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Theorem slmdass 28194
Description: Semiring left module vector sum is associative. (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
Hypotheses
Ref Expression
slmdvacl.v  |-  V  =  ( Base `  W
)
slmdvacl.a  |-  .+  =  ( +g  `  W )
Assertion
Ref Expression
slmdass  |-  ( ( W  e. SLMod  /\  ( X  e.  V  /\  Y  e.  V  /\  Z  e.  V )
)  ->  ( ( X  .+  Y )  .+  Z )  =  ( X  .+  ( Y 
.+  Z ) ) )

Proof of Theorem slmdass
StepHypRef Expression
1 slmdmnd 28187 . 2  |-  ( W  e. SLMod  ->  W  e.  Mnd )
2 slmdvacl.v . . 3  |-  V  =  ( Base `  W
)
3 slmdvacl.a . . 3  |-  .+  =  ( +g  `  W )
42, 3mndass 16252 . 2  |-  ( ( W  e.  Mnd  /\  ( X  e.  V  /\  Y  e.  V  /\  Z  e.  V
) )  ->  (
( X  .+  Y
)  .+  Z )  =  ( X  .+  ( Y  .+  Z ) ) )
51, 4sylan 469 1  |-  ( ( W  e. SLMod  /\  ( X  e.  V  /\  Y  e.  V  /\  Z  e.  V )
)  ->  ( ( X  .+  Y )  .+  Z )  =  ( X  .+  ( Y 
.+  Z ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   ` cfv 5568  (class class class)co 6277   Basecbs 14839   +g cplusg 14907   Mndcmnd 16241  SLModcslmd 28181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-nul 4524
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-iota 5532  df-fv 5576  df-ov 6280  df-sgrp 16233  df-mnd 16243  df-cmn 17122  df-slmd 28182
This theorem is referenced by: (None)
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