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Theorem slmdass 27418
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 27411 . 2  |-  ( W  e. SLMod  ->  W  e.  Mnd )
2 slmdvacl.v . . 3  |-  V  =  ( Base `  W
)
3 slmdvacl.a . . 3  |-  .+  =  ( +g  `  W )
42, 3mndass 15734 . 2  |-  ( ( W  e.  Mnd  /\  ( X  e.  V  /\  Y  e.  V  /\  Z  e.  V
) )  ->  (
( X  .+  Y
)  .+  Z )  =  ( X  .+  ( Y  .+  Z ) ) )
51, 4sylan 471 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 369    /\ w3a 973    = wceq 1379    e. wcel 1767   ` cfv 5586  (class class class)co 6282   Basecbs 14486   +g cplusg 14551   Mndcmnd 15722  SLModcslmd 27405
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-nul 4576  ax-pow 4625
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-iota 5549  df-fv 5594  df-ov 6285  df-mnd 15728  df-cmn 16596  df-slmd 27406
This theorem is referenced by: (None)
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