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Theorem riscer 28794
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 28789 . . 3  |-  ~=r  =  { <. r ,  s
>.  |  ( (
r  e.  RingOps  /\  s  e.  RingOps )  /\  E. f  f  e.  (
r  RngIso  s ) ) }
21relopabi 4965 . 2  |-  Rel  ~=r
3 eqid 2443 . 2  |-  dom  ~=r  =  dom  ~=r
4 vex 2975 . . . . . . 7  |-  r  e. 
_V
5 vex 2975 . . . . . . 7  |-  s  e. 
_V
64, 5isrisc 28791 . . . . . 6  |-  ( r 
~=r  s  <->  ( (
r  e.  RingOps  /\  s  e.  RingOps )  /\  E. f  f  e.  (
r  RngIso  s ) ) )
7 rngoisocnv 28787 . . . . . . . . . 10  |-  ( ( r  e.  RingOps  /\  s  e.  RingOps  /\  f  e.  ( r  RngIso  s ) )  ->  `' f  e.  ( s  RngIso  r ) )
873expia 1189 . . . . . . . . 9  |-  ( ( r  e.  RingOps  /\  s  e.  RingOps )  ->  (
f  e.  ( r 
RngIso  s )  ->  `' f  e.  ( s  RngIso  r ) ) )
9 risci 28793 . . . . . . . . . . 11  |-  ( ( s  e.  RingOps  /\  r  e.  RingOps  /\  `' f  e.  ( s  RngIso  r ) )  ->  s  ~=r  r )
1093expia 1189 . . . . . . . . . 10  |-  ( ( s  e.  RingOps  /\  r  e.  RingOps )  ->  ( `' f  e.  (
s  RngIso  r )  -> 
s  ~=r  r )
)
1110ancoms 453 . . . . . . . . 9  |-  ( ( r  e.  RingOps  /\  s  e.  RingOps )  ->  ( `' f  e.  (
s  RngIso  r )  -> 
s  ~=r  r )
)
128, 11syld 44 . . . . . . . 8  |-  ( ( r  e.  RingOps  /\  s  e.  RingOps )  ->  (
f  e.  ( r 
RngIso  s )  ->  s  ~=r  r ) )
1312exlimdv 1690 . . . . . . 7  |-  ( ( r  e.  RingOps  /\  s  e.  RingOps )  ->  ( E. f  f  e.  ( r  RngIso  s )  ->  s  ~=r  r
) )
1413imp 429 . . . . . 6  |-  ( ( ( r  e.  RingOps  /\  s  e.  RingOps )  /\  E. f  f  e.  ( r  RngIso  s ) )  ->  s  ~=r  r
)
156, 14sylbi 195 . . . . 5  |-  ( r 
~=r  s  ->  s  ~=r  r )
16 vex 2975 . . . . . . 7  |-  t  e. 
_V
175, 16isrisc 28791 . . . . . 6  |-  ( s 
~=r  t  <->  ( (
s  e.  RingOps  /\  t  e.  RingOps )  /\  E. g  g  e.  (
s  RngIso  t ) ) )
18 eeanv 1932 . . . . . . . . . . 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 28788 . . . . . . . . . . . . . 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 434 . . . . . . . . . . . . 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 28793 . . . . . . . . . . . . . . 15  |-  ( ( r  e.  RingOps  /\  t  e.  RingOps  /\  ( g  o.  f )  e.  ( r  RngIso  t ) )  ->  r  ~=r  t
)
22213expia 1189 . . . . . . . . . . . . . 14  |-  ( ( r  e.  RingOps  /\  t  e.  RingOps )  ->  (
( g  o.  f
)  e.  ( r 
RngIso  t )  ->  r  ~=r  t ) )
23223adant2 1007 . . . . . . . . . . . . 13  |-  ( ( r  e.  RingOps  /\  s  e.  RingOps  /\  t  e.  RingOps )  ->  ( ( g  o.  f )  e.  ( r  RngIso  t )  ->  r  ~=r  t
) )
2420, 23syld 44 . . . . . . . . . . . 12  |-  ( ( r  e.  RingOps  /\  s  e.  RingOps  /\  t  e.  RingOps )  ->  ( ( f  e.  ( r  RngIso  s )  /\  g  e.  ( s  RngIso  t ) )  ->  r  ~=r  t ) )
2524exlimdvv 1691 . . . . . . . . . . 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 218 . . . . . . . . . 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 1188 . . . . . . . . 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 714 . . . . . . . 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 429 . . . . . . 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 822 . . . . . 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 479 . . . . 5  |-  ( ( r  ~=r  s  /\  s  ~=r  t )  -> 
r  ~=r  t )
3215, 31pm3.2i 455 . . . 4  |-  ( ( r  ~=r  s  ->  s 
~=r  r )  /\  ( ( r  ~=r  s  /\  s  ~=r  t
)  ->  r  ~=r  t ) )
3332ax-gen 1591 . . 3  |-  A. t
( ( r  ~=r  s  ->  s  ~=r  r
)  /\  ( (
r  ~=r  s  /\  s  ~=r  t )  -> 
r  ~=r  t )
)
3433gen2 1592 . 2  |-  A. r A. s A. t ( ( r  ~=r  s  ->  s  ~=r  r )  /\  ( ( r  ~=r  s  /\  s  ~=r  t
)  ->  r  ~=r  t ) )
35 dfer2 7102 . 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 1170 1  |-  ~=r  Er  dom  ~=r
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965   A.wal 1367    = wceq 1369   E.wex 1586    e. wcel 1756   class class class wbr 4292   `'ccnv 4839   dom cdm 4840    o. ccom 4844   Rel wrel 4845  (class class class)co 6091    Er wer 7098   RingOpscrngo 23862    RngIso crngiso 28767    ~=r crisc 28768
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-1st 6577  df-2nd 6578  df-er 7101  df-map 7216  df-grpo 23678  df-gid 23679  df-ablo 23769  df-ass 23800  df-exid 23802  df-mgm 23806  df-sgr 23818  df-mndo 23825  df-rngo 23863  df-rngohom 28769  df-rngoiso 28782  df-risc 28789
This theorem is referenced by: (None)
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