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Theorem riscer 26595
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 26590 . . 3  |-  ~=r  =  { <. r ,  s
>.  |  ( (
r  e.  RingOps  /\  s  e.  RingOps )  /\  E. f  f  e.  (
r  RngIso  s ) ) }
21relopabi 4992 . 2  |-  Rel  ~=r
3 eqid 2435 . 2  |-  dom  ~=r  =  dom  ~=r
4 vex 2951 . . . . . . 7  |-  r  e. 
_V
5 vex 2951 . . . . . . 7  |-  s  e. 
_V
64, 5isrisc 26592 . . . . . 6  |-  ( r 
~=r  s  <->  ( (
r  e.  RingOps  /\  s  e.  RingOps )  /\  E. f  f  e.  (
r  RngIso  s ) ) )
7 rngoisocnv 26588 . . . . . . . . . 10  |-  ( ( r  e.  RingOps  /\  s  e.  RingOps  /\  f  e.  ( r  RngIso  s ) )  ->  `' f  e.  ( s  RngIso  r ) )
873expia 1155 . . . . . . . . 9  |-  ( ( r  e.  RingOps  /\  s  e.  RingOps )  ->  (
f  e.  ( r 
RngIso  s )  ->  `' f  e.  ( s  RngIso  r ) ) )
9 risci 26594 . . . . . . . . . . 11  |-  ( ( s  e.  RingOps  /\  r  e.  RingOps  /\  `' f  e.  ( s  RngIso  r ) )  ->  s  ~=r  r )
1093expia 1155 . . . . . . . . . 10  |-  ( ( s  e.  RingOps  /\  r  e.  RingOps )  ->  ( `' f  e.  (
s  RngIso  r )  -> 
s  ~=r  r )
)
1110ancoms 440 . . . . . . . . 9  |-  ( ( r  e.  RingOps  /\  s  e.  RingOps )  ->  ( `' f  e.  (
s  RngIso  r )  -> 
s  ~=r  r )
)
128, 11syld 42 . . . . . . . 8  |-  ( ( r  e.  RingOps  /\  s  e.  RingOps )  ->  (
f  e.  ( r 
RngIso  s )  ->  s  ~=r  r ) )
1312exlimdv 1646 . . . . . . 7  |-  ( ( r  e.  RingOps  /\  s  e.  RingOps )  ->  ( E. f  f  e.  ( r  RngIso  s )  ->  s  ~=r  r
) )
1413imp 419 . . . . . 6  |-  ( ( ( r  e.  RingOps  /\  s  e.  RingOps )  /\  E. f  f  e.  ( r  RngIso  s ) )  ->  s  ~=r  r
)
156, 14sylbi 188 . . . . 5  |-  ( r 
~=r  s  ->  s  ~=r  r )
16 vex 2951 . . . . . . 7  |-  t  e. 
_V
175, 16isrisc 26592 . . . . . 6  |-  ( s 
~=r  t  <->  ( (
s  e.  RingOps  /\  t  e.  RingOps )  /\  E. g  g  e.  (
s  RngIso  t ) ) )
18 eeanv 1937 . . . . . . . . . . 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 26589 . . . . . . . . . . . . . 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 424 . . . . . . . . . . . . 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 26594 . . . . . . . . . . . . . . 15  |-  ( ( r  e.  RingOps  /\  t  e.  RingOps  /\  ( g  o.  f )  e.  ( r  RngIso  t ) )  ->  r  ~=r  t
)
22213expia 1155 . . . . . . . . . . . . . 14  |-  ( ( r  e.  RingOps  /\  t  e.  RingOps )  ->  (
( g  o.  f
)  e.  ( r 
RngIso  t )  ->  r  ~=r  t ) )
23223adant2 976 . . . . . . . . . . . . 13  |-  ( ( r  e.  RingOps  /\  s  e.  RingOps  /\  t  e.  RingOps )  ->  ( ( g  o.  f )  e.  ( r  RngIso  t )  ->  r  ~=r  t
) )
2420, 23syld 42 . . . . . . . . . . . 12  |-  ( ( r  e.  RingOps  /\  s  e.  RingOps  /\  t  e.  RingOps )  ->  ( ( f  e.  ( r  RngIso  s )  /\  g  e.  ( s  RngIso  t ) )  ->  r  ~=r  t ) )
2524exlimdvv 1647 . . . . . . . . . . 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 210 . . . . . . . . . 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 1154 . . . . . . . . 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 696 . . . . . . . 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 419 . . . . . . 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 800 . . . . . 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 466 . . . . 5  |-  ( ( r  ~=r  s  /\  s  ~=r  t )  -> 
r  ~=r  t )
3215, 31pm3.2i 442 . . . 4  |-  ( ( r  ~=r  s  ->  s 
~=r  r )  /\  ( ( r  ~=r  s  /\  s  ~=r  t
)  ->  r  ~=r  t ) )
3332ax-gen 1555 . . 3  |-  A. t
( ( r  ~=r  s  ->  s  ~=r  r
)  /\  ( (
r  ~=r  s  /\  s  ~=r  t )  -> 
r  ~=r  t )
)
3433gen2 1556 . 2  |-  A. r A. s A. t ( ( r  ~=r  s  ->  s  ~=r  r )  /\  ( ( r  ~=r  s  /\  s  ~=r  t
)  ->  r  ~=r  t ) )
35 dfer2 6898 . 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 1136 1  |-  ~=r  Er  dom  ~=r
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936   A.wal 1549   E.wex 1550    = wceq 1652    e. wcel 1725   class class class wbr 4204   `'ccnv 4869   dom cdm 4870    o. ccom 4874   Rel wrel 4875  (class class class)co 6073    Er wer 6894   RingOpscrngo 21955    RngIso crngiso 26568    ~=r crisc 26569
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-er 6897  df-map 7012  df-grpo 21771  df-gid 21772  df-ablo 21862  df-ass 21893  df-exid 21895  df-mgm 21899  df-sgr 21911  df-mndo 21918  df-rngo 21956  df-rngohom 26570  df-rngoiso 26583  df-risc 26590
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