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Theorem isriscg 31650
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 2474 . . . 4  |-  ( r  =  R  ->  (
r  e.  RingOps  <->  R  e.  RingOps ) )
21anbi1d 703 . . 3  |-  ( r  =  R  ->  (
( r  e.  RingOps  /\  s  e.  RingOps )  <->  ( R  e.  RingOps  /\  s  e.  RingOps ) ) )
3 oveq1 6241 . . . . 5  |-  ( r  =  R  ->  (
r  RngIso  s )  =  ( R  RngIso  s ) )
43eleq2d 2472 . . . 4  |-  ( r  =  R  ->  (
f  e.  ( r 
RngIso  s )  <->  f  e.  ( R  RngIso  s ) ) )
54exbidv 1735 . . 3  |-  ( r  =  R  ->  ( E. f  f  e.  ( r  RngIso  s )  <->  E. f  f  e.  ( R  RngIso  s ) ) )
62, 5anbi12d 709 . 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 2474 . . . 4  |-  ( s  =  S  ->  (
s  e.  RingOps  <->  S  e.  RingOps ) )
87anbi2d 702 . . 3  |-  ( s  =  S  ->  (
( R  e.  RingOps  /\  s  e.  RingOps )  <->  ( R  e.  RingOps  /\  S  e.  RingOps ) ) )
9 oveq2 6242 . . . . 5  |-  ( s  =  S  ->  ( R  RngIso  s )  =  ( R  RngIso  S ) )
109eleq2d 2472 . . . 4  |-  ( s  =  S  ->  (
f  e.  ( R 
RngIso  s )  <->  f  e.  ( R  RngIso  S ) ) )
1110exbidv 1735 . . 3  |-  ( s  =  S  ->  ( E. f  f  e.  ( R  RngIso  s )  <->  E. f  f  e.  ( R  RngIso  S ) ) )
128, 11anbi12d 709 . 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 31649 . 2  |-  ~=R  =  { <. r ,  s
>.  |  ( (
r  e.  RingOps  /\  s  e.  RingOps )  /\  E. f  f  e.  (
r  RngIso  s ) ) }
146, 12, 13brabg 4708 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405   E.wex 1633    e. wcel 1842   class class class wbr 4394  (class class class)co 6234   RingOpscrngo 25671    RngIso crngiso 31627    ~=R crisc 31628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-rex 2759  df-rab 2762  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-iota 5489  df-fv 5533  df-ov 6237  df-risc 31649
This theorem is referenced by:  isrisc  31651  risc  31652
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