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Theorem riscer 31930
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 31925 . . 3  |-  ~=R  =  { <. r ,  s
>.  |  ( (
r  e.  RingOps  /\  s  e.  RingOps )  /\  E. f  f  e.  (
r  RngIso  s ) ) }
21relopabi 4979 . 2  |-  Rel  ~=R
3 eqid 2429 . 2  |-  dom  ~=R  =  dom  ~=R
4 vex 3090 . . . . . . 7  |-  r  e. 
_V
5 vex 3090 . . . . . . 7  |-  s  e. 
_V
64, 5isrisc 31927 . . . . . 6  |-  ( r 
~=R  s  <->  ( (
r  e.  RingOps  /\  s  e.  RingOps )  /\  E. f  f  e.  (
r  RngIso  s ) ) )
7 rngoisocnv 31923 . . . . . . . . . 10  |-  ( ( r  e.  RingOps  /\  s  e.  RingOps  /\  f  e.  ( r  RngIso  s ) )  ->  `' f  e.  ( s  RngIso  r ) )
873expia 1207 . . . . . . . . 9  |-  ( ( r  e.  RingOps  /\  s  e.  RingOps )  ->  (
f  e.  ( r 
RngIso  s )  ->  `' f  e.  ( s  RngIso  r ) ) )
9 risci 31929 . . . . . . . . . . 11  |-  ( ( s  e.  RingOps  /\  r  e.  RingOps  /\  `' f  e.  ( s  RngIso  r ) )  ->  s  ~=R  r )
1093expia 1207 . . . . . . . . . 10  |-  ( ( s  e.  RingOps  /\  r  e.  RingOps )  ->  ( `' f  e.  (
s  RngIso  r )  -> 
s  ~=R  r )
)
1110ancoms 454 . . . . . . . . 9  |-  ( ( r  e.  RingOps  /\  s  e.  RingOps )  ->  ( `' f  e.  (
s  RngIso  r )  -> 
s  ~=R  r )
)
128, 11syld 45 . . . . . . . 8  |-  ( ( r  e.  RingOps  /\  s  e.  RingOps )  ->  (
f  e.  ( r 
RngIso  s )  ->  s  ~=R  r ) )
1312exlimdv 1771 . . . . . . 7  |-  ( ( r  e.  RingOps  /\  s  e.  RingOps )  ->  ( E. f  f  e.  ( r  RngIso  s )  ->  s  ~=R  r
) )
1413imp 430 . . . . . 6  |-  ( ( ( r  e.  RingOps  /\  s  e.  RingOps )  /\  E. f  f  e.  ( r  RngIso  s ) )  ->  s  ~=R  r
)
156, 14sylbi 198 . . . . 5  |-  ( r 
~=R  s  ->  s  ~=R  r )
16 vex 3090 . . . . . . 7  |-  t  e. 
_V
175, 16isrisc 31927 . . . . . 6  |-  ( s 
~=R  t  <->  ( (
s  e.  RingOps  /\  t  e.  RingOps )  /\  E. g  g  e.  (
s  RngIso  t ) ) )
18 eeanv 2045 . . . . . . . . . . 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 31924 . . . . . . . . . . . . . 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 435 . . . . . . . . . . . . 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 31929 . . . . . . . . . . . . . . 15  |-  ( ( r  e.  RingOps  /\  t  e.  RingOps  /\  ( g  o.  f )  e.  ( r  RngIso  t ) )  ->  r  ~=R  t
)
22213expia 1207 . . . . . . . . . . . . . 14  |-  ( ( r  e.  RingOps  /\  t  e.  RingOps )  ->  (
( g  o.  f
)  e.  ( r 
RngIso  t )  ->  r  ~=R  t ) )
23223adant2 1024 . . . . . . . . . . . . 13  |-  ( ( r  e.  RingOps  /\  s  e.  RingOps  /\  t  e.  RingOps )  ->  ( ( g  o.  f )  e.  ( r  RngIso  t )  ->  r  ~=R  t
) )
2420, 23syld 45 . . . . . . . . . . . 12  |-  ( ( r  e.  RingOps  /\  s  e.  RingOps  /\  t  e.  RingOps )  ->  ( ( f  e.  ( r  RngIso  s )  /\  g  e.  ( s  RngIso  t ) )  ->  r  ~=R  t ) )
2524exlimdvv 1772 . . . . . . . . . . 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 221 . . . . . . . . . 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 1206 . . . . . . . . 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 719 . . . . . . . 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 430 . . . . . . 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 833 . . . . . 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 481 . . . . 5  |-  ( ( r  ~=R  s  /\  s  ~=R  t )  -> 
r  ~=R  t )
3215, 31pm3.2i 456 . . . 4  |-  ( ( r  ~=R  s  ->  s 
~=R  r )  /\  ( ( r  ~=R  s  /\  s  ~=R  t
)  ->  r  ~=R  t ) )
3332ax-gen 1665 . . 3  |-  A. t
( ( r  ~=R  s  ->  s  ~=R  r
)  /\  ( (
r  ~=R  s  /\  s  ~=R  t )  -> 
r  ~=R  t )
)
3433gen2 1666 . 2  |-  A. r A. s A. t ( ( r  ~=R  s  ->  s  ~=R  r )  /\  ( ( r  ~=R  s  /\  s  ~=R  t
)  ->  r  ~=R  t ) )
35 dfer2 7372 . 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 1187 1  |-  ~=R  Er  dom  ~=R
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982   A.wal 1435    = wceq 1437   E.wex 1659    e. wcel 1870   class class class wbr 4426   `'ccnv 4853   dom cdm 4854    o. ccom 4858   Rel wrel 4859  (class class class)co 6305    Er wer 7368   RingOpscrngo 25948    RngIso crngiso 31903    ~=R crisc 31904
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-er 7371  df-map 7482  df-grpo 25764  df-gid 25765  df-ablo 25855  df-ass 25886  df-exid 25888  df-mgmOLD 25892  df-sgrOLD 25904  df-mndo 25911  df-rngo 25949  df-rngohom 31905  df-rngoiso 31918  df-risc 31925
This theorem is referenced by: (None)
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