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Theorem mnd4g 14378
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 14377 . . 3  |-  ( ph  ->  ( Y  .+  ( Z  .+  W ) )  =  ( Z  .+  ( Y  .+  W ) ) )
98oveq2d 5874 . 2  |-  ( ph  ->  ( X  .+  ( Y  .+  ( Z  .+  W ) ) )  =  ( X  .+  ( Z  .+  ( Y 
.+  W ) ) ) )
10 mnd4g.2 . . 3  |-  ( ph  ->  X  e.  B )
111, 2mndcl 14372 . . . 4  |-  ( ( G  e.  Mnd  /\  Z  e.  B  /\  W  e.  B )  ->  ( Z  .+  W
)  e.  B )
123, 5, 6, 11syl3anc 1182 . . 3  |-  ( ph  ->  ( Z  .+  W
)  e.  B )
131, 2mndass 14373 . . 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 1184 . 2  |-  ( ph  ->  ( ( X  .+  Y )  .+  ( Z  .+  W ) )  =  ( X  .+  ( Y  .+  ( Z 
.+  W ) ) ) )
151, 2mndcl 14372 . . . 4  |-  ( ( G  e.  Mnd  /\  Y  e.  B  /\  W  e.  B )  ->  ( Y  .+  W
)  e.  B )
163, 4, 6, 15syl3anc 1182 . . 3  |-  ( ph  ->  ( Y  .+  W
)  e.  B )
171, 2mndass 14373 . . 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 1184 . 2  |-  ( ph  ->  ( ( X  .+  Z )  .+  ( Y  .+  W ) )  =  ( X  .+  ( Z  .+  ( Y 
.+  W ) ) ) )
199, 14, 183eqtr4d 2325 1  |-  ( ph  ->  ( ( X  .+  Y )  .+  ( Z  .+  W ) )  =  ( ( X 
.+  Z )  .+  ( Y  .+  W ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   Mndcmnd 14361
This theorem is referenced by:  lsmsubm  14964  pj1ghm  15012  cmn4  15108  gsumzaddlem  15203
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
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