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Theorem risci 29980
Description: Determine that two rings are isomorphic. (Contributed by Jeff Madsen, 16-Jun-2011.)
Assertion
Ref Expression
risci  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngIso  S ) )  ->  R  ~=R  S
)

Proof of Theorem risci
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 elex2 3118 . . 3  |-  ( F  e.  ( R  RngIso  S )  ->  E. f 
f  e.  ( R 
RngIso  S ) )
2 risc 29979 . . 3  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps )  ->  ( R  ~=R  S  <->  E. f 
f  e.  ( R 
RngIso  S ) ) )
31, 2syl5ibr 221 . 2  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps )  ->  ( F  e.  ( R  RngIso  S )  ->  R  ~=R 
S ) )
433impia 1188 1  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngIso  S ) )  ->  R  ~=R  S
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968   E.wex 1591    e. wcel 1762   class class class wbr 4440  (class class class)co 6275   RingOpscrngo 25039    RngIso crngiso 29954    ~=R crisc 29955
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-rex 2813  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-iota 5542  df-fv 5587  df-ov 6278  df-risc 29976
This theorem is referenced by:  riscer  29981
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