MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mnd4g Structured version   Unicode version

Theorem mnd4g 14693
Description: Commutative/associative law for commutative 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 )
mnd4g.5  |-  ( ph  ->  W  e.  B )
mnd4g.6  |-  ( ph  ->  ( Y  .+  Z
)  =  ( Z 
.+  Y ) )
Assertion
Ref Expression
mnd4g  |-  ( ph  ->  ( ( X  .+  Y )  .+  ( Z  .+  W ) )  =  ( ( X 
.+  Z )  .+  ( Y  .+  W ) ) )

Proof of Theorem mnd4g
StepHypRef Expression
1 mndlem1.b . . . 4  |-  B  =  ( Base `  G
)
2 mndlem1.p . . . 4  |-  .+  =  ( +g  `  G )
3 mnd4g.1 . . . 4  |-  ( ph  ->  G  e.  Mnd )
4 mnd4g.3 . . . 4  |-  ( ph  ->  Y  e.  B )
5 mnd4g.4 . . . 4  |-  ( ph  ->  Z  e.  B )
6 mnd4g.5 . . . 4  |-  ( ph  ->  W  e.  B )
7 mnd4g.6 . . . 4  |-  ( ph  ->  ( Y  .+  Z
)  =  ( Z 
.+  Y ) )
81, 2, 3, 4, 5, 6, 7mnd12g 14692 . . 3  |-  ( ph  ->  ( Y  .+  ( Z  .+  W ) )  =  ( Z  .+  ( Y  .+  W ) ) )
98oveq2d 6089 . 2  |-  ( ph  ->  ( X  .+  ( Y  .+  ( Z  .+  W ) ) )  =  ( X  .+  ( Z  .+  ( Y 
.+  W ) ) ) )
10 mnd4g.2 . . 3  |-  ( ph  ->  X  e.  B )
111, 2mndcl 14687 . . . 4  |-  ( ( G  e.  Mnd  /\  Z  e.  B  /\  W  e.  B )  ->  ( Z  .+  W
)  e.  B )
123, 5, 6, 11syl3anc 1184 . . 3  |-  ( ph  ->  ( Z  .+  W
)  e.  B )
131, 2mndass 14688 . . 3  |-  ( ( G  e.  Mnd  /\  ( X  e.  B  /\  Y  e.  B  /\  ( Z  .+  W
)  e.  B ) )  ->  ( ( X  .+  Y )  .+  ( Z  .+  W ) )  =  ( X 
.+  ( Y  .+  ( Z  .+  W ) ) ) )
143, 10, 4, 12, 13syl13anc 1186 . 2  |-  ( ph  ->  ( ( X  .+  Y )  .+  ( Z  .+  W ) )  =  ( X  .+  ( Y  .+  ( Z 
.+  W ) ) ) )
151, 2mndcl 14687 . . . 4  |-  ( ( G  e.  Mnd  /\  Y  e.  B  /\  W  e.  B )  ->  ( Y  .+  W
)  e.  B )
163, 4, 6, 15syl3anc 1184 . . 3  |-  ( ph  ->  ( Y  .+  W
)  e.  B )
171, 2mndass 14688 . . 3  |-  ( ( G  e.  Mnd  /\  ( X  e.  B  /\  Z  e.  B  /\  ( Y  .+  W
)  e.  B ) )  ->  ( ( X  .+  Z )  .+  ( Y  .+  W ) )  =  ( X 
.+  ( Z  .+  ( Y  .+  W ) ) ) )
183, 10, 5, 16, 17syl13anc 1186 . 2  |-  ( ph  ->  ( ( X  .+  Z )  .+  ( Y  .+  W ) )  =  ( X  .+  ( Z  .+  ( Y 
.+  W ) ) ) )
199, 14, 183eqtr4d 2477 1  |-  ( ph  ->  ( ( X  .+  Y )  .+  ( Z  .+  W ) )  =  ( ( X 
.+  Z )  .+  ( Y  .+  W ) ) )
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:  lsmsubm  15279  pj1ghm  15327  cmn4  15423  gsumzaddlem  15518
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
  Copyright terms: Public domain W3C validator