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Theorem sgrpass 12706
Description: A semigroup operation is associative. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 30-Jan-2020.)
Hypotheses
Ref Expression
sgrpass.b  |-  B  =  ( Base `  G
)
sgrpass.o  |-  .o.  =  ( +g  `  G )
Assertion
Ref Expression
sgrpass  |-  ( ( G  e. Smgrp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( ( X  .o.  Y )  .o. 
Z )  =  ( X  .o.  ( Y  .o.  Z ) ) )

Proof of Theorem sgrpass
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sgrpass.b . . . 4  |-  B  =  ( Base `  G
)
2 sgrpass.o . . . 4  |-  .o.  =  ( +g  `  G )
31, 2issgrp 12701 . . 3  |-  ( G  e. Smgrp 
<->  ( G  e. Mgm  /\  A. x  e.  B  A. y  e.  B  A. z  e.  B  (
( x  .o.  y
)  .o.  z )  =  ( x  .o.  ( y  .o.  z
) ) ) )
4 oveq1 5876 . . . . . . 7  |-  ( x  =  X  ->  (
x  .o.  y )  =  ( X  .o.  y ) )
54oveq1d 5884 . . . . . 6  |-  ( x  =  X  ->  (
( x  .o.  y
)  .o.  z )  =  ( ( X  .o.  y )  .o.  z ) )
6 oveq1 5876 . . . . . 6  |-  ( x  =  X  ->  (
x  .o.  ( y  .o.  z ) )  =  ( X  .o.  (
y  .o.  z )
) )
75, 6eqeq12d 2192 . . . . 5  |-  ( x  =  X  ->  (
( ( x  .o.  y )  .o.  z
)  =  ( x  .o.  ( y  .o.  z ) )  <->  ( ( X  .o.  y )  .o.  z )  =  ( X  .o.  ( y  .o.  z ) ) ) )
8 oveq2 5877 . . . . . . 7  |-  ( y  =  Y  ->  ( X  .o.  y )  =  ( X  .o.  Y
) )
98oveq1d 5884 . . . . . 6  |-  ( y  =  Y  ->  (
( X  .o.  y
)  .o.  z )  =  ( ( X  .o.  Y )  .o.  z ) )
10 oveq1 5876 . . . . . . 7  |-  ( y  =  Y  ->  (
y  .o.  z )  =  ( Y  .o.  z ) )
1110oveq2d 5885 . . . . . 6  |-  ( y  =  Y  ->  ( X  .o.  ( y  .o.  z ) )  =  ( X  .o.  ( Y  .o.  z ) ) )
129, 11eqeq12d 2192 . . . . 5  |-  ( y  =  Y  ->  (
( ( X  .o.  y )  .o.  z
)  =  ( X  .o.  ( y  .o.  z ) )  <->  ( ( X  .o.  Y )  .o.  z )  =  ( X  .o.  ( Y  .o.  z ) ) ) )
13 oveq2 5877 . . . . . 6  |-  ( z  =  Z  ->  (
( X  .o.  Y
)  .o.  z )  =  ( ( X  .o.  Y )  .o. 
Z ) )
14 oveq2 5877 . . . . . . 7  |-  ( z  =  Z  ->  ( Y  .o.  z )  =  ( Y  .o.  Z
) )
1514oveq2d 5885 . . . . . 6  |-  ( z  =  Z  ->  ( X  .o.  ( Y  .o.  z ) )  =  ( X  .o.  ( Y  .o.  Z ) ) )
1613, 15eqeq12d 2192 . . . . 5  |-  ( z  =  Z  ->  (
( ( X  .o.  Y )  .o.  z
)  =  ( X  .o.  ( Y  .o.  z ) )  <->  ( ( X  .o.  Y )  .o. 
Z )  =  ( X  .o.  ( Y  .o.  Z ) ) ) )
177, 12, 16rspc3v 2857 . . . 4  |-  ( ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  ->  ( A. x  e.  B  A. y  e.  B  A. z  e.  B  ( ( x  .o.  y )  .o.  z )  =  ( x  .o.  ( y  .o.  z ) )  ->  ( ( X  .o.  Y )  .o. 
Z )  =  ( X  .o.  ( Y  .o.  Z ) ) ) )
1817com12 30 . . 3  |-  ( A. x  e.  B  A. y  e.  B  A. z  e.  B  (
( x  .o.  y
)  .o.  z )  =  ( x  .o.  ( y  .o.  z
) )  ->  (
( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  ->  ( ( X  .o.  Y )  .o. 
Z )  =  ( X  .o.  ( Y  .o.  Z ) ) ) )
193, 18simplbiim 387 . 2  |-  ( G  e. Smgrp  ->  ( ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  ->  (
( X  .o.  Y
)  .o.  Z )  =  ( X  .o.  ( Y  .o.  Z ) ) ) )
2019imp 124 1  |-  ( ( G  e. Smgrp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( ( X  .o.  Y )  .o. 
Z )  =  ( X  .o.  ( Y  .o.  Z ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    /\ w3a 978    = wceq 1353    e. wcel 2148   A.wral 2455   ` cfv 5212  (class class class)co 5869   Basecbs 12445   +g cplusg 12518  Mgmcmgm 12665  Smgrpcsgrp 12699
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-cnex 7893  ax-resscn 7894  ax-1re 7896  ax-addrcl 7899
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-iota 5174  df-fun 5214  df-fn 5215  df-fv 5220  df-ov 5872  df-inn 8909  df-2 8967  df-ndx 12448  df-slot 12449  df-base 12451  df-plusg 12531  df-sgrp 12700
This theorem is referenced by:  mndass  12717  dfgrp2  12792  dfgrp3mlem  12857  dfgrp3me  12859  mulgnndir  12900
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