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Theorem mnd4g 14394
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 14393 . . 3  |-  ( ph  ->  ( Y  .+  ( Z  .+  W ) )  =  ( Z  .+  ( Y  .+  W ) ) )
98oveq2d 5890 . 2  |-  ( ph  ->  ( X  .+  ( Y  .+  ( Z  .+  W ) ) )  =  ( X  .+  ( Z  .+  ( Y 
.+  W ) ) ) )
10 mnd4g.2 . . 3  |-  ( ph  ->  X  e.  B )
111, 2mndcl 14388 . . . 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 14389 . . 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 14388 . . . 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 14389 . . 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 2338 1  |-  ( ph  ->  ( ( X  .+  Y )  .+  ( Z  .+  W ) )  =  ( ( X 
.+  Z )  .+  ( Y  .+  W ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224   Mndcmnd 14377
This theorem is referenced by:  lsmsubm  14980  pj1ghm  15028  cmn4  15124  gsumzaddlem  15219
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-nul 4165  ax-pow 4204
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-ov 5877  df-mnd 14383
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