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Theorem rngoideu 21972
Description: The unit element of a ring is unique. (Contributed by NM, 4-Apr-2009.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
ringi.1  |-  G  =  ( 1st `  R
)
ringi.2  |-  H  =  ( 2nd `  R
)
ringi.3  |-  X  =  ran  G
Assertion
Ref Expression
rngoideu  |-  ( R  e.  RingOps  ->  E! u  e.  X  A. x  e.  X  ( ( u H x )  =  x  /\  ( x H u )  =  x ) )
Distinct variable groups:    x, u, G    u, H, x    u, X, x    u, R, x

Proof of Theorem rngoideu
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 ringi.1 . . . . 5  |-  G  =  ( 1st `  R
)
2 ringi.2 . . . . 5  |-  H  =  ( 2nd `  R
)
3 ringi.3 . . . . 5  |-  X  =  ran  G
41, 2, 3rngoi 21968 . . . 4  |-  ( R  e.  RingOps  ->  ( ( G  e.  AbelOp  /\  H :
( X  X.  X
) --> X )  /\  ( A. u  e.  X  A. x  e.  X  A. y  e.  X  ( ( ( u H x ) H y )  =  ( u H ( x H y ) )  /\  ( u H ( x G y ) )  =  ( ( u H x ) G ( u H y ) )  /\  ( ( u G x ) H y )  =  ( ( u H y ) G ( x H y ) ) )  /\  E. u  e.  X  A. x  e.  X  ( (
u H x )  =  x  /\  (
x H u )  =  x ) ) ) )
5 simprr 734 . . . 4  |-  ( ( ( G  e.  AbelOp  /\  H : ( X  X.  X ) --> X )  /\  ( A. u  e.  X  A. x  e.  X  A. y  e.  X  (
( ( u H x ) H y )  =  ( u H ( x H y ) )  /\  ( u H ( x G y ) )  =  ( ( u H x ) G ( u H y ) )  /\  ( ( u G x ) H y )  =  ( ( u H y ) G ( x H y ) ) )  /\  E. u  e.  X  A. x  e.  X  ( ( u H x )  =  x  /\  ( x H u )  =  x ) ) )  ->  E. u  e.  X  A. x  e.  X  ( ( u H x )  =  x  /\  ( x H u )  =  x ) )
64, 5syl 16 . . 3  |-  ( R  e.  RingOps  ->  E. u  e.  X  A. x  e.  X  ( ( u H x )  =  x  /\  ( x H u )  =  x ) )
7 simpl 444 . . . . . . . 8  |-  ( ( ( u H x )  =  x  /\  ( x H u )  =  x )  ->  ( u H x )  =  x )
87ralimi 2781 . . . . . . 7  |-  ( A. x  e.  X  (
( u H x )  =  x  /\  ( x H u )  =  x )  ->  A. x  e.  X  ( u H x )  =  x )
9 oveq2 6089 . . . . . . . . 9  |-  ( x  =  y  ->  (
u H x )  =  ( u H y ) )
10 id 20 . . . . . . . . 9  |-  ( x  =  y  ->  x  =  y )
119, 10eqeq12d 2450 . . . . . . . 8  |-  ( x  =  y  ->  (
( u H x )  =  x  <->  ( u H y )  =  y ) )
1211rspcv 3048 . . . . . . 7  |-  ( y  e.  X  ->  ( A. x  e.  X  ( u H x )  =  x  -> 
( u H y )  =  y ) )
138, 12syl5 30 . . . . . 6  |-  ( y  e.  X  ->  ( A. x  e.  X  ( ( u H x )  =  x  /\  ( x H u )  =  x )  ->  ( u H y )  =  y ) )
14 simpr 448 . . . . . . . 8  |-  ( ( ( y H x )  =  x  /\  ( x H y )  =  x )  ->  ( x H y )  =  x )
1514ralimi 2781 . . . . . . 7  |-  ( A. x  e.  X  (
( y H x )  =  x  /\  ( x H y )  =  x )  ->  A. x  e.  X  ( x H y )  =  x )
16 oveq1 6088 . . . . . . . . 9  |-  ( x  =  u  ->  (
x H y )  =  ( u H y ) )
17 id 20 . . . . . . . . 9  |-  ( x  =  u  ->  x  =  u )
1816, 17eqeq12d 2450 . . . . . . . 8  |-  ( x  =  u  ->  (
( x H y )  =  x  <->  ( u H y )  =  u ) )
1918rspcv 3048 . . . . . . 7  |-  ( u  e.  X  ->  ( A. x  e.  X  ( x H y )  =  x  -> 
( u H y )  =  u ) )
2015, 19syl5 30 . . . . . 6  |-  ( u  e.  X  ->  ( A. x  e.  X  ( ( y H x )  =  x  /\  ( x H y )  =  x )  ->  ( u H y )  =  u ) )
2113, 20im2anan9r 810 . . . . 5  |-  ( ( u  e.  X  /\  y  e.  X )  ->  ( ( A. x  e.  X  ( (
u H x )  =  x  /\  (
x H u )  =  x )  /\  A. x  e.  X  ( ( y H x )  =  x  /\  ( x H y )  =  x ) )  ->  ( (
u H y )  =  y  /\  (
u H y )  =  u ) ) )
22 eqtr2 2454 . . . . . 6  |-  ( ( ( u H y )  =  y  /\  ( u H y )  =  u )  ->  y  =  u )
2322eqcomd 2441 . . . . 5  |-  ( ( ( u H y )  =  y  /\  ( u H y )  =  u )  ->  u  =  y )
2421, 23syl6 31 . . . 4  |-  ( ( u  e.  X  /\  y  e.  X )  ->  ( ( A. x  e.  X  ( (
u H x )  =  x  /\  (
x H u )  =  x )  /\  A. x  e.  X  ( ( y H x )  =  x  /\  ( x H y )  =  x ) )  ->  u  =  y ) )
2524rgen2a 2772 . . 3  |-  A. u  e.  X  A. y  e.  X  ( ( A. x  e.  X  ( ( u H x )  =  x  /\  ( x H u )  =  x )  /\  A. x  e.  X  ( (
y H x )  =  x  /\  (
x H y )  =  x ) )  ->  u  =  y )
266, 25jctir 525 . 2  |-  ( R  e.  RingOps  ->  ( E. u  e.  X  A. x  e.  X  ( (
u H x )  =  x  /\  (
x H u )  =  x )  /\  A. u  e.  X  A. y  e.  X  (
( A. x  e.  X  ( ( u H x )  =  x  /\  ( x H u )  =  x )  /\  A. x  e.  X  (
( y H x )  =  x  /\  ( x H y )  =  x ) )  ->  u  =  y ) ) )
27 oveq1 6088 . . . . . 6  |-  ( u  =  y  ->  (
u H x )  =  ( y H x ) )
2827eqeq1d 2444 . . . . 5  |-  ( u  =  y  ->  (
( u H x )  =  x  <->  ( y H x )  =  x ) )
29 oveq2 6089 . . . . . 6  |-  ( u  =  y  ->  (
x H u )  =  ( x H y ) )
3029eqeq1d 2444 . . . . 5  |-  ( u  =  y  ->  (
( x H u )  =  x  <->  ( x H y )  =  x ) )
3128, 30anbi12d 692 . . . 4  |-  ( u  =  y  ->  (
( ( u H x )  =  x  /\  ( x H u )  =  x )  <->  ( ( y H x )  =  x  /\  ( x H y )  =  x ) ) )
3231ralbidv 2725 . . 3  |-  ( u  =  y  ->  ( A. x  e.  X  ( ( u H x )  =  x  /\  ( x H u )  =  x )  <->  A. x  e.  X  ( ( y H x )  =  x  /\  ( x H y )  =  x ) ) )
3332reu4 3128 . 2  |-  ( E! u  e.  X  A. x  e.  X  (
( u H x )  =  x  /\  ( x H u )  =  x )  <-> 
( E. u  e.  X  A. x  e.  X  ( ( u H x )  =  x  /\  ( x H u )  =  x )  /\  A. u  e.  X  A. y  e.  X  (
( A. x  e.  X  ( ( u H x )  =  x  /\  ( x H u )  =  x )  /\  A. x  e.  X  (
( y H x )  =  x  /\  ( x H y )  =  x ) )  ->  u  =  y ) ) )
3426, 33sylibr 204 1  |-  ( R  e.  RingOps  ->  E! u  e.  X  A. x  e.  X  ( ( u H x )  =  x  /\  ( x H u )  =  x ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2705   E.wrex 2706   E!wreu 2707    X. cxp 4876   ran crn 4879   -->wf 5450   ` cfv 5454  (class class class)co 6081   1stc1st 6347   2ndc2nd 6348   AbelOpcablo 21869   RingOpscrngo 21963
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 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462  df-ov 6084  df-1st 6349  df-2nd 6350  df-rngo 21964
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