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Theorem riscer 26722
Description: Ring isomorphism is an equivalence relation. (Contributed by Jeff Madsen, 16-Jun-2011.) (Revised by Mario Carneiro, 12-Aug-2015.)
Assertion
Ref Expression
riscer  |-  ~=r  Er  dom  ~=r

Proof of Theorem riscer
Dummy variables  f 
g  r  s  t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-risc 26717 . . 3  |-  ~=r  =  { <. r ,  s
>.  |  ( (
r  e.  RingOps  /\  s  e.  RingOps )  /\  E. f  f  e.  (
r  RngIso  s ) ) }
21relopabi 4827 . 2  |-  Rel  ~=r
3 eqid 2296 . 2  |-  dom  ~=r  =  dom  ~=r
4 vex 2804 . . . . . . 7  |-  r  e. 
_V
5 vex 2804 . . . . . . 7  |-  s  e. 
_V
64, 5isrisc 26719 . . . . . 6  |-  ( r 
~=r  s  <->  ( (
r  e.  RingOps  /\  s  e.  RingOps )  /\  E. f  f  e.  (
r  RngIso  s ) ) )
7 rngoisocnv 26715 . . . . . . . . . 10  |-  ( ( r  e.  RingOps  /\  s  e.  RingOps  /\  f  e.  ( r  RngIso  s ) )  ->  `' f  e.  ( s  RngIso  r ) )
873expia 1153 . . . . . . . . 9  |-  ( ( r  e.  RingOps  /\  s  e.  RingOps )  ->  (
f  e.  ( r 
RngIso  s )  ->  `' f  e.  ( s  RngIso  r ) ) )
9 risci 26721 . . . . . . . . . . 11  |-  ( ( s  e.  RingOps  /\  r  e.  RingOps  /\  `' f  e.  ( s  RngIso  r ) )  ->  s  ~=r  r )
1093expia 1153 . . . . . . . . . 10  |-  ( ( s  e.  RingOps  /\  r  e.  RingOps )  ->  ( `' f  e.  (
s  RngIso  r )  -> 
s  ~=r  r )
)
1110ancoms 439 . . . . . . . . 9  |-  ( ( r  e.  RingOps  /\  s  e.  RingOps )  ->  ( `' f  e.  (
s  RngIso  r )  -> 
s  ~=r  r )
)
128, 11syld 40 . . . . . . . 8  |-  ( ( r  e.  RingOps  /\  s  e.  RingOps )  ->  (
f  e.  ( r 
RngIso  s )  ->  s  ~=r  r ) )
1312exlimdv 1626 . . . . . . 7  |-  ( ( r  e.  RingOps  /\  s  e.  RingOps )  ->  ( E. f  f  e.  ( r  RngIso  s )  ->  s  ~=r  r
) )
1413imp 418 . . . . . 6  |-  ( ( ( r  e.  RingOps  /\  s  e.  RingOps )  /\  E. f  f  e.  ( r  RngIso  s ) )  ->  s  ~=r  r
)
156, 14sylbi 187 . . . . 5  |-  ( r 
~=r  s  ->  s  ~=r  r )
16 vex 2804 . . . . . . 7  |-  t  e. 
_V
175, 16isrisc 26719 . . . . . 6  |-  ( s 
~=r  t  <->  ( (
s  e.  RingOps  /\  t  e.  RingOps )  /\  E. g  g  e.  (
s  RngIso  t ) ) )
18 eeanv 1866 . . . . . . . . . . 11  |-  ( E. f E. g ( f  e.  ( r 
RngIso  s )  /\  g  e.  ( s  RngIso  t ) )  <->  ( E. f 
f  e.  ( r 
RngIso  s )  /\  E. g  g  e.  (
s  RngIso  t ) ) )
19 rngoisoco 26716 . . . . . . . . . . . . . 14  |-  ( ( ( r  e.  RingOps  /\  s  e.  RingOps  /\  t  e.  RingOps )  /\  (
f  e.  ( r 
RngIso  s )  /\  g  e.  ( s  RngIso  t ) ) )  ->  (
g  o.  f )  e.  ( r  RngIso  t ) )
2019ex 423 . . . . . . . . . . . . 13  |-  ( ( r  e.  RingOps  /\  s  e.  RingOps  /\  t  e.  RingOps )  ->  ( ( f  e.  ( r  RngIso  s )  /\  g  e.  ( s  RngIso  t ) )  ->  ( g  o.  f )  e.  ( r  RngIso  t ) ) )
21 risci 26721 . . . . . . . . . . . . . . 15  |-  ( ( r  e.  RingOps  /\  t  e.  RingOps  /\  ( g  o.  f )  e.  ( r  RngIso  t ) )  ->  r  ~=r  t
)
22213expia 1153 . . . . . . . . . . . . . 14  |-  ( ( r  e.  RingOps  /\  t  e.  RingOps )  ->  (
( g  o.  f
)  e.  ( r 
RngIso  t )  ->  r  ~=r  t ) )
23223adant2 974 . . . . . . . . . . . . 13  |-  ( ( r  e.  RingOps  /\  s  e.  RingOps  /\  t  e.  RingOps )  ->  ( ( g  o.  f )  e.  ( r  RngIso  t )  ->  r  ~=r  t
) )
2420, 23syld 40 . . . . . . . . . . . 12  |-  ( ( r  e.  RingOps  /\  s  e.  RingOps  /\  t  e.  RingOps )  ->  ( ( f  e.  ( r  RngIso  s )  /\  g  e.  ( s  RngIso  t ) )  ->  r  ~=r  t ) )
2524exlimdvv 1627 . . . . . . . . . . 11  |-  ( ( r  e.  RingOps  /\  s  e.  RingOps  /\  t  e.  RingOps )  ->  ( E. f E. g ( f  e.  ( r  RngIso  s )  /\  g  e.  ( s  RngIso  t ) )  ->  r  ~=r  t
) )
2618, 25syl5bir 209 . . . . . . . . . 10  |-  ( ( r  e.  RingOps  /\  s  e.  RingOps  /\  t  e.  RingOps )  ->  ( ( E. f  f  e.  ( r  RngIso  s )  /\  E. g  g  e.  ( s  RngIso  t ) )  ->  r  ~=r  t
) )
27263expb 1152 . . . . . . . . 9  |-  ( ( r  e.  RingOps  /\  (
s  e.  RingOps  /\  t  e.  RingOps ) )  -> 
( ( E. f 
f  e.  ( r 
RngIso  s )  /\  E. g  g  e.  (
s  RngIso  t ) )  ->  r  ~=r  t
) )
2827adantlr 695 . . . . . . . 8  |-  ( ( ( r  e.  RingOps  /\  s  e.  RingOps )  /\  ( s  e.  RingOps  /\  t  e.  RingOps ) )  ->  ( ( E. f  f  e.  ( r  RngIso  s )  /\  E. g  g  e.  ( s  RngIso  t ) )  ->  r  ~=r  t
) )
2928imp 418 . . . . . . 7  |-  ( ( ( ( r  e.  RingOps 
/\  s  e.  RingOps )  /\  ( s  e.  RingOps 
/\  t  e.  RingOps ) )  /\  ( E. f  f  e.  ( r  RngIso  s )  /\  E. g  g  e.  ( s  RngIso  t ) ) )  ->  r  ~=r  t )
3029an4s 799 . . . . . 6  |-  ( ( ( ( r  e.  RingOps 
/\  s  e.  RingOps )  /\  E. f  f  e.  ( r  RngIso  s ) )  /\  (
( s  e.  RingOps  /\  t  e.  RingOps )  /\  E. g  g  e.  ( s  RngIso  t ) ) )  ->  r  ~=r  t )
316, 17, 30syl2anb 465 . . . . 5  |-  ( ( r  ~=r  s  /\  s  ~=r  t )  -> 
r  ~=r  t )
3215, 31pm3.2i 441 . . . 4  |-  ( ( r  ~=r  s  ->  s 
~=r  r )  /\  ( ( r  ~=r  s  /\  s  ~=r  t
)  ->  r  ~=r  t ) )
3332ax-gen 1536 . . 3  |-  A. t
( ( r  ~=r  s  ->  s  ~=r  r
)  /\  ( (
r  ~=r  s  /\  s  ~=r  t )  -> 
r  ~=r  t )
)
3433gen2 1537 . 2  |-  A. r A. s A. t ( ( r  ~=r  s  ->  s  ~=r  r )  /\  ( ( r  ~=r  s  /\  s  ~=r  t
)  ->  r  ~=r  t ) )
35 dfer2 6677 . 2  |-  (  ~=r  Er 
dom  ~=r  <->  ( Rel  ~=r  /\  dom  ~=r  =  dom  ~=r  /\  A. r A. s A. t
( ( r  ~=r  s  ->  s  ~=r  r
)  /\  ( (
r  ~=r  s  /\  s  ~=r  t )  -> 
r  ~=r  t )
) ) )
362, 3, 34, 35mpbir3an 1134 1  |-  ~=r  Er  dom  ~=r
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934   A.wal 1530   E.wex 1531    = wceq 1632    e. wcel 1696   class class class wbr 4039   `'ccnv 4704   dom cdm 4705    o. ccom 4709   Rel wrel 4710  (class class class)co 5874    Er wer 6673   RingOpscrngo 21058    RngIso crngiso 26695    ~=r crisc 26696
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-er 6676  df-map 6790  df-grpo 20874  df-gid 20875  df-ablo 20965  df-ass 20996  df-exid 20998  df-mgm 21002  df-sgr 21014  df-mndo 21021  df-rngo 21059  df-rngohom 26697  df-rngoiso 26710  df-risc 26717
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