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Theorem rngoideu 21067
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 21063 . . . 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 733 . . . 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 15 . . 3  |-  ( R  e.  RingOps  ->  E. u  e.  X  A. x  e.  X  ( ( u H x )  =  x  /\  ( x H u )  =  x ) )
7 simpl 443 . . . . . . . 8  |-  ( ( ( u H x )  =  x  /\  ( x H u )  =  x )  ->  ( u H x )  =  x )
87ralimi 2631 . . . . . . 7  |-  ( A. x  e.  X  (
( u H x )  =  x  /\  ( x H u )  =  x )  ->  A. x  e.  X  ( u H x )  =  x )
9 oveq2 5882 . . . . . . . . 9  |-  ( x  =  y  ->  (
u H x )  =  ( u H y ) )
10 id 19 . . . . . . . . 9  |-  ( x  =  y  ->  x  =  y )
119, 10eqeq12d 2310 . . . . . . . 8  |-  ( x  =  y  ->  (
( u H x )  =  x  <->  ( u H y )  =  y ) )
1211rspcv 2893 . . . . . . 7  |-  ( y  e.  X  ->  ( A. x  e.  X  ( u H x )  =  x  -> 
( u H y )  =  y ) )
138, 12syl5 28 . . . . . 6  |-  ( y  e.  X  ->  ( A. x  e.  X  ( ( u H x )  =  x  /\  ( x H u )  =  x )  ->  ( u H y )  =  y ) )
14 simpr 447 . . . . . . . 8  |-  ( ( ( y H x )  =  x  /\  ( x H y )  =  x )  ->  ( x H y )  =  x )
1514ralimi 2631 . . . . . . 7  |-  ( A. x  e.  X  (
( y H x )  =  x  /\  ( x H y )  =  x )  ->  A. x  e.  X  ( x H y )  =  x )
16 oveq1 5881 . . . . . . . . 9  |-  ( x  =  u  ->  (
x H y )  =  ( u H y ) )
17 id 19 . . . . . . . . 9  |-  ( x  =  u  ->  x  =  u )
1816, 17eqeq12d 2310 . . . . . . . 8  |-  ( x  =  u  ->  (
( x H y )  =  x  <->  ( u H y )  =  u ) )
1918rspcv 2893 . . . . . . 7  |-  ( u  e.  X  ->  ( A. x  e.  X  ( x H y )  =  x  -> 
( u H y )  =  u ) )
2015, 19syl5 28 . . . . . 6  |-  ( u  e.  X  ->  ( A. x  e.  X  ( ( y H x )  =  x  /\  ( x H y )  =  x )  ->  ( u H y )  =  u ) )
2113, 20im2anan9r 809 . . . . 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 2314 . . . . . 6  |-  ( ( ( u H y )  =  y  /\  ( u H y )  =  u )  ->  y  =  u )
2322eqcomd 2301 . . . . 5  |-  ( ( ( u H y )  =  y  /\  ( u H y )  =  u )  ->  u  =  y )
2421, 23syl6 29 . . . 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 2622 . . 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 524 . 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 5881 . . . . . 6  |-  ( u  =  y  ->  (
u H x )  =  ( y H x ) )
2827eqeq1d 2304 . . . . 5  |-  ( u  =  y  ->  (
( u H x )  =  x  <->  ( y H x )  =  x ) )
29 oveq2 5882 . . . . . 6  |-  ( u  =  y  ->  (
x H u )  =  ( x H y ) )
3029eqeq1d 2304 . . . . 5  |-  ( u  =  y  ->  (
( x H u )  =  x  <->  ( x H y )  =  x ) )
3128, 30anbi12d 691 . . . 4  |-  ( u  =  y  ->  (
( ( u H x )  =  x  /\  ( x H u )  =  x )  <->  ( ( y H x )  =  x  /\  ( x H y )  =  x ) ) )
3231ralbidv 2576 . . 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 2972 . 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 203 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 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   E!wreu 2558    X. cxp 4703   ran crn 4706   -->wf 5267   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137   AbelOpcablo 20964   RingOpscrngo 21058
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-1st 6138  df-2nd 6139  df-rngo 21059
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