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Theorem reacomsmgrp1 25446
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
reacomsmgrp1  |-  ( ( G  e.  ( SemiGrp  i^i 
Com1 )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A G ( B G C ) )  =  ( B G ( A G C ) ) )

Proof of Theorem reacomsmgrp1
StepHypRef Expression
1 inss1 3402 . . . 4  |-  ( SemiGrp  i^i 
Com1 )  C_  SemiGrp
21sseli 3189 . . 3  |-  ( G  e.  ( SemiGrp  i^i  Com1 )  ->  G  e.  SemiGrp )
3 reacomsmgrp1.1 . . . . 5  |-  X  =  dom  dom  G
43smgrpass2 25444 . . . 4  |-  ( ( G  e.  SemiGrp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A G B ) G C )  =  ( A G ( B G C ) ) )
54eqcomd 2301 . . 3  |-  ( ( G  e.  SemiGrp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A G ( B G C ) )  =  ( ( A G B ) G C ) )
62, 5sylan 457 . 2  |-  ( ( G  e.  ( SemiGrp  i^i 
Com1 )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A G ( B G C ) )  =  ( ( A G B ) G C ) )
7 inss2 3403 . . . . . 6  |-  ( SemiGrp  i^i 
Com1 )  C_  Com1
87sseli 3189 . . . . 5  |-  ( G  e.  ( SemiGrp  i^i  Com1 )  ->  G  e.  Com1 )
98adantr 451 . . . 4  |-  ( ( G  e.  ( SemiGrp  i^i 
Com1 )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  G  e.  Com1 )
10 simpr1 961 . . . 4  |-  ( ( G  e.  ( SemiGrp  i^i 
Com1 )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  A  e.  X )
11 simpr2 962 . . . 4  |-  ( ( G  e.  ( SemiGrp  i^i 
Com1 )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  B  e.  X )
123iscomb 25437 . . . 4  |-  ( ( G  e.  Com1  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  =  ( B G A ) )
139, 10, 11, 12syl3anc 1182 . . 3  |-  ( ( G  e.  ( SemiGrp  i^i 
Com1 )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A G B )  =  ( B G A ) )
1413oveq1d 5889 . 2  |-  ( ( G  e.  ( SemiGrp  i^i 
Com1 )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A G B ) G C )  =  ( ( B G A ) G C ) )
15 simp2 956 . . . 4  |-  ( ( A  e.  X  /\  B  e.  X  /\  C  e.  X )  ->  B  e.  X )
16 simp1 955 . . . 4  |-  ( ( A  e.  X  /\  B  e.  X  /\  C  e.  X )  ->  A  e.  X )
17 simp3 957 . . . 4  |-  ( ( A  e.  X  /\  B  e.  X  /\  C  e.  X )  ->  C  e.  X )
1815, 16, 173jca 1132 . . 3  |-  ( ( A  e.  X  /\  B  e.  X  /\  C  e.  X )  ->  ( B  e.  X  /\  A  e.  X  /\  C  e.  X
) )
193smgrpass2 25444 . . 3  |-  ( ( G  e.  SemiGrp  /\  ( B  e.  X  /\  A  e.  X  /\  C  e.  X )
)  ->  ( ( B G A ) G C )  =  ( B G ( A G C ) ) )
202, 18, 19syl2an 463 . 2  |-  ( ( G  e.  ( SemiGrp  i^i 
Com1 )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( B G A ) G C )  =  ( B G ( A G C ) ) )
216, 14, 203eqtrd 2332 1  |-  ( ( G  e.  ( SemiGrp  i^i 
Com1 )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A G ( B G C ) )  =  ( B G ( A G C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    i^i cin 3164   dom cdm 4705  (class class class)co 5874   SemiGrpcsem 21013   Com1ccm1 25434
This theorem is referenced by:  resgcom  25454
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-ass 20996  df-mgm 21002  df-sgr 21014  df-com1 25435
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