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Theorem exidu1 20993
Description: Unicity of the left and right identity element of a magma when it exists. (Contributed by FL, 12-Dec-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
Hypothesis
Ref Expression
exidu1.1  |-  X  =  ran  G
Assertion
Ref Expression
exidu1  |-  ( 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 exidu1
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 exidu1.1 . . 3  |-  X  =  ran  G
21isexid2 20992 . 2  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  E. u  e.  X  A. x  e.  X  ( ( u G x )  =  x  /\  ( x G u )  =  x ) )
3 simpl 443 . . . . . . . 8  |-  ( ( ( u G x )  =  x  /\  ( x G u )  =  x )  ->  ( u G x )  =  x )
43ralimi 2618 . . . . . . 7  |-  ( A. x  e.  X  (
( u G x )  =  x  /\  ( x G u )  =  x )  ->  A. x  e.  X  ( u G x )  =  x )
5 oveq2 5866 . . . . . . . . 9  |-  ( x  =  y  ->  (
u G x )  =  ( u G y ) )
6 id 19 . . . . . . . . 9  |-  ( x  =  y  ->  x  =  y )
75, 6eqeq12d 2297 . . . . . . . 8  |-  ( x  =  y  ->  (
( u G x )  =  x  <->  ( u G y )  =  y ) )
87rspcv 2880 . . . . . . 7  |-  ( y  e.  X  ->  ( A. x  e.  X  ( u G x )  =  x  -> 
( u G y )  =  y ) )
94, 8syl5 28 . . . . . 6  |-  ( y  e.  X  ->  ( A. x  e.  X  ( ( u G x )  =  x  /\  ( x G u )  =  x )  ->  ( u G y )  =  y ) )
10 simpr 447 . . . . . . . 8  |-  ( ( ( y G x )  =  x  /\  ( x G y )  =  x )  ->  ( x G y )  =  x )
1110ralimi 2618 . . . . . . 7  |-  ( A. x  e.  X  (
( y G x )  =  x  /\  ( x G y )  =  x )  ->  A. x  e.  X  ( x G y )  =  x )
12 oveq1 5865 . . . . . . . . 9  |-  ( x  =  u  ->  (
x G y )  =  ( u G y ) )
13 id 19 . . . . . . . . 9  |-  ( x  =  u  ->  x  =  u )
1412, 13eqeq12d 2297 . . . . . . . 8  |-  ( x  =  u  ->  (
( x G y )  =  x  <->  ( u G y )  =  u ) )
1514rspcv 2880 . . . . . . 7  |-  ( u  e.  X  ->  ( A. x  e.  X  ( x G y )  =  x  -> 
( u G y )  =  u ) )
1611, 15syl5 28 . . . . . 6  |-  ( u  e.  X  ->  ( A. x  e.  X  ( ( y G x )  =  x  /\  ( x G y )  =  x )  ->  ( u G y )  =  u ) )
179, 16im2anan9r 809 . . . . 5  |-  ( ( u  e.  X  /\  y  e.  X )  ->  ( ( A. x  e.  X  ( (
u G x )  =  x  /\  (
x G u )  =  x )  /\  A. x  e.  X  ( ( y G x )  =  x  /\  ( x G y )  =  x ) )  ->  ( (
u G y )  =  y  /\  (
u G y )  =  u ) ) )
18 eqtr2 2301 . . . . . 6  |-  ( ( ( u G y )  =  y  /\  ( u G y )  =  u )  ->  y  =  u )
1918eqcomd 2288 . . . . 5  |-  ( ( ( u G y )  =  y  /\  ( u G y )  =  u )  ->  u  =  y )
2017, 19syl6 29 . . . 4  |-  ( ( u  e.  X  /\  y  e.  X )  ->  ( ( A. x  e.  X  ( (
u G x )  =  x  /\  (
x G u )  =  x )  /\  A. x  e.  X  ( ( y G x )  =  x  /\  ( x G y )  =  x ) )  ->  u  =  y ) )
2120rgen2a 2609 . . 3  |-  A. u  e.  X  A. y  e.  X  ( ( A. x  e.  X  ( ( u G x )  =  x  /\  ( x G u )  =  x )  /\  A. x  e.  X  ( (
y G x )  =  x  /\  (
x G y )  =  x ) )  ->  u  =  y )
2221a1i 10 . 2  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  A. u  e.  X  A. y  e.  X  ( ( A. x  e.  X  ( (
u G x )  =  x  /\  (
x G u )  =  x )  /\  A. x  e.  X  ( ( y G x )  =  x  /\  ( x G y )  =  x ) )  ->  u  =  y ) )
23 oveq1 5865 . . . . . 6  |-  ( u  =  y  ->  (
u G x )  =  ( y G x ) )
2423eqeq1d 2291 . . . . 5  |-  ( u  =  y  ->  (
( u G x )  =  x  <->  ( y G x )  =  x ) )
25 oveq2 5866 . . . . . 6  |-  ( u  =  y  ->  (
x G u )  =  ( x G y ) )
2625eqeq1d 2291 . . . . 5  |-  ( u  =  y  ->  (
( x G u )  =  x  <->  ( x G y )  =  x ) )
2724, 26anbi12d 691 . . . 4  |-  ( u  =  y  ->  (
( ( u G x )  =  x  /\  ( x G u )  =  x )  <->  ( ( y G x )  =  x  /\  ( x G y )  =  x ) ) )
2827ralbidv 2563 . . 3  |-  ( u  =  y  ->  ( A. x  e.  X  ( ( u G x )  =  x  /\  ( x G u )  =  x )  <->  A. x  e.  X  ( ( y G x )  =  x  /\  ( x G y )  =  x ) ) )
2928reu4 2959 . 2  |-  ( E! u  e.  X  A. x  e.  X  (
( u G x )  =  x  /\  ( x G u )  =  x )  <-> 
( E. u  e.  X  A. x  e.  X  ( ( u G x )  =  x  /\  ( x G u )  =  x )  /\  A. u  e.  X  A. y  e.  X  (
( A. x  e.  X  ( ( u G x )  =  x  /\  ( x G u )  =  x )  /\  A. x  e.  X  (
( y G x )  =  x  /\  ( x G y )  =  x ) )  ->  u  =  y ) ) )
302, 22, 29sylanbrc 645 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 set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   E!wreu 2545    i^i cin 3151   ran crn 4690  (class class class)co 5858    ExId cexid 20981   Magmacmagm 20985
This theorem is referenced by:  iorlid  20995  cmpidelt  20996  ununr  25420  exidresid  26569
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-ov 5861  df-exid 20982  df-mgm 20986
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