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Theorem mnd32g 14691
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 6089 . 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 14688 . . 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 14688 . . 3  |-  ( ( G  e.  Mnd  /\  ( X  e.  B  /\  Z  e.  B  /\  Y  e.  B
) )  ->  (
( X  .+  Z
)  .+  Y )  =  ( X  .+  ( Z  .+  Y ) ) )
123, 4, 6, 5, 11syl13anc 1186 . 2  |-  ( ph  ->  ( ( X  .+  Z )  .+  Y
)  =  ( X 
.+  ( Z  .+  Y ) ) )
132, 10, 123eqtr4d 2477 1  |-  ( ph  ->  ( ( X  .+  Y )  .+  Z
)  =  ( ( X  .+  Z ) 
.+  Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   ` cfv 5446  (class class class)co 6073   Basecbs 13461   +g cplusg 13521   Mndcmnd 14676
This theorem is referenced by:  cmn32  15422
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-nul 4330  ax-pow 4369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454  df-ov 6076  df-mnd 14682
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