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Theorem mnd12g 14621
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 6029 . 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 14617 . . 3  |-  ( ( G  e.  Mnd  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  .+  Y
)  .+  Z )  =  ( X  .+  ( Y  .+  Z ) ) )
103, 4, 5, 6, 9syl13anc 1186 . 2  |-  ( ph  ->  ( ( X  .+  Y )  .+  Z
)  =  ( X 
.+  ( Y  .+  Z ) ) )
117, 8mndass 14617 . . 3  |-  ( ( G  e.  Mnd  /\  ( Y  e.  B  /\  X  e.  B  /\  Z  e.  B
) )  ->  (
( Y  .+  X
)  .+  Z )  =  ( Y  .+  ( X  .+  Z ) ) )
123, 5, 4, 6, 11syl13anc 1186 . 2  |-  ( ph  ->  ( ( Y  .+  X )  .+  Z
)  =  ( Y 
.+  ( X  .+  Z ) ) )
132, 10, 123eqtr3d 2421 1  |-  ( ph  ->  ( X  .+  ( Y  .+  Z ) )  =  ( Y  .+  ( X  .+  Z ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717   ` cfv 5388  (class class class)co 6014   Basecbs 13390   +g cplusg 13450   Mndcmnd 14605
This theorem is referenced by:  mnd4g  14622  cmn12  15353
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2362  ax-nul 4273  ax-pow 4312
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2236  df-mo 2237  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2506  df-ne 2546  df-ral 2648  df-rex 2649  df-rab 2652  df-v 2895  df-sbc 3099  df-dif 3260  df-un 3262  df-in 3264  df-ss 3271  df-nul 3566  df-if 3677  df-sn 3757  df-pr 3758  df-op 3760  df-uni 3952  df-br 4148  df-iota 5352  df-fv 5396  df-ov 6017  df-mnd 14611
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