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Theorem lmodass 15957
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 15949 . 2  |-  ( W  e.  LMod  ->  W  e. 
Grp )
2 lmodvacl.v . . 3  |-  V  =  ( Base `  W
)
3 lmodvacl.a . . 3  |-  .+  =  ( +g  `  W )
42, 3grpass 14811 . 2  |-  ( ( W  e.  Grp  /\  ( X  e.  V  /\  Y  e.  V  /\  Z  e.  V
) )  ->  (
( X  .+  Y
)  .+  Z )  =  ( X  .+  ( Y  .+  Z ) ) )
51, 4sylan 458 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 359    /\ w3a 936    = wceq 1652    e. wcel 1725   ` cfv 5446  (class class class)co 6073   Basecbs 13461   +g cplusg 13521   Grpcgrp 14677   LModclmod 15942
This theorem is referenced by:  lmodvneg1  15979  lmodcom  15982  baerlem5alem1  32443  mapdh6gN  32477  mapdh6hN  32478  hdmap1l6g  32552  hdmap1l6h  32553
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-nul 4330  ax-pow 4369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454  df-ov 6076  df-mnd 14682  df-grp 14804  df-lmod 15944
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