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Theorem isriscg 26498
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 2472 . . . 4  |-  ( r  =  R  ->  (
r  e.  RingOps  <->  R  e.  RingOps ) )
21anbi1d 686 . . 3  |-  ( r  =  R  ->  (
( r  e.  RingOps  /\  s  e.  RingOps )  <->  ( R  e.  RingOps  /\  s  e.  RingOps ) ) )
3 oveq1 6055 . . . . 5  |-  ( r  =  R  ->  (
r  RngIso  s )  =  ( R  RngIso  s ) )
43eleq2d 2479 . . . 4  |-  ( r  =  R  ->  (
f  e.  ( r 
RngIso  s )  <->  f  e.  ( R  RngIso  s ) ) )
54exbidv 1633 . . 3  |-  ( r  =  R  ->  ( E. f  f  e.  ( r  RngIso  s )  <->  E. f  f  e.  ( R  RngIso  s ) ) )
62, 5anbi12d 692 . 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 2472 . . . 4  |-  ( s  =  S  ->  (
s  e.  RingOps  <->  S  e.  RingOps ) )
87anbi2d 685 . . 3  |-  ( s  =  S  ->  (
( R  e.  RingOps  /\  s  e.  RingOps )  <->  ( R  e.  RingOps  /\  S  e.  RingOps ) ) )
9 oveq2 6056 . . . . 5  |-  ( s  =  S  ->  ( R  RngIso  s )  =  ( R  RngIso  S ) )
109eleq2d 2479 . . . 4  |-  ( s  =  S  ->  (
f  e.  ( R 
RngIso  s )  <->  f  e.  ( R  RngIso  S ) ) )
1110exbidv 1633 . . 3  |-  ( s  =  S  ->  ( E. f  f  e.  ( R  RngIso  s )  <->  E. f  f  e.  ( R  RngIso  S ) ) )
128, 11anbi12d 692 . 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 26497 . 2  |-  ~=r  =  { <. r ,  s
>.  |  ( (
r  e.  RingOps  /\  s  e.  RingOps )  /\  E. f  f  e.  (
r  RngIso  s ) ) }
146, 12, 13brabg 4442 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 177    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1721   class class class wbr 4180  (class class class)co 6048   RingOpscrngo 21924    RngIso crngiso 26475    ~=r crisc 26476
This theorem is referenced by:  isrisc  26499  risc  26500
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-rex 2680  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-iota 5385  df-fv 5429  df-ov 6051  df-risc 26497
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