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Theorem rngoid 21066
Description: The multiplication operation of a unital ring has (one or more) identity elements. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Mario Carneiro, 22-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
rngoid  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  E. u  e.  X  ( (
u H A )  =  A  /\  ( A H u )  =  A ) )
Distinct variable groups:    u, G    u, H    u, X    u, A    u, R

Proof of Theorem rngoid
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ringi.1 . . . . . 6  |-  G  =  ( 1st `  R
)
2 ringi.2 . . . . . 6  |-  H  =  ( 2nd `  R
)
3 ringi.3 . . . . . 6  |-  X  =  ran  G
41, 2, 3rngoi 21063 . . . . 5  |-  ( 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 ) ) ) )
54simprd 449 . . . 4  |-  ( R  e.  RingOps  ->  ( 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 ) ) )
65simprd 449 . . 3  |-  ( R  e.  RingOps  ->  E. u  e.  X  A. x  e.  X  ( ( u H x )  =  x  /\  ( x H u )  =  x ) )
7 r19.12 2669 . . 3  |-  ( E. u  e.  X  A. x  e.  X  (
( u H x )  =  x  /\  ( x H u )  =  x )  ->  A. x  e.  X  E. u  e.  X  ( ( u H x )  =  x  /\  ( x H u )  =  x ) )
86, 7syl 15 . 2  |-  ( R  e.  RingOps  ->  A. x  e.  X  E. u  e.  X  ( ( u H x )  =  x  /\  ( x H u )  =  x ) )
9 oveq2 5882 . . . . . 6  |-  ( x  =  A  ->  (
u H x )  =  ( u H A ) )
10 id 19 . . . . . 6  |-  ( x  =  A  ->  x  =  A )
119, 10eqeq12d 2310 . . . . 5  |-  ( x  =  A  ->  (
( u H x )  =  x  <->  ( u H A )  =  A ) )
12 oveq1 5881 . . . . . 6  |-  ( x  =  A  ->  (
x H u )  =  ( A H u ) )
1312, 10eqeq12d 2310 . . . . 5  |-  ( x  =  A  ->  (
( x H u )  =  x  <->  ( A H u )  =  A ) )
1411, 13anbi12d 691 . . . 4  |-  ( x  =  A  ->  (
( ( u H x )  =  x  /\  ( x H u )  =  x )  <->  ( ( u H A )  =  A  /\  ( A H u )  =  A ) ) )
1514rexbidv 2577 . . 3  |-  ( x  =  A  ->  ( E. u  e.  X  ( ( u H x )  =  x  /\  ( x H u )  =  x )  <->  E. u  e.  X  ( ( u H A )  =  A  /\  ( A H u )  =  A ) ) )
1615rspccva 2896 . 2  |-  ( ( A. x  e.  X  E. u  e.  X  ( ( u H x )  =  x  /\  ( x H u )  =  x )  /\  A  e.  X )  ->  E. u  e.  X  ( (
u H A )  =  A  /\  ( A H u )  =  A ) )
178, 16sylan 457 1  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  E. u  e.  X  ( (
u H A )  =  A  /\  ( A H u )  =  A ) )
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    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 is referenced by:  rngo2  21071
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-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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