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Theorem rngoideu 23806
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 23802 . . . 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 751 . . . 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 454 . . . . . . . 8  |-  ( ( ( u H x )  =  x  /\  ( x H u )  =  x )  ->  ( u H x )  =  x )
87ralimi 2789 . . . . . . 7  |-  ( A. x  e.  X  (
( u H x )  =  x  /\  ( x H u )  =  x )  ->  A. x  e.  X  ( u H x )  =  x )
9 oveq2 6098 . . . . . . . . 9  |-  ( x  =  y  ->  (
u H x )  =  ( u H y ) )
10 id 22 . . . . . . . . 9  |-  ( x  =  y  ->  x  =  y )
119, 10eqeq12d 2455 . . . . . . . 8  |-  ( x  =  y  ->  (
( u H x )  =  x  <->  ( u H y )  =  y ) )
1211rspcv 3066 . . . . . . 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 458 . . . . . . . 8  |-  ( ( ( y H x )  =  x  /\  ( x H y )  =  x )  ->  ( x H y )  =  x )
1514ralimi 2789 . . . . . . 7  |-  ( A. x  e.  X  (
( y H x )  =  x  /\  ( x H y )  =  x )  ->  A. x  e.  X  ( x H y )  =  x )
16 oveq1 6097 . . . . . . . . 9  |-  ( x  =  u  ->  (
x H y )  =  ( u H y ) )
17 id 22 . . . . . . . . 9  |-  ( x  =  u  ->  x  =  u )
1816, 17eqeq12d 2455 . . . . . . . 8  |-  ( x  =  u  ->  (
( x H y )  =  x  <->  ( u H y )  =  u ) )
1918rspcv 3066 . . . . . . 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 827 . . . . 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 2459 . . . . . 6  |-  ( ( ( u H y )  =  y  /\  ( u H y )  =  u )  ->  y  =  u )
2322eqcomd 2446 . . . . 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 2780 . . 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 535 . 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 6097 . . . . . 6  |-  ( u  =  y  ->  (
u H x )  =  ( y H x ) )
2827eqeq1d 2449 . . . . 5  |-  ( u  =  y  ->  (
( u H x )  =  x  <->  ( y H x )  =  x ) )
29 oveq2 6098 . . . . . 6  |-  ( u  =  y  ->  (
x H u )  =  ( x H y ) )
3029eqeq1d 2449 . . . . 5  |-  ( u  =  y  ->  (
( x H u )  =  x  <->  ( x H y )  =  x ) )
3128, 30anbi12d 705 . . . 4  |-  ( u  =  y  ->  (
( ( u H x )  =  x  /\  ( x H u )  =  x )  <->  ( ( y H x )  =  x  /\  ( x H y )  =  x ) ) )
3231ralbidv 2733 . . 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 3150 . 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 960    = wceq 1364    e. wcel 1761   A.wral 2713   E.wrex 2714   E!wreu 2715    X. cxp 4834   ran crn 4837   -->wf 5411   ` cfv 5415  (class class class)co 6090   1stc1st 6574   2ndc2nd 6575   AbelOpcablo 23703   RingOpscrngo 23797
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-fv 5423  df-ov 6093  df-1st 6576  df-2nd 6577  df-rngo 23798
This theorem is referenced by: (None)
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