df-mnd 14367  df-grp 14489  df-lmod 15629
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