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Theorem issgrp 16130
Description: The predicate "is a semigroup". (Contributed by FL, 2-Nov-2009.) (Revised by by AV, 6-Jan-2020.)
Hypotheses
Ref Expression
issgrp.b  |-  B  =  ( Base `  M
)
issgrp.o  |-  .o.  =  ( +g  `  M )
Assertion
Ref Expression
issgrp  |-  ( M  e. SGrp 
<->  ( M  e. Mgm  /\  A. x  e.  B  A. y  e.  B  A. z  e.  B  (
( x  .o.  y
)  .o.  z )  =  ( x  .o.  ( y  .o.  z
) ) ) )
Distinct variable groups:    x, B, y, z    x, M, y, z    x,  .o. , y, z

Proof of Theorem issgrp
Dummy variables  b 
g  o are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvex 5882 . . . 4  |-  ( Base `  g )  e.  _V
21a1i 11 . . 3  |-  ( g  =  M  ->  ( Base `  g )  e. 
_V )
3 fveq2 5872 . . . 4  |-  ( g  =  M  ->  ( Base `  g )  =  ( Base `  M
) )
4 issgrp.b . . . 4  |-  B  =  ( Base `  M
)
53, 4syl6eqr 2516 . . 3  |-  ( g  =  M  ->  ( Base `  g )  =  B )
6 fvex 5882 . . . . 5  |-  ( +g  `  g )  e.  _V
76a1i 11 . . . 4  |-  ( ( g  =  M  /\  b  =  B )  ->  ( +g  `  g
)  e.  _V )
8 fveq2 5872 . . . . . 6  |-  ( g  =  M  ->  ( +g  `  g )  =  ( +g  `  M
) )
98adantr 465 . . . . 5  |-  ( ( g  =  M  /\  b  =  B )  ->  ( +g  `  g
)  =  ( +g  `  M ) )
10 issgrp.o . . . . 5  |-  .o.  =  ( +g  `  M )
119, 10syl6eqr 2516 . . . 4  |-  ( ( g  =  M  /\  b  =  B )  ->  ( +g  `  g
)  =  .o.  )
12 simplr 755 . . . . 5  |-  ( ( ( g  =  M  /\  b  =  B )  /\  o  =  .o.  )  ->  b  =  B )
13 id 22 . . . . . . . . . 10  |-  ( o  =  .o.  ->  o  =  .o.  )
14 oveq 6302 . . . . . . . . . 10  |-  ( o  =  .o.  ->  (
x o y )  =  ( x  .o.  y ) )
15 eqidd 2458 . . . . . . . . . 10  |-  ( o  =  .o.  ->  z  =  z )
1613, 14, 15oveq123d 6317 . . . . . . . . 9  |-  ( o  =  .o.  ->  (
( x o y ) o z )  =  ( ( x  .o.  y )  .o.  z ) )
17 eqidd 2458 . . . . . . . . . 10  |-  ( o  =  .o.  ->  x  =  x )
18 oveq 6302 . . . . . . . . . 10  |-  ( o  =  .o.  ->  (
y o z )  =  ( y  .o.  z ) )
1913, 17, 18oveq123d 6317 . . . . . . . . 9  |-  ( o  =  .o.  ->  (
x o ( y o z ) )  =  ( x  .o.  ( y  .o.  z
) ) )
2016, 19eqeq12d 2479 . . . . . . . 8  |-  ( o  =  .o.  ->  (
( ( x o y ) o z )  =  ( x o ( y o z ) )  <->  ( (
x  .o.  y )  .o.  z )  =  ( x  .o.  ( y  .o.  z ) ) ) )
2120adantl 466 . . . . . . 7  |-  ( ( ( g  =  M  /\  b  =  B )  /\  o  =  .o.  )  ->  (
( ( x o y ) o z )  =  ( x o ( y o z ) )  <->  ( (
x  .o.  y )  .o.  z )  =  ( x  .o.  ( y  .o.  z ) ) ) )
2212, 21raleqbidv 3068 . . . . . 6  |-  ( ( ( g  =  M  /\  b  =  B )  /\  o  =  .o.  )  ->  ( A. z  e.  b 
( ( x o y ) o z )  =  ( x o ( y o z ) )  <->  A. z  e.  B  ( (
x  .o.  y )  .o.  z )  =  ( x  .o.  ( y  .o.  z ) ) ) )
2312, 22raleqbidv 3068 . . . . 5  |-  ( ( ( g  =  M  /\  b  =  B )  /\  o  =  .o.  )  ->  ( A. y  e.  b  A. z  e.  b 
( ( x o y ) o z )  =  ( x o ( y o z ) )  <->  A. y  e.  B  A. z  e.  B  ( (
x  .o.  y )  .o.  z )  =  ( x  .o.  ( y  .o.  z ) ) ) )
2412, 23raleqbidv 3068 . . . 4  |-  ( ( ( g  =  M  /\  b  =  B )  /\  o  =  .o.  )  ->  ( A. x  e.  b  A. y  e.  b  A. z  e.  b 
( ( x o y ) o z )  =  ( x o ( y o z ) )  <->  A. x  e.  B  A. y  e.  B  A. z  e.  B  ( (
x  .o.  y )  .o.  z )  =  ( x  .o.  ( y  .o.  z ) ) ) )
257, 11, 24sbcied2 3365 . . 3  |-  ( ( g  =  M  /\  b  =  B )  ->  ( [. ( +g  `  g )  /  o ]. A. x  e.  b 
A. y  e.  b 
A. z  e.  b  ( ( x o y ) o z )  =  ( x o ( y o z ) )  <->  A. x  e.  B  A. y  e.  B  A. z  e.  B  ( (
x  .o.  y )  .o.  z )  =  ( x  .o.  ( y  .o.  z ) ) ) )
262, 5, 25sbcied2 3365 . 2  |-  ( g  =  M  ->  ( [. ( Base `  g
)  /  b ]. [. ( +g  `  g
)  /  o ]. A. x  e.  b  A. y  e.  b  A. z  e.  b 
( ( x o y ) o z )  =  ( x o ( y o z ) )  <->  A. x  e.  B  A. y  e.  B  A. z  e.  B  ( (
x  .o.  y )  .o.  z )  =  ( x  .o.  ( y  .o.  z ) ) ) )
27 df-sgrp 16129 . 2  |- SGrp  =  {
g  e. Mgm  |  [. ( Base `  g )  / 
b ]. [. ( +g  `  g )  /  o ]. A. x  e.  b 
A. y  e.  b 
A. z  e.  b  ( ( x o y ) o z )  =  ( x o ( y o z ) ) }
2826, 27elrab2 3259 1  |-  ( M  e. SGrp 
<->  ( M  e. Mgm  /\  A. x  e.  B  A. y  e.  B  A. z  e.  B  (
( x  .o.  y
)  .o.  z )  =  ( x  .o.  ( y  .o.  z
) ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   A.wral 2807   _Vcvv 3109   [.wsbc 3327   ` cfv 5594  (class class class)co 6296   Basecbs 14735   +g cplusg 14803  Mgmcmgm 16088  SGrpcsgrp 16128
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-nul 4586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-iota 5557  df-fv 5602  df-ov 6299  df-sgrp 16129
This theorem is referenced by:  issgrpv  16131  issgrpn0  16132  isnsgrp  16133  sgrpmgm  16134  sgrpass  16135  sgrp1  16136  sgrp2nmndlem4  16264  copissgrp  32801  nnsgrp  32810  sgrpplusgaopALT  32824  sgrp2sgrp  32857  lidlmsgrp  32919  2zrngasgrp  32933  2zrngmsgrp  32940
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