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Theorem isriscg 28961
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 2526 . . . 4  |-  ( r  =  R  ->  (
r  e.  RingOps  <->  R  e.  RingOps ) )
21anbi1d 704 . . 3  |-  ( r  =  R  ->  (
( r  e.  RingOps  /\  s  e.  RingOps )  <->  ( R  e.  RingOps  /\  s  e.  RingOps ) ) )
3 oveq1 6210 . . . . 5  |-  ( r  =  R  ->  (
r  RngIso  s )  =  ( R  RngIso  s ) )
43eleq2d 2524 . . . 4  |-  ( r  =  R  ->  (
f  e.  ( r 
RngIso  s )  <->  f  e.  ( R  RngIso  s ) ) )
54exbidv 1681 . . 3  |-  ( r  =  R  ->  ( E. f  f  e.  ( r  RngIso  s )  <->  E. f  f  e.  ( R  RngIso  s ) ) )
62, 5anbi12d 710 . 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 2526 . . . 4  |-  ( s  =  S  ->  (
s  e.  RingOps  <->  S  e.  RingOps ) )
87anbi2d 703 . . 3  |-  ( s  =  S  ->  (
( R  e.  RingOps  /\  s  e.  RingOps )  <->  ( R  e.  RingOps  /\  S  e.  RingOps ) ) )
9 oveq2 6211 . . . . 5  |-  ( s  =  S  ->  ( R  RngIso  s )  =  ( R  RngIso  S ) )
109eleq2d 2524 . . . 4  |-  ( s  =  S  ->  (
f  e.  ( R 
RngIso  s )  <->  f  e.  ( R  RngIso  S ) ) )
1110exbidv 1681 . . 3  |-  ( s  =  S  ->  ( E. f  f  e.  ( R  RngIso  s )  <->  E. f  f  e.  ( R  RngIso  S ) ) )
128, 11anbi12d 710 . 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 28960 . 2  |-  ~=r  =  { <. r ,  s
>.  |  ( (
r  e.  RingOps  /\  s  e.  RingOps )  /\  E. f  f  e.  (
r  RngIso  s ) ) }
146, 12, 13brabg 4719 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370   E.wex 1587    e. wcel 1758   class class class wbr 4403  (class class class)co 6203   RingOpscrngo 24041    RngIso crngiso 28938    ~=r crisc 28939
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 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-iota 5492  df-fv 5537  df-ov 6206  df-risc 28960
This theorem is referenced by:  isrisc  28962  risc  28963
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