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Theorem isriscg 26638
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 2502 . . . 4  |-  ( r  =  R  ->  (
r  e.  RingOps  <->  R  e.  RingOps ) )
21anbi1d 687 . . 3  |-  ( r  =  R  ->  (
( r  e.  RingOps  /\  s  e.  RingOps )  <->  ( R  e.  RingOps  /\  s  e.  RingOps ) ) )
3 oveq1 6117 . . . . 5  |-  ( r  =  R  ->  (
r  RngIso  s )  =  ( R  RngIso  s ) )
43eleq2d 2509 . . . 4  |-  ( r  =  R  ->  (
f  e.  ( r 
RngIso  s )  <->  f  e.  ( R  RngIso  s ) ) )
54exbidv 1637 . . 3  |-  ( r  =  R  ->  ( E. f  f  e.  ( r  RngIso  s )  <->  E. f  f  e.  ( R  RngIso  s ) ) )
62, 5anbi12d 693 . 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 2502 . . . 4  |-  ( s  =  S  ->  (
s  e.  RingOps  <->  S  e.  RingOps ) )
87anbi2d 686 . . 3  |-  ( s  =  S  ->  (
( R  e.  RingOps  /\  s  e.  RingOps )  <->  ( R  e.  RingOps  /\  S  e.  RingOps ) ) )
9 oveq2 6118 . . . . 5  |-  ( s  =  S  ->  ( R  RngIso  s )  =  ( R  RngIso  S ) )
109eleq2d 2509 . . . 4  |-  ( s  =  S  ->  (
f  e.  ( R 
RngIso  s )  <->  f  e.  ( R  RngIso  S ) ) )
1110exbidv 1637 . . 3  |-  ( s  =  S  ->  ( E. f  f  e.  ( R  RngIso  s )  <->  E. f  f  e.  ( R  RngIso  S ) ) )
128, 11anbi12d 693 . 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 26637 . 2  |-  ~=r  =  { <. r ,  s
>.  |  ( (
r  e.  RingOps  /\  s  e.  RingOps )  /\  E. f  f  e.  (
r  RngIso  s ) ) }
146, 12, 13brabg 4503 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 178    /\ wa 360   E.wex 1551    = wceq 1653    e. wcel 1727   class class class wbr 4237  (class class class)co 6110   RingOpscrngo 21994    RngIso crngiso 26615    ~=r crisc 26616
This theorem is referenced by:  isrisc  26639  risc  26640
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pr 4432
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-rex 2717  df-rab 2720  df-v 2964  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-br 4238  df-opab 4292  df-iota 5447  df-fv 5491  df-ov 6113  df-risc 26637
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