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Theorem exidu1 21906
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 21905 . 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 444 . . . . . . . 8  |-  ( ( ( u G x )  =  x  /\  ( x G u )  =  x )  ->  ( u G x )  =  x )
43ralimi 2773 . . . . . . 7  |-  ( A. x  e.  X  (
( u G x )  =  x  /\  ( x G u )  =  x )  ->  A. x  e.  X  ( u G x )  =  x )
5 oveq2 6081 . . . . . . . . 9  |-  ( x  =  y  ->  (
u G x )  =  ( u G y ) )
6 id 20 . . . . . . . . 9  |-  ( x  =  y  ->  x  =  y )
75, 6eqeq12d 2449 . . . . . . . 8  |-  ( x  =  y  ->  (
( u G x )  =  x  <->  ( u G y )  =  y ) )
87rspcv 3040 . . . . . . 7  |-  ( y  e.  X  ->  ( A. x  e.  X  ( u G x )  =  x  -> 
( u G y )  =  y ) )
94, 8syl5 30 . . . . . 6  |-  ( y  e.  X  ->  ( A. x  e.  X  ( ( u G x )  =  x  /\  ( x G u )  =  x )  ->  ( u G y )  =  y ) )
10 simpr 448 . . . . . . . 8  |-  ( ( ( y G x )  =  x  /\  ( x G y )  =  x )  ->  ( x G y )  =  x )
1110ralimi 2773 . . . . . . 7  |-  ( A. x  e.  X  (
( y G x )  =  x  /\  ( x G y )  =  x )  ->  A. x  e.  X  ( x G y )  =  x )
12 oveq1 6080 . . . . . . . . 9  |-  ( x  =  u  ->  (
x G y )  =  ( u G y ) )
13 id 20 . . . . . . . . 9  |-  ( x  =  u  ->  x  =  u )
1412, 13eqeq12d 2449 . . . . . . . 8  |-  ( x  =  u  ->  (
( x G y )  =  x  <->  ( u G y )  =  u ) )
1514rspcv 3040 . . . . . . 7  |-  ( u  e.  X  ->  ( A. x  e.  X  ( x G y )  =  x  -> 
( u G y )  =  u ) )
1611, 15syl5 30 . . . . . 6  |-  ( u  e.  X  ->  ( A. x  e.  X  ( ( y G x )  =  x  /\  ( x G y )  =  x )  ->  ( u G y )  =  u ) )
179, 16im2anan9r 810 . . . . 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 2453 . . . . . 6  |-  ( ( ( u G y )  =  y  /\  ( u G y )  =  u )  ->  y  =  u )
1918eqcomd 2440 . . . . 5  |-  ( ( ( u G y )  =  y  /\  ( u G y )  =  u )  ->  u  =  y )
2017, 19syl6 31 . . . 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 2764 . . 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 11 . 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 6080 . . . . . 6  |-  ( u  =  y  ->  (
u G x )  =  ( y G x ) )
2423eqeq1d 2443 . . . . 5  |-  ( u  =  y  ->  (
( u G x )  =  x  <->  ( y G x )  =  x ) )
25 oveq2 6081 . . . . . 6  |-  ( u  =  y  ->  (
x G u )  =  ( x G y ) )
2625eqeq1d 2443 . . . . 5  |-  ( u  =  y  ->  (
( x G u )  =  x  <->  ( x G y )  =  x ) )
2724, 26anbi12d 692 . . . 4  |-  ( u  =  y  ->  (
( ( u G x )  =  x  /\  ( x G u )  =  x )  <->  ( ( y G x )  =  x  /\  ( x G y )  =  x ) ) )
2827ralbidv 2717 . . 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 3120 . 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 646 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 359    = wceq 1652    e. wcel 1725   A.wral 2697   E.wrex 2698   E!wreu 2699    i^i cin 3311   ran crn 4871  (class class class)co 6073    ExId cexid 21894   Magmacmagm 21898
This theorem is referenced by:  iorlid  21908  cmpidelt  21909  exidresid  26545
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fo 5452  df-fv 5454  df-ov 6076  df-exid 21895  df-mgm 21899
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