Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  risci Unicode version

Theorem risci 26287
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 2904 . . 3  |-  ( F  e.  ( R  RngIso  S )  ->  E. f 
f  e.  ( R 
RngIso  S ) )
2 risc 26286 . . 3  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps )  ->  ( R  ~=r  S  <->  E. f 
f  e.  ( R 
RngIso  S ) ) )
31, 2syl5ibr 213 . 2  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps )  ->  ( F  e.  ( R  RngIso  S )  ->  R  ~=r 
S ) )
433impia 1150 1  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngIso  S ) )  ->  R  ~=r  S
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936   E.wex 1547    e. wcel 1717   class class class wbr 4146  (class class class)co 6013   RingOpscrngo 21804    RngIso crngiso 26261    ~=r crisc 26262
This theorem is referenced by:  riscer  26288
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 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-rex 2648  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-iota 5351  df-fv 5395  df-ov 6016  df-risc 26283
  Copyright terms: Public domain W3C validator