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Theorem isrisc 28934
Description: The ring isomorphism relation. (Contributed by Jeff Madsen, 16-Jun-2011.)
Hypotheses
Ref Expression
isrisc.1  |-  R  e. 
_V
isrisc.2  |-  S  e. 
_V
Assertion
Ref Expression
isrisc  |-  ( R 
~=r  S  <->  ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  E. f  f  e.  ( R  RngIso  S ) ) )
Distinct variable groups:    R, f    S, f

Proof of Theorem isrisc
StepHypRef Expression
1 isrisc.1 . 2  |-  R  e. 
_V
2 isrisc.2 . 2  |-  S  e. 
_V
3 isriscg 28933 . 2  |-  ( ( R  e.  _V  /\  S  e.  _V )  ->  ( R  ~=r  S  <->  ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  E. f  f  e.  ( R  RngIso  S ) ) ) )
41, 2, 3mp2an 672 1  |-  ( R 
~=r  S  <->  ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  E. f  f  e.  ( R  RngIso  S ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369   E.wex 1587    e. wcel 1758   _Vcvv 3072   class class class wbr 4395  (class class class)co 6195   RingOpscrngo 24009    RngIso crngiso 28910    ~=r crisc 28911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pr 4634
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-rex 2802  df-rab 2805  df-v 3074  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-br 4396  df-opab 4454  df-iota 5484  df-fv 5529  df-ov 6198  df-risc 28932
This theorem is referenced by:  riscer  28937
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