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Theorem risci 31652
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 3070 . . 3  |-  ( F  e.  ( R  RngIso  S )  ->  E. f 
f  e.  ( R 
RngIso  S ) )
2 risc 31651 . . 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 1194 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 367    /\ w3a 974   E.wex 1633    e. wcel 1842   class class class wbr 4394  (class class class)co 6277   RingOpscrngo 25777    RngIso crngiso 31626    ~=R crisc 31627
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 5532  df-fv 5576  df-ov 6280  df-risc 31648
This theorem is referenced by:  riscer  31653
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