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Theorem riscer 26619
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 26614 . . 3  |-  ~=r  =  { <. r ,  s
>.  |  ( (
r  e.  RingOps  /\  s  e.  RingOps )  /\  E. f  f  e.  (
r  RngIso  s ) ) }
21relopabi 4811 . 2  |-  Rel  ~=r
3 eqid 2283 . 2  |-  dom  ~=r  =  dom  ~=r
4 vex 2791 . . . . . . 7  |-  r  e. 
_V
5 vex 2791 . . . . . . 7  |-  s  e. 
_V
64, 5isrisc 26616 . . . . . 6  |-  ( r 
~=r  s  <->  ( (
r  e.  RingOps  /\  s  e.  RingOps )  /\  E. f  f  e.  (
r  RngIso  s ) ) )
7 rngoisocnv 26612 . . . . . . . . . 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 26618 . . . . . . . . . . 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 1664 . . . . . . 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 2791 . . . . . . 7  |-  t  e. 
_V
175, 16isrisc 26616 . . . . . 6  |-  ( s 
~=r  t  <->  ( (
s  e.  RingOps  /\  t  e.  RingOps )  /\  E. g  g  e.  (
s  RngIso  t ) ) )
18 eeanv 1854 . . . . . . . . . . 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 26613 . . . . . . . . . . . . . 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 26618 . . . . . . . . . . . . . . 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 1668 . . . . . . . . . . 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 1533 . . 3  |-  A. t
( ( r  ~=r  s  ->  s  ~=r  r
)  /\  ( (
r  ~=r  s  /\  s  ~=r  t )  -> 
r  ~=r  t )
)
3433gen2 1534 . 2  |-  A. r A. s A. t ( ( r  ~=r  s  ->  s  ~=r  r )  /\  ( ( r  ~=r  s  /\  s  ~=r  t
)  ->  r  ~=r  t ) )
35 dfer2 6661 . 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 1527   E.wex 1528    = wceq 1623    e. wcel 1684   class class class wbr 4023   `'ccnv 4688   dom cdm 4689    o. ccom 4693   Rel wrel 4694  (class class class)co 5858    Er wer 6657   RingOpscrngo 21042    RngIso crngiso 26592    ~=r crisc 26593
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-er 6660  df-map 6774  df-grpo 20858  df-gid 20859  df-ablo 20949  df-ass 20980  df-exid 20982  df-mgm 20986  df-sgr 20998  df-mndo 21005  df-rngo 21043  df-rngohom 26594  df-rngoiso 26607  df-risc 26614
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