Users' Mathboxes Mathbox for Frédéric Liné < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  reacomsmgrp3 Unicode version

Theorem reacomsmgrp3 25345
Description: Rearrangement of three terms in a commutative semi-group. (Contributed by FL, 14-Sep-2010.)
Hypothesis
Ref Expression
reacomsmgrp1.1  |-  X  =  dom  dom  G
Assertion
Ref Expression
reacomsmgrp3  |-  ( ( G  e.  ( SemiGrp  i^i 
Com1 )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A G B ) G C )  =  ( ( A G C ) G B ) )

Proof of Theorem reacomsmgrp3
StepHypRef Expression
1 inss2 3390 . . . . 5  |-  ( SemiGrp  i^i 
Com1 )  C_  Com1
21sseli 3176 . . . 4  |-  ( G  e.  ( SemiGrp  i^i  Com1 )  ->  G  e.  Com1 )
32adantr 451 . . 3  |-  ( ( G  e.  ( SemiGrp  i^i 
Com1 )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  G  e.  Com1 )
4 inss1 3389 . . . . . . 7  |-  ( SemiGrp  i^i 
Com1 )  C_  SemiGrp
54sseli 3176 . . . . . 6  |-  ( G  e.  ( SemiGrp  i^i  Com1 )  ->  G  e.  SemiGrp )
6 smgrpismgm 20999 . . . . . 6  |-  ( G  e.  SemiGrp  ->  G  e.  Magma )
75, 6syl 15 . . . . 5  |-  ( G  e.  ( SemiGrp  i^i  Com1 )  ->  G  e.  Magma )
87adantr 451 . . . 4  |-  ( ( G  e.  ( SemiGrp  i^i 
Com1 )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  G  e.  Magma
)
9 simpr1 961 . . . 4  |-  ( ( G  e.  ( SemiGrp  i^i 
Com1 )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  A  e.  X )
10 simpr2 962 . . . 4  |-  ( ( G  e.  ( SemiGrp  i^i 
Com1 )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  B  e.  X )
11 reacomsmgrp1.1 . . . . 5  |-  X  =  dom  dom  G
1211clmgm 20988 . . . 4  |-  ( ( G  e.  Magma  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  e.  X )
138, 9, 10, 12syl3anc 1182 . . 3  |-  ( ( G  e.  ( SemiGrp  i^i 
Com1 )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A G B )  e.  X
)
14 simpr3 963 . . 3  |-  ( ( G  e.  ( SemiGrp  i^i 
Com1 )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  C  e.  X )
1511iscomb 25334 . . 3  |-  ( ( G  e.  Com1  /\  ( A G B )  e.  X  /\  C  e.  X )  ->  (
( A G B ) G C )  =  ( C G ( A G B ) ) )
163, 13, 14, 15syl3anc 1182 . 2  |-  ( ( G  e.  ( SemiGrp  i^i 
Com1 )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A G B ) G C )  =  ( C G ( A G B ) ) )
17 simp3 957 . . . 4  |-  ( ( A  e.  X  /\  B  e.  X  /\  C  e.  X )  ->  C  e.  X )
18 simp1 955 . . . 4  |-  ( ( A  e.  X  /\  B  e.  X  /\  C  e.  X )  ->  A  e.  X )
19 simp2 956 . . . 4  |-  ( ( A  e.  X  /\  B  e.  X  /\  C  e.  X )  ->  B  e.  X )
2017, 18, 193jca 1132 . . 3  |-  ( ( A  e.  X  /\  B  e.  X  /\  C  e.  X )  ->  ( C  e.  X  /\  A  e.  X  /\  B  e.  X
) )
2111reacomsmgrp2 25344 . . 3  |-  ( ( G  e.  ( SemiGrp  i^i 
Com1 )  /\  ( C  e.  X  /\  A  e.  X  /\  B  e.  X )
)  ->  ( C G ( A G B ) )  =  ( B G ( A G C ) ) )
2220, 21sylan2 460 . 2  |-  ( ( G  e.  ( SemiGrp  i^i 
Com1 )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( C G ( A G B ) )  =  ( B G ( A G C ) ) )
2311clmgm 20988 . . . 4  |-  ( ( G  e.  Magma  /\  A  e.  X  /\  C  e.  X )  ->  ( A G C )  e.  X )
248, 9, 14, 23syl3anc 1182 . . 3  |-  ( ( G  e.  ( SemiGrp  i^i 
Com1 )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A G C )  e.  X
)
2511iscomb 25334 . . 3  |-  ( ( G  e.  Com1  /\  B  e.  X  /\  ( A G C )  e.  X )  ->  ( B G ( A G C ) )  =  ( ( A G C ) G B ) )
263, 10, 24, 25syl3anc 1182 . 2  |-  ( ( G  e.  ( SemiGrp  i^i 
Com1 )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( B G ( A G C ) )  =  ( ( A G C ) G B ) )
2716, 22, 263eqtrd 2319 1  |-  ( ( G  e.  ( SemiGrp  i^i 
Com1 )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A G B ) G C )  =  ( ( A G C ) G B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    i^i cin 3151   dom cdm 4689  (class class class)co 5858   Magmacmagm 20985   SemiGrpcsem 20997   Com1ccm1 25331
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-ass 20980  df-mgm 20986  df-sgr 20998  df-com1 25332
  Copyright terms: Public domain W3C validator