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Theorem isriscg 25938
Description: The ring isomorphism relation. (Contributed by Jeff Madsen, 16-Jun-2011.)
Assertion
Ref Expression
isriscg  |-  ( ( R  e.  A  /\  S  e.  B )  ->  ( R  ~=r  S  <->  ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  E. f  f  e.  ( R  RngIso  S ) ) ) )
Distinct variable groups:    R, f    S, f
Allowed substitution hints:    A( f)    B( f)

Proof of Theorem isriscg
Dummy variables  r 
s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1 2418 . . . 4  |-  ( r  =  R  ->  (
r  e.  RingOps  <->  R  e.  RingOps ) )
21anbi1d 685 . . 3  |-  ( r  =  R  ->  (
( r  e.  RingOps  /\  s  e.  RingOps )  <->  ( R  e.  RingOps  /\  s  e.  RingOps ) ) )
3 oveq1 5949 . . . . 5  |-  ( r  =  R  ->  (
r  RngIso  s )  =  ( R  RngIso  s ) )
43eleq2d 2425 . . . 4  |-  ( r  =  R  ->  (
f  e.  ( r 
RngIso  s )  <->  f  e.  ( R  RngIso  s ) ) )
54exbidv 1626 . . 3  |-  ( r  =  R  ->  ( E. f  f  e.  ( r  RngIso  s )  <->  E. f  f  e.  ( R  RngIso  s ) ) )
62, 5anbi12d 691 . 2  |-  ( r  =  R  ->  (
( ( r  e.  RingOps 
/\  s  e.  RingOps )  /\  E. f  f  e.  ( r  RngIso  s ) )  <->  ( ( R  e.  RingOps  /\  s  e.  RingOps )  /\  E. f  f  e.  ( R  RngIso  s ) ) ) )
7 eleq1 2418 . . . 4  |-  ( s  =  S  ->  (
s  e.  RingOps  <->  S  e.  RingOps ) )
87anbi2d 684 . . 3  |-  ( s  =  S  ->  (
( R  e.  RingOps  /\  s  e.  RingOps )  <->  ( R  e.  RingOps  /\  S  e.  RingOps ) ) )
9 oveq2 5950 . . . . 5  |-  ( s  =  S  ->  ( R  RngIso  s )  =  ( R  RngIso  S ) )
109eleq2d 2425 . . . 4  |-  ( s  =  S  ->  (
f  e.  ( R 
RngIso  s )  <->  f  e.  ( R  RngIso  S ) ) )
1110exbidv 1626 . . 3  |-  ( s  =  S  ->  ( E. f  f  e.  ( R  RngIso  s )  <->  E. f  f  e.  ( R  RngIso  S ) ) )
128, 11anbi12d 691 . 2  |-  ( s  =  S  ->  (
( ( R  e.  RingOps 
/\  s  e.  RingOps )  /\  E. f  f  e.  ( R  RngIso  s ) )  <->  ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  E. f  f  e.  ( R  RngIso  S ) ) ) )
13 df-risc 25937 . 2  |-  ~=r  =  { <. r ,  s
>.  |  ( (
r  e.  RingOps  /\  s  e.  RingOps )  /\  E. f  f  e.  (
r  RngIso  s ) ) }
146, 12, 13brabg 4363 1  |-  ( ( R  e.  A  /\  S  e.  B )  ->  ( R  ~=r  S  <->  ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  E. f  f  e.  ( R  RngIso  S ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1541    = wceq 1642    e. wcel 1710   class class class wbr 4102  (class class class)co 5942   RingOpscrngo 21148    RngIso crngiso 25915    ~=r crisc 25916
This theorem is referenced by:  isrisc  25939  risc  25940
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pr 4293
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-rex 2625  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-br 4103  df-opab 4157  df-iota 5298  df-fv 5342  df-ov 5945  df-risc 25937
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