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Theorem mnd32g 15512
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 )
mnd32g.5  |-  ( ph  ->  ( Y  .+  Z
)  =  ( Z 
.+  Y ) )
Assertion
Ref Expression
mnd32g  |-  ( ph  ->  ( ( X  .+  Y )  .+  Z
)  =  ( ( X  .+  Z ) 
.+  Y ) )

Proof of Theorem mnd32g
StepHypRef Expression
1 mnd32g.5 . . 3  |-  ( ph  ->  ( Y  .+  Z
)  =  ( Z 
.+  Y ) )
21oveq2d 6192 . 2  |-  ( ph  ->  ( X  .+  ( Y  .+  Z ) )  =  ( X  .+  ( Z  .+  Y ) ) )
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 15509 . . 3  |-  ( ( G  e.  Mnd  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  .+  Y
)  .+  Z )  =  ( X  .+  ( Y  .+  Z ) ) )
103, 4, 5, 6, 9syl13anc 1221 . 2  |-  ( ph  ->  ( ( X  .+  Y )  .+  Z
)  =  ( X 
.+  ( Y  .+  Z ) ) )
117, 8mndass 15509 . . 3  |-  ( ( G  e.  Mnd  /\  ( X  e.  B  /\  Z  e.  B  /\  Y  e.  B
) )  ->  (
( X  .+  Z
)  .+  Y )  =  ( X  .+  ( Z  .+  Y ) ) )
123, 4, 6, 5, 11syl13anc 1221 . 2  |-  ( ph  ->  ( ( X  .+  Z )  .+  Y
)  =  ( X 
.+  ( Z  .+  Y ) ) )
132, 10, 123eqtr4d 2500 1  |-  ( ph  ->  ( ( X  .+  Y )  .+  Z
)  =  ( ( X  .+  Z ) 
.+  Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1757   ` cfv 5502  (class class class)co 6176   Basecbs 14262   +g cplusg 14326   Mndcmnd 15497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-nul 4505  ax-pow 4554
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-ral 2797  df-rex 2798  df-rab 2801  df-v 3056  df-sbc 3271  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-nul 3722  df-if 3876  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4176  df-br 4377  df-iota 5465  df-fv 5510  df-ov 6179  df-mnd 15503
This theorem is referenced by:  cmn32  16385
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