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Theorem mnd4g 15739
Description: Commutative/associative law for commutative monoids, with an explicit commutativity hypothesis. (Contributed by Mario Carneiro, 21-Apr-2016.)
Hypotheses
Ref Expression
mndlem1.b  |-  B  =  ( Base `  G
)
mndlem1.p  |-  .+  =  ( +g  `  G )
mnd4g.1  |-  ( ph  ->  G  e.  Mnd )
mnd4g.2  |-  ( ph  ->  X  e.  B )
mnd4g.3  |-  ( ph  ->  Y  e.  B )
mnd4g.4  |-  ( ph  ->  Z  e.  B )
mnd4g.5  |-  ( ph  ->  W  e.  B )
mnd4g.6  |-  ( ph  ->  ( Y  .+  Z
)  =  ( Z 
.+  Y ) )
Assertion
Ref Expression
mnd4g  |-  ( ph  ->  ( ( X  .+  Y )  .+  ( Z  .+  W ) )  =  ( ( X 
.+  Z )  .+  ( Y  .+  W ) ) )

Proof of Theorem mnd4g
StepHypRef Expression
1 mndlem1.b . . . 4  |-  B  =  ( Base `  G
)
2 mndlem1.p . . . 4  |-  .+  =  ( +g  `  G )
3 mnd4g.1 . . . 4  |-  ( ph  ->  G  e.  Mnd )
4 mnd4g.3 . . . 4  |-  ( ph  ->  Y  e.  B )
5 mnd4g.4 . . . 4  |-  ( ph  ->  Z  e.  B )
6 mnd4g.5 . . . 4  |-  ( ph  ->  W  e.  B )
7 mnd4g.6 . . . 4  |-  ( ph  ->  ( Y  .+  Z
)  =  ( Z 
.+  Y ) )
81, 2, 3, 4, 5, 6, 7mnd12g 15738 . . 3  |-  ( ph  ->  ( Y  .+  ( Z  .+  W ) )  =  ( Z  .+  ( Y  .+  W ) ) )
98oveq2d 6298 . 2  |-  ( ph  ->  ( X  .+  ( Y  .+  ( Z  .+  W ) ) )  =  ( X  .+  ( Z  .+  ( Y 
.+  W ) ) ) )
10 mnd4g.2 . . 3  |-  ( ph  ->  X  e.  B )
111, 2mndcl 15733 . . . 4  |-  ( ( G  e.  Mnd  /\  Z  e.  B  /\  W  e.  B )  ->  ( Z  .+  W
)  e.  B )
123, 5, 6, 11syl3anc 1228 . . 3  |-  ( ph  ->  ( Z  .+  W
)  e.  B )
131, 2mndass 15734 . . 3  |-  ( ( G  e.  Mnd  /\  ( X  e.  B  /\  Y  e.  B  /\  ( Z  .+  W
)  e.  B ) )  ->  ( ( X  .+  Y )  .+  ( Z  .+  W ) )  =  ( X 
.+  ( Y  .+  ( Z  .+  W ) ) ) )
143, 10, 4, 12, 13syl13anc 1230 . 2  |-  ( ph  ->  ( ( X  .+  Y )  .+  ( Z  .+  W ) )  =  ( X  .+  ( Y  .+  ( Z 
.+  W ) ) ) )
151, 2mndcl 15733 . . . 4  |-  ( ( G  e.  Mnd  /\  Y  e.  B  /\  W  e.  B )  ->  ( Y  .+  W
)  e.  B )
163, 4, 6, 15syl3anc 1228 . . 3  |-  ( ph  ->  ( Y  .+  W
)  e.  B )
171, 2mndass 15734 . . 3  |-  ( ( G  e.  Mnd  /\  ( X  e.  B  /\  Z  e.  B  /\  ( Y  .+  W
)  e.  B ) )  ->  ( ( X  .+  Z )  .+  ( Y  .+  W ) )  =  ( X 
.+  ( Z  .+  ( Y  .+  W ) ) ) )
183, 10, 5, 16, 17syl13anc 1230 . 2  |-  ( ph  ->  ( ( X  .+  Z )  .+  ( Y  .+  W ) )  =  ( X  .+  ( Z  .+  ( Y 
.+  W ) ) ) )
199, 14, 183eqtr4d 2518 1  |-  ( ph  ->  ( ( X  .+  Y )  .+  ( Z  .+  W ) )  =  ( ( X 
.+  Z )  .+  ( Y  .+  W ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   ` cfv 5586  (class class class)co 6282   Basecbs 14486   +g cplusg 14551   Mndcmnd 15722
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
This theorem is referenced by:  lsmsubm  16469  pj1ghm  16517  cmn4  16613  gsumzaddlem  16725  gsumzaddlemOLD  16727
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