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Theorem mnd4g 16263
Description: Commutative/associative law for commutative monoids, with an explicit commutativity hypothesis. (Contributed by Mario Carneiro, 21-Apr-2016.)
Hypotheses
Ref Expression
mndcl.b  |-  B  =  ( Base `  G
)
mndcl.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 mndcl.b . . . 4  |-  B  =  ( Base `  G
)
2 mndcl.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 16262 . . 3  |-  ( ph  ->  ( Y  .+  ( Z  .+  W ) )  =  ( Z  .+  ( Y  .+  W ) ) )
98oveq2d 6296 . 2  |-  ( ph  ->  ( X  .+  ( Y  .+  ( Z  .+  W ) ) )  =  ( X  .+  ( Z  .+  ( Y 
.+  W ) ) ) )
10 mnd4g.2 . . 3  |-  ( ph  ->  X  e.  B )
111, 2mndcl 16255 . . . 4  |-  ( ( G  e.  Mnd  /\  Z  e.  B  /\  W  e.  B )  ->  ( Z  .+  W
)  e.  B )
123, 5, 6, 11syl3anc 1232 . . 3  |-  ( ph  ->  ( Z  .+  W
)  e.  B )
131, 2mndass 16256 . . 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 1234 . 2  |-  ( ph  ->  ( ( X  .+  Y )  .+  ( Z  .+  W ) )  =  ( X  .+  ( Y  .+  ( Z 
.+  W ) ) ) )
151, 2mndcl 16255 . . . 4  |-  ( ( G  e.  Mnd  /\  Y  e.  B  /\  W  e.  B )  ->  ( Y  .+  W
)  e.  B )
163, 4, 6, 15syl3anc 1232 . . 3  |-  ( ph  ->  ( Y  .+  W
)  e.  B )
171, 2mndass 16256 . . 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 1234 . 2  |-  ( ph  ->  ( ( X  .+  Z )  .+  ( Y  .+  W ) )  =  ( X  .+  ( Z  .+  ( Y 
.+  W ) ) ) )
199, 14, 183eqtr4d 2455 1  |-  ( ph  ->  ( ( X  .+  Y )  .+  ( Z  .+  W ) )  =  ( ( X 
.+  Z )  .+  ( Y  .+  W ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1407    e. wcel 1844   ` cfv 5571  (class class class)co 6280   Basecbs 14843   +g cplusg 14911   Mndcmnd 16245
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-nul 4527
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-br 4398  df-iota 5535  df-fv 5579  df-ov 6283  df-mgm 16198  df-sgrp 16237  df-mnd 16247
This theorem is referenced by:  lsmsubm  16999  pj1ghm  17047  cmn4  17143  gsumzaddlem  17260  gsumzaddlemOLD  17262
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