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Theorem issgrp 12701
Description: The predicate "is a semigroup". (Contributed by FL, 2-Nov-2009.) (Revised by AV, 6-Jan-2020.)
Hypotheses
Ref Expression
issgrp.b  |-  B  =  ( Base `  M
)
issgrp.o  |-  .o.  =  ( +g  `  M )
Assertion
Ref Expression
issgrp  |-  ( M  e. Smgrp 
<->  ( 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 basfn 12502 . . . . 5  |-  Base  Fn  _V
2 vex 2740 . . . . 5  |-  g  e. 
_V
3 funfvex 5528 . . . . . 6  |-  ( ( Fun  Base  /\  g  e.  dom  Base )  ->  ( Base `  g )  e. 
_V )
43funfni 5312 . . . . 5  |-  ( (
Base  Fn  _V  /\  g  e.  _V )  ->  ( Base `  g )  e. 
_V )
51, 2, 4mp2an 426 . . . 4  |-  ( Base `  g )  e.  _V
65a1i 9 . . 3  |-  ( g  =  M  ->  ( Base `  g )  e. 
_V )
7 fveq2 5511 . . . 4  |-  ( g  =  M  ->  ( Base `  g )  =  ( Base `  M
) )
8 issgrp.b . . . 4  |-  B  =  ( Base `  M
)
97, 8eqtr4di 2228 . . 3  |-  ( g  =  M  ->  ( Base `  g )  =  B )
10 plusgslid 12551 . . . . . . 7  |-  ( +g  = Slot  ( +g  `  ndx )  /\  ( +g  `  ndx )  e.  NN )
1110slotex 12472 . . . . . 6  |-  ( g  e.  _V  ->  ( +g  `  g )  e. 
_V )
1211elv 2741 . . . . 5  |-  ( +g  `  g )  e.  _V
1312a1i 9 . . . 4  |-  ( ( g  =  M  /\  b  =  B )  ->  ( +g  `  g
)  e.  _V )
14 fveq2 5511 . . . . . 6  |-  ( g  =  M  ->  ( +g  `  g )  =  ( +g  `  M
) )
1514adantr 276 . . . . 5  |-  ( ( g  =  M  /\  b  =  B )  ->  ( +g  `  g
)  =  ( +g  `  M ) )
16 issgrp.o . . . . 5  |-  .o.  =  ( +g  `  M )
1715, 16eqtr4di 2228 . . . 4  |-  ( ( g  =  M  /\  b  =  B )  ->  ( +g  `  g
)  =  .o.  )
18 simplr 528 . . . . 5  |-  ( ( ( g  =  M  /\  b  =  B )  /\  o  =  .o.  )  ->  b  =  B )
19 id 19 . . . . . . . . . 10  |-  ( o  =  .o.  ->  o  =  .o.  )
20 oveq 5875 . . . . . . . . . 10  |-  ( o  =  .o.  ->  (
x o y )  =  ( x  .o.  y ) )
21 eqidd 2178 . . . . . . . . . 10  |-  ( o  =  .o.  ->  z  =  z )
2219, 20, 21oveq123d 5890 . . . . . . . . 9  |-  ( o  =  .o.  ->  (
( x o y ) o z )  =  ( ( x  .o.  y )  .o.  z ) )
23 eqidd 2178 . . . . . . . . . 10  |-  ( o  =  .o.  ->  x  =  x )
24 oveq 5875 . . . . . . . . . 10  |-  ( o  =  .o.  ->  (
y o z )  =  ( y  .o.  z ) )
2519, 23, 24oveq123d 5890 . . . . . . . . 9  |-  ( o  =  .o.  ->  (
x o ( y o z ) )  =  ( x  .o.  ( y  .o.  z
) ) )
2622, 25eqeq12d 2192 . . . . . . . 8  |-  ( o  =  .o.  ->  (
( ( x o y ) o z )  =  ( x o ( y o z ) )  <->  ( (
x  .o.  y )  .o.  z )  =  ( x  .o.  ( y  .o.  z ) ) ) )
2726adantl 277 . . . . . . 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 ) ) ) )
2818, 27raleqbidv 2684 . . . . . 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 ) ) ) )
2918, 28raleqbidv 2684 . . . . 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 ) ) ) )
3018, 29raleqbidv 2684 . . . 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 ) ) ) )
3113, 17, 30sbcied2 3000 . . 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 ) ) ) )
326, 9, 31sbcied2 3000 . 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 ) ) ) )
33 df-sgrp 12700 . 2  |- Smgrp  =  {
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 ) ) }
3432, 33elrab2 2896 1  |-  ( M  e. Smgrp 
<->  ( 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 set class
Syntax hints:    /\ wa 104    <-> wb 105    = wceq 1353    e. wcel 2148   A.wral 2455   _Vcvv 2737   [.wsbc 2962    Fn wfn 5207   ` 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:  issgrpv  12702  issgrpn0  12703  isnsgrp  12704  sgrpmgm  12705  sgrpass  12706  sgrp0  12707  sgrp1  12708
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