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Theorem grpoidinv2 20901
Description: A group's properties using the explicit identity element. (Contributed by NM, 5-Feb-2010.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
grpoidval.1  |-  X  =  ran  G
grpoidval.2  |-  U  =  (GId `  G )
Assertion
Ref Expression
grpoidinv2  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( ( U G A )  =  A  /\  ( A G U )  =  A )  /\  E. y  e.  X  ( (
y G A )  =  U  /\  ( A G y )  =  U ) ) )
Distinct variable groups:    y, A    y, G    y, U    y, X

Proof of Theorem grpoidinv2
Dummy variables  x  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 grpoidval.1 . . . . . . 7  |-  X  =  ran  G
2 grpoidval.2 . . . . . . 7  |-  U  =  (GId `  G )
31, 2grpoidval 20899 . . . . . 6  |-  ( G  e.  GrpOp  ->  U  =  ( iota_ u  e.  X A. x  e.  X  ( u G x )  =  x ) )
41grpoideu 20892 . . . . . . 7  |-  ( G  e.  GrpOp  ->  E! u  e.  X  A. x  e.  X  ( u G x )  =  x )
5 riotacl2 6334 . . . . . . 7  |-  ( E! u  e.  X  A. x  e.  X  (
u G x )  =  x  ->  ( iota_ u  e.  X A. x  e.  X  (
u G x )  =  x )  e. 
{ u  e.  X  |  A. x  e.  X  ( u G x )  =  x }
)
64, 5syl 15 . . . . . 6  |-  ( G  e.  GrpOp  ->  ( iota_ u  e.  X A. x  e.  X  ( u G x )  =  x )  e.  {
u  e.  X  |  A. x  e.  X  ( u G x )  =  x }
)
73, 6eqeltrd 2370 . . . . 5  |-  ( G  e.  GrpOp  ->  U  e.  { u  e.  X  |  A. x  e.  X  ( u G x )  =  x }
)
8 simpll 730 . . . . . . . . . . 11  |-  ( ( ( ( u G x )  =  x  /\  ( x G u )  =  x )  /\  E. y  e.  X  ( (
y G x )  =  u  /\  (
x G y )  =  u ) )  ->  ( u G x )  =  x )
98ralimi 2631 . . . . . . . . . 10  |-  ( A. x  e.  X  (
( ( u G x )  =  x  /\  ( x G u )  =  x )  /\  E. y  e.  X  ( (
y G x )  =  u  /\  (
x G y )  =  u ) )  ->  A. x  e.  X  ( u G x )  =  x )
109rgenw 2623 . . . . . . . . 9  |-  A. u  e.  X  ( A. x  e.  X  (
( ( u G x )  =  x  /\  ( x G u )  =  x )  /\  E. y  e.  X  ( (
y G x )  =  u  /\  (
x G y )  =  u ) )  ->  A. x  e.  X  ( u G x )  =  x )
1110a1i 10 . . . . . . . 8  |-  ( G  e.  GrpOp  ->  A. u  e.  X  ( A. x  e.  X  (
( ( u G x )  =  x  /\  ( x G u )  =  x )  /\  E. y  e.  X  ( (
y G x )  =  u  /\  (
x G y )  =  u ) )  ->  A. x  e.  X  ( u G x )  =  x ) )
121grpoidinv 20891 . . . . . . . 8  |-  ( G  e.  GrpOp  ->  E. u  e.  X  A. x  e.  X  ( (
( u G x )  =  x  /\  ( x G u )  =  x )  /\  E. y  e.  X  ( ( y G x )  =  u  /\  ( x G y )  =  u ) ) )
1311, 12, 43jca 1132 . . . . . . 7  |-  ( G  e.  GrpOp  ->  ( A. u  e.  X  ( A. x  e.  X  ( ( ( u G x )  =  x  /\  ( x G u )  =  x )  /\  E. y  e.  X  (
( y G x )  =  u  /\  ( x G y )  =  u ) )  ->  A. x  e.  X  ( u G x )  =  x )  /\  E. u  e.  X  A. x  e.  X  (
( ( u G x )  =  x  /\  ( x G u )  =  x )  /\  E. y  e.  X  ( (
y G x )  =  u  /\  (
x G y )  =  u ) )  /\  E! u  e.  X  A. x  e.  X  ( u G x )  =  x ) )
14 reupick2 3467 . . . . . . 7  |-  ( ( ( A. u  e.  X  ( A. x  e.  X  ( (
( u G x )  =  x  /\  ( x G u )  =  x )  /\  E. y  e.  X  ( ( y G x )  =  u  /\  ( x G y )  =  u ) )  ->  A. x  e.  X  ( u G x )  =  x )  /\  E. u  e.  X  A. x  e.  X  ( ( ( u G x )  =  x  /\  (
x G u )  =  x )  /\  E. y  e.  X  ( ( y G x )  =  u  /\  ( x G y )  =  u ) )  /\  E! u  e.  X  A. x  e.  X  ( u G x )  =  x )  /\  u  e.  X )  ->  ( A. x  e.  X  ( u G x )  =  x  <->  A. x  e.  X  ( (
( u G x )  =  x  /\  ( x G u )  =  x )  /\  E. y  e.  X  ( ( y G x )  =  u  /\  ( x G y )  =  u ) ) ) )
1513, 14sylan 457 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  u  e.  X )  ->  ( A. x  e.  X  ( u G x )  =  x  <->  A. x  e.  X  ( (
( u G x )  =  x  /\  ( x G u )  =  x )  /\  E. y  e.  X  ( ( y G x )  =  u  /\  ( x G y )  =  u ) ) ) )
1615rabbidva 2792 . . . . 5  |-  ( G  e.  GrpOp  ->  { u  e.  X  |  A. x  e.  X  (
u G x )  =  x }  =  { u  e.  X  |  A. x  e.  X  ( ( ( u G x )  =  x  /\  ( x G u )  =  x )  /\  E. y  e.  X  (
( y G x )  =  u  /\  ( x G y )  =  u ) ) } )
177, 16eleqtrd 2372 . . . 4  |-  ( G  e.  GrpOp  ->  U  e.  { u  e.  X  |  A. x  e.  X  ( ( ( u G x )  =  x  /\  ( x G u )  =  x )  /\  E. y  e.  X  (
( y G x )  =  u  /\  ( x G y )  =  u ) ) } )
18 oveq1 5881 . . . . . . . . 9  |-  ( u  =  U  ->  (
u G x )  =  ( U G x ) )
1918eqeq1d 2304 . . . . . . . 8  |-  ( u  =  U  ->  (
( u G x )  =  x  <->  ( U G x )  =  x ) )
20 oveq2 5882 . . . . . . . . 9  |-  ( u  =  U  ->  (
x G u )  =  ( x G U ) )
2120eqeq1d 2304 . . . . . . . 8  |-  ( u  =  U  ->  (
( x G u )  =  x  <->  ( x G U )  =  x ) )
2219, 21anbi12d 691 . . . . . . 7  |-  ( u  =  U  ->  (
( ( u G x )  =  x  /\  ( x G u )  =  x )  <->  ( ( U G x )  =  x  /\  ( x G U )  =  x ) ) )
23 eqeq2 2305 . . . . . . . . 9  |-  ( u  =  U  ->  (
( y G x )  =  u  <->  ( y G x )  =  U ) )
24 eqeq2 2305 . . . . . . . . 9  |-  ( u  =  U  ->  (
( x G y )  =  u  <->  ( x G y )  =  U ) )
2523, 24anbi12d 691 . . . . . . . 8  |-  ( u  =  U  ->  (
( ( y G x )  =  u  /\  ( x G y )  =  u )  <->  ( ( y G x )  =  U  /\  ( x G y )  =  U ) ) )
2625rexbidv 2577 . . . . . . 7  |-  ( u  =  U  ->  ( E. y  e.  X  ( ( y G x )  =  u  /\  ( x G y )  =  u )  <->  E. y  e.  X  ( ( y G x )  =  U  /\  ( x G y )  =  U ) ) )
2722, 26anbi12d 691 . . . . . 6  |-  ( u  =  U  ->  (
( ( ( u G x )  =  x  /\  ( x G u )  =  x )  /\  E. y  e.  X  (
( y G x )  =  u  /\  ( x G y )  =  u ) )  <->  ( ( ( U G x )  =  x  /\  (
x G U )  =  x )  /\  E. y  e.  X  ( ( y G x )  =  U  /\  ( x G y )  =  U ) ) ) )
2827ralbidv 2576 . . . . 5  |-  ( u  =  U  ->  ( A. x  e.  X  ( ( ( u G x )  =  x  /\  ( x G u )  =  x )  /\  E. y  e.  X  (
( y G x )  =  u  /\  ( x G y )  =  u ) )  <->  A. x  e.  X  ( ( ( U G x )  =  x  /\  ( x G U )  =  x )  /\  E. y  e.  X  (
( y G x )  =  U  /\  ( x G y )  =  U ) ) ) )
2928elrab 2936 . . . 4  |-  ( U  e.  { u  e.  X  |  A. x  e.  X  ( (
( u G x )  =  x  /\  ( x G u )  =  x )  /\  E. y  e.  X  ( ( y G x )  =  u  /\  ( x G y )  =  u ) ) }  <-> 
( U  e.  X  /\  A. x  e.  X  ( ( ( U G x )  =  x  /\  ( x G U )  =  x )  /\  E. y  e.  X  (
( y G x )  =  U  /\  ( x G y )  =  U ) ) ) )
3017, 29sylib 188 . . 3  |-  ( G  e.  GrpOp  ->  ( U  e.  X  /\  A. x  e.  X  ( (
( U G x )  =  x  /\  ( x G U )  =  x )  /\  E. y  e.  X  ( ( y G x )  =  U  /\  ( x G y )  =  U ) ) ) )
3130simprd 449 . 2  |-  ( G  e.  GrpOp  ->  A. x  e.  X  ( (
( U G x )  =  x  /\  ( x G U )  =  x )  /\  E. y  e.  X  ( ( y G x )  =  U  /\  ( x G y )  =  U ) ) )
32 oveq2 5882 . . . . . 6  |-  ( x  =  A  ->  ( U G x )  =  ( U G A ) )
33 id 19 . . . . . 6  |-  ( x  =  A  ->  x  =  A )
3432, 33eqeq12d 2310 . . . . 5  |-  ( x  =  A  ->  (
( U G x )  =  x  <->  ( U G A )  =  A ) )
35 oveq1 5881 . . . . . 6  |-  ( x  =  A  ->  (
x G U )  =  ( A G U ) )
3635, 33eqeq12d 2310 . . . . 5  |-  ( x  =  A  ->  (
( x G U )  =  x  <->  ( A G U )  =  A ) )
3734, 36anbi12d 691 . . . 4  |-  ( x  =  A  ->  (
( ( U G x )  =  x  /\  ( x G U )  =  x )  <->  ( ( U G A )  =  A  /\  ( A G U )  =  A ) ) )
38 oveq2 5882 . . . . . . 7  |-  ( x  =  A  ->  (
y G x )  =  ( y G A ) )
3938eqeq1d 2304 . . . . . 6  |-  ( x  =  A  ->  (
( y G x )  =  U  <->  ( y G A )  =  U ) )
40 oveq1 5881 . . . . . . 7  |-  ( x  =  A  ->  (
x G y )  =  ( A G y ) )
4140eqeq1d 2304 . . . . . 6  |-  ( x  =  A  ->  (
( x G y )  =  U  <->  ( A G y )  =  U ) )
4239, 41anbi12d 691 . . . . 5  |-  ( x  =  A  ->  (
( ( y G x )  =  U  /\  ( x G y )  =  U )  <->  ( ( y G A )  =  U  /\  ( A G y )  =  U ) ) )
4342rexbidv 2577 . . . 4  |-  ( x  =  A  ->  ( E. y  e.  X  ( ( y G x )  =  U  /\  ( x G y )  =  U )  <->  E. y  e.  X  ( ( y G A )  =  U  /\  ( A G y )  =  U ) ) )
4437, 43anbi12d 691 . . 3  |-  ( x  =  A  ->  (
( ( ( U G x )  =  x  /\  ( x G U )  =  x )  /\  E. y  e.  X  (
( y G x )  =  U  /\  ( x G y )  =  U ) )  <->  ( ( ( U G A )  =  A  /\  ( A G U )  =  A )  /\  E. y  e.  X  (
( y G A )  =  U  /\  ( A G y )  =  U ) ) ) )
4544rspccva 2896 . 2  |-  ( ( A. x  e.  X  ( ( ( U G x )  =  x  /\  ( x G U )  =  x )  /\  E. y  e.  X  (
( y G x )  =  U  /\  ( x G y )  =  U ) )  /\  A  e.  X )  ->  (
( ( U G A )  =  A  /\  ( A G U )  =  A )  /\  E. y  e.  X  ( (
y G A )  =  U  /\  ( A G y )  =  U ) ) )
4631, 45sylan 457 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( ( U G A )  =  A  /\  ( A G U )  =  A )  /\  E. y  e.  X  ( (
y G A )  =  U  /\  ( A G y )  =  U ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   E!wreu 2558   {crab 2560   ran crn 4706   ` cfv 5271  (class class class)co 5874   iota_crio 6313   GrpOpcgr 20869  GIdcgi 20870
This theorem is referenced by:  grpolid  20902  grporid  20903  grporcan  20904  grpoinveu  20905  grpoinv  20910
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-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  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-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  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-fo 5277  df-fv 5279  df-ov 5877  df-riota 6320  df-grpo 20874  df-gid 20875
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