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Theorem exidu1 21046
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 21045 . 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 2652 . . . . . . 7  |-  ( A. x  e.  X  (
( u G x )  =  x  /\  ( x G u )  =  x )  ->  A. x  e.  X  ( u G x )  =  x )
5 oveq2 5908 . . . . . . . . 9  |-  ( x  =  y  ->  (
u G x )  =  ( u G y ) )
6 id 19 . . . . . . . . 9  |-  ( x  =  y  ->  x  =  y )
75, 6eqeq12d 2330 . . . . . . . 8  |-  ( x  =  y  ->  (
( u G x )  =  x  <->  ( u G y )  =  y ) )
87rspcv 2914 . . . . . . 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 2652 . . . . . . 7  |-  ( A. x  e.  X  (
( y G x )  =  x  /\  ( x G y )  =  x )  ->  A. x  e.  X  ( x G y )  =  x )
12 oveq1 5907 . . . . . . . . 9  |-  ( x  =  u  ->  (
x G y )  =  ( u G y ) )
13 id 19 . . . . . . . . 9  |-  ( x  =  u  ->  x  =  u )
1412, 13eqeq12d 2330 . . . . . . . 8  |-  ( x  =  u  ->  (
( x G y )  =  x  <->  ( u G y )  =  u ) )
1514rspcv 2914 . . . . . . 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 2334 . . . . . 6  |-  ( ( ( u G y )  =  y  /\  ( u G y )  =  u )  ->  y  =  u )
1918eqcomd 2321 . . . . 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 2643 . . 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 5907 . . . . . 6  |-  ( u  =  y  ->  (
u G x )  =  ( y G x ) )
2423eqeq1d 2324 . . . . 5  |-  ( u  =  y  ->  (
( u G x )  =  x  <->  ( y G x )  =  x ) )
25 oveq2 5908 . . . . . 6  |-  ( u  =  y  ->  (
x G u )  =  ( x G y ) )
2625eqeq1d 2324 . . . . 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 2597 . . 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 2993 . 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 1633    e. wcel 1701   A.wral 2577   E.wrex 2578   E!wreu 2579    i^i cin 3185   ran crn 4727  (class class class)co 5900    ExId cexid 21034   Magmacmagm 21038
This theorem is referenced by:  iorlid  21048  cmpidelt  21049  exidresid  25717
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-fo 5298  df-fv 5300  df-ov 5903  df-exid 21035  df-mgm 21039
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