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Theorem mnd12g 15738
Description: Commutative/associative law for 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 )
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 6297 . 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 mndlem1.b . . . 4  |-  B  =  ( Base `  G
)
8 mndlem1.p . . . 4  |-  .+  =  ( +g  `  G )
97, 8mndass 15734 . . 3  |-  ( ( G  e.  Mnd  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  .+  Y
)  .+  Z )  =  ( X  .+  ( Y  .+  Z ) ) )
103, 4, 5, 6, 9syl13anc 1230 . 2  |-  ( ph  ->  ( ( X  .+  Y )  .+  Z
)  =  ( X 
.+  ( Y  .+  Z ) ) )
117, 8mndass 15734 . . 3  |-  ( ( G  e.  Mnd  /\  ( Y  e.  B  /\  X  e.  B  /\  Z  e.  B
) )  ->  (
( Y  .+  X
)  .+  Z )  =  ( Y  .+  ( X  .+  Z ) ) )
123, 5, 4, 6, 11syl13anc 1230 . 2  |-  ( ph  ->  ( ( Y  .+  X )  .+  Z
)  =  ( Y 
.+  ( X  .+  Z ) ) )
132, 10, 123eqtr3d 2516 1  |-  ( ph  ->  ( X  .+  ( Y  .+  Z ) )  =  ( Y  .+  ( X  .+  Z ) ) )
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:  mnd4g  15739  cmn12  16614
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