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Theorem isexid2 23812
Description: If  G  e.  ( Magma  i^i  ExId  ), then it has a left and right identity element that belongs to the range of the operation. (Contributed by FL, 12-Dec-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
Hypothesis
Ref Expression
isexid2.1  |-  X  =  ran  G
Assertion
Ref Expression
isexid2  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  E. u  e.  X  A. x  e.  X  ( ( u G x )  =  x  /\  ( x G u )  =  x ) )
Distinct variable groups:    u, G, x    u, X, x

Proof of Theorem isexid2
StepHypRef Expression
1 isexid2.1 . 2  |-  X  =  ran  G
2 rngopid 23810 . . . . 5  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  ran  G  =  dom  dom  G )
3 elin 3539 . . . . . . 7  |-  ( G  e.  ( Magma  i^i  ExId  )  <-> 
( G  e.  Magma  /\  G  e.  ExId  )
)
4 eqid 2443 . . . . . . . . . . 11  |-  dom  dom  G  =  dom  dom  G
54isexid 23804 . . . . . . . . . 10  |-  ( G  e.  ExId  ->  ( G  e.  ExId  <->  E. u  e.  dom  dom 
G A. x  e. 
dom  dom  G ( ( u G x )  =  x  /\  (
x G u )  =  x ) ) )
65ibi 241 . . . . . . . . 9  |-  ( G  e.  ExId  ->  E. u  e.  dom  dom  G A. x  e.  dom  dom  G
( ( u G x )  =  x  /\  ( x G u )  =  x ) )
76a1d 25 . . . . . . . 8  |-  ( G  e.  ExId  ->  ( X  =  dom  dom  G  ->  E. u  e.  dom  dom 
G A. x  e. 
dom  dom  G ( ( u G x )  =  x  /\  (
x G u )  =  x ) ) )
87adantl 466 . . . . . . 7  |-  ( ( G  e.  Magma  /\  G  e.  ExId  )  ->  ( X  =  dom  dom  G  ->  E. u  e.  dom  dom 
G A. x  e. 
dom  dom  G ( ( u G x )  =  x  /\  (
x G u )  =  x ) ) )
93, 8sylbi 195 . . . . . 6  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  ( X  =  dom  dom  G  ->  E. u  e.  dom  dom  G A. x  e.  dom  dom 
G ( ( u G x )  =  x  /\  ( x G u )  =  x ) ) )
10 eqeq2 2452 . . . . . . 7  |-  ( ran 
G  =  dom  dom  G  ->  ( X  =  ran  G  <->  X  =  dom  dom  G ) )
11 raleq 2917 . . . . . . . 8  |-  ( ran 
G  =  dom  dom  G  ->  ( A. x  e.  ran  G ( ( u G x )  =  x  /\  (
x G u )  =  x )  <->  A. x  e.  dom  dom  G (
( u G x )  =  x  /\  ( x G u )  =  x ) ) )
1211rexeqbi1dv 2926 . . . . . . 7  |-  ( ran 
G  =  dom  dom  G  ->  ( E. u  e.  ran  G A. x  e.  ran  G ( ( u G x )  =  x  /\  (
x G u )  =  x )  <->  E. u  e.  dom  dom  G A. x  e.  dom  dom  G
( ( u G x )  =  x  /\  ( x G u )  =  x ) ) )
1310, 12imbi12d 320 . . . . . 6  |-  ( ran 
G  =  dom  dom  G  ->  ( ( X  =  ran  G  ->  E. u  e.  ran  G A. x  e.  ran  G ( ( u G x )  =  x  /\  ( x G u )  =  x ) )  <->  ( X  =  dom  dom  G  ->  E. u  e.  dom  dom  G A. x  e.  dom  dom 
G ( ( u G x )  =  x  /\  ( x G u )  =  x ) ) ) )
149, 13syl5ibr 221 . . . . 5  |-  ( ran 
G  =  dom  dom  G  ->  ( G  e.  ( Magma  i^i  ExId  )  ->  ( X  =  ran  G  ->  E. u  e.  ran  G A. x  e.  ran  G ( ( u G x )  =  x  /\  ( x G u )  =  x ) ) ) )
152, 14mpcom 36 . . . 4  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  ( X  =  ran  G  ->  E. u  e.  ran  G A. x  e.  ran  G ( ( u G x )  =  x  /\  (
x G u )  =  x ) ) )
1615com12 31 . . 3  |-  ( X  =  ran  G  -> 
( G  e.  (
Magma  i^i  ExId  )  ->  E. u  e.  ran  G A. x  e.  ran  G ( ( u G x )  =  x  /\  ( x G u )  =  x ) ) )
17 raleq 2917 . . . 4  |-  ( X  =  ran  G  -> 
( A. x  e.  X  ( ( u G x )  =  x  /\  ( x G u )  =  x )  <->  A. x  e.  ran  G ( ( u G x )  =  x  /\  (
x G u )  =  x ) ) )
1817rexeqbi1dv 2926 . . 3  |-  ( X  =  ran  G  -> 
( E. u  e.  X  A. x  e.  X  ( ( u G x )  =  x  /\  ( x G u )  =  x )  <->  E. u  e.  ran  G A. x  e.  ran  G ( ( u G x )  =  x  /\  (
x G u )  =  x ) ) )
1916, 18sylibrd 234 . 2  |-  ( X  =  ran  G  -> 
( G  e.  (
Magma  i^i  ExId  )  ->  E. u  e.  X  A. x  e.  X  (
( u G x )  =  x  /\  ( x G u )  =  x ) ) )
201, 19ax-mp 5 1  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  E. u  e.  X  A. x  e.  X  ( ( u G x )  =  x  /\  ( x G u )  =  x ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2715   E.wrex 2716    i^i cin 3327   dom cdm 4840   ran crn 4841  (class class class)co 6091    ExId cexid 23801   Magmacmagm 23805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-fo 5424  df-fv 5426  df-ov 6094  df-exid 23802  df-mgm 23806
This theorem is referenced by:  exidu1  23813
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