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Theorem mnd12g 16260
Description: Commutative/associative law for monoids, with an explicit commutativity hypothesis. (Contributed by Mario Carneiro, 21-Apr-2016.)
Hypotheses
Ref Expression
mndcl.b  |-  B  =  ( Base `  G
)
mndcl.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 )
mnd12g.5  |-  ( ph  ->  ( X  .+  Y
)  =  ( Y 
.+  X ) )
Assertion
Ref Expression
mnd12g  |-  ( ph  ->  ( X  .+  ( Y  .+  Z ) )  =  ( Y  .+  ( X  .+  Z ) ) )

Proof of Theorem mnd12g
StepHypRef Expression
1 mnd12g.5 . . 3  |-  ( ph  ->  ( X  .+  Y
)  =  ( Y 
.+  X ) )
21oveq1d 6293 . 2  |-  ( ph  ->  ( ( X  .+  Y )  .+  Z
)  =  ( ( Y  .+  X ) 
.+  Z ) )
3 mnd4g.1 . . 3  |-  ( ph  ->  G  e.  Mnd )
4 mnd4g.2 . . 3  |-  ( ph  ->  X  e.  B )
5 mnd4g.3 . . 3  |-  ( ph  ->  Y  e.  B )
6 mnd4g.4 . . 3  |-  ( ph  ->  Z  e.  B )
7 mndcl.b . . . 4  |-  B  =  ( Base `  G
)
8 mndcl.p . . . 4  |-  .+  =  ( +g  `  G )
97, 8mndass 16254 . . 3  |-  ( ( G  e.  Mnd  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  .+  Y
)  .+  Z )  =  ( X  .+  ( Y  .+  Z ) ) )
103, 4, 5, 6, 9syl13anc 1232 . 2  |-  ( ph  ->  ( ( X  .+  Y )  .+  Z
)  =  ( X 
.+  ( Y  .+  Z ) ) )
117, 8mndass 16254 . . 3  |-  ( ( G  e.  Mnd  /\  ( Y  e.  B  /\  X  e.  B  /\  Z  e.  B
) )  ->  (
( Y  .+  X
)  .+  Z )  =  ( Y  .+  ( X  .+  Z ) ) )
123, 5, 4, 6, 11syl13anc 1232 . 2  |-  ( ph  ->  ( ( Y  .+  X )  .+  Z
)  =  ( Y 
.+  ( X  .+  Z ) ) )
132, 10, 123eqtr3d 2451 1  |-  ( ph  ->  ( X  .+  ( Y  .+  Z ) )  =  ( Y  .+  ( X  .+  Z ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1405    e. wcel 1842   ` cfv 5569  (class class class)co 6278   Basecbs 14841   +g cplusg 14909   Mndcmnd 16243
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 4525
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 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-iota 5533  df-fv 5577  df-ov 6281  df-sgrp 16235  df-mnd 16245
This theorem is referenced by:  mnd4g  16261  cmn12  17142
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