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Theorem rngoideu 25217
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 25213 . . . 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 756 . . . 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 457 . . . . . . . 8  |-  ( ( ( u H x )  =  x  /\  ( x H u )  =  x )  ->  ( u H x )  =  x )
87ralimi 2860 . . . . . . 7  |-  ( A. x  e.  X  (
( u H x )  =  x  /\  ( x H u )  =  x )  ->  A. x  e.  X  ( u H x )  =  x )
9 oveq2 6303 . . . . . . . . 9  |-  ( x  =  y  ->  (
u H x )  =  ( u H y ) )
10 id 22 . . . . . . . . 9  |-  ( x  =  y  ->  x  =  y )
119, 10eqeq12d 2489 . . . . . . . 8  |-  ( x  =  y  ->  (
( u H x )  =  x  <->  ( u H y )  =  y ) )
1211rspcv 3215 . . . . . . 7  |-  ( y  e.  X  ->  ( A. x  e.  X  ( u H x )  =  x  -> 
( u H y )  =  y ) )
138, 12syl5 32 . . . . . 6  |-  ( y  e.  X  ->  ( A. x  e.  X  ( ( u H x )  =  x  /\  ( x H u )  =  x )  ->  ( u H y )  =  y ) )
14 simpr 461 . . . . . . . 8  |-  ( ( ( y H x )  =  x  /\  ( x H y )  =  x )  ->  ( x H y )  =  x )
1514ralimi 2860 . . . . . . 7  |-  ( A. x  e.  X  (
( y H x )  =  x  /\  ( x H y )  =  x )  ->  A. x  e.  X  ( x H y )  =  x )
16 oveq1 6302 . . . . . . . . 9  |-  ( x  =  u  ->  (
x H y )  =  ( u H y ) )
17 id 22 . . . . . . . . 9  |-  ( x  =  u  ->  x  =  u )
1816, 17eqeq12d 2489 . . . . . . . 8  |-  ( x  =  u  ->  (
( x H y )  =  x  <->  ( u H y )  =  u ) )
1918rspcv 3215 . . . . . . 7  |-  ( u  e.  X  ->  ( A. x  e.  X  ( x H y )  =  x  -> 
( u H y )  =  u ) )
2015, 19syl5 32 . . . . . 6  |-  ( u  e.  X  ->  ( A. x  e.  X  ( ( y H x )  =  x  /\  ( x H y )  =  x )  ->  ( u H y )  =  u ) )
2113, 20im2anan9r 834 . . . . 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 2494 . . . . . 6  |-  ( ( ( u H y )  =  y  /\  ( u H y )  =  u )  ->  y  =  u )
2322eqcomd 2475 . . . . 5  |-  ( ( ( u H y )  =  y  /\  ( u H y )  =  u )  ->  u  =  y )
2421, 23syl6 33 . . . 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 2894 . . 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 538 . 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 6302 . . . . . 6  |-  ( u  =  y  ->  (
u H x )  =  ( y H x ) )
2827eqeq1d 2469 . . . . 5  |-  ( u  =  y  ->  (
( u H x )  =  x  <->  ( y H x )  =  x ) )
29 oveq2 6303 . . . . . 6  |-  ( u  =  y  ->  (
x H u )  =  ( x H y ) )
3029eqeq1d 2469 . . . . 5  |-  ( u  =  y  ->  (
( x H u )  =  x  <->  ( x H y )  =  x ) )
3128, 30anbi12d 710 . . . 4  |-  ( u  =  y  ->  (
( ( u H x )  =  x  /\  ( x H u )  =  x )  <->  ( ( y H x )  =  x  /\  ( x H y )  =  x ) ) )
3231ralbidv 2906 . . 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 3302 . 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 212 1  |-  ( R  e.  RingOps  ->  E! u  e.  X  A. x  e.  X  ( ( u H x )  =  x  /\  ( x H u )  =  x ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2817   E.wrex 2818   E!wreu 2819    X. cxp 5003   ran crn 5006   -->wf 5590   ` cfv 5594  (class class class)co 6295   1stc1st 6793   2ndc2nd 6794   AbelOpcablo 25114   RingOpscrngo 25208
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-ov 6298  df-1st 6795  df-2nd 6796  df-rngo 25209
This theorem is referenced by: (None)
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