Theorem riscer 33787

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.

Step	Hyp	Ref	Expression
1		df-risc 33782	. . 3 |- ~=R = { <. r , s >. | ( ( r e. RingOps /\ s e. RingOps ) /\ E. f f e. ( r RngIso s ) ) }
2	1	relopabi 5245	. 2 |- Rel ~=R
3		eqid 2622	. 2 |- dom ~=R = dom ~=R
4		vex 3203	. . . . . . 7 |- r e. _V
5		vex 3203	. . . . . . 7 |- s e. _V
6	4, 5	isrisc 33784	. . . . . 6 |- ( r ~=R s <-> ( ( r e. RingOps /\ s e. RingOps ) /\ E. f f e. ( r RngIso s ) ) )
7		rngoisocnv 33780	. . . . . . . . . 10 |- ( ( r e. RingOps /\ s e. RingOps /\ f e. ( r RngIso s ) ) -> `' f e. ( s RngIso r ) )
8	7	3expia 1267	. . . . . . . . 9 |- ( ( r e. RingOps /\ s e. RingOps ) -> ( f e. ( r RngIso s ) -> `' f e. ( s RngIso r ) ) )
9		risci 33786	. . . . . . . . . . 11 |- ( ( s e. RingOps /\ r e. RingOps /\ `' f e. ( s RngIso r ) ) -> s ~=R r )
10	9	3expia 1267	. . . . . . . . . 10 |- ( ( s e. RingOps /\ r e. RingOps ) -> ( `' f e. ( s RngIso r ) -> s ~=R r ) )
11	10	ancoms 469	. . . . . . . . 9 |- ( ( r e. RingOps /\ s e. RingOps ) -> ( `' f e. ( s RngIso r ) -> s ~=R r ) )
12	8, 11	syld 47	. . . . . . . 8 |- ( ( r e. RingOps /\ s e. RingOps ) -> ( f e. ( r RngIso s ) -> s ~=R r ) )
13	12	exlimdv 1861	. . . . . . 7 |- ( ( r e. RingOps /\ s e. RingOps ) -> ( E. f f e. ( r RngIso s ) -> s ~=R r ) )
14	13	imp 445	. . . . . 6 |- ( ( ( r e. RingOps /\ s e. RingOps ) /\ E. f f e. ( r RngIso s ) ) -> s ~=R r )
15	6, 14	sylbi 207	. . . . 5 |- ( r ~=R s -> s ~=R r )
16		vex 3203	. . . . . . 7 |- t e. _V
17	5, 16	isrisc 33784	. . . . . 6 |- ( s ~=R t <-> ( ( s e. RingOps /\ t e. RingOps ) /\ E. g g e. ( s RngIso t ) ) )
18		eeanv 2182	. . . . . . . . . . 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 33781	. . . . . . . . . . . . . 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 ) )
20	19	ex 450	. . . . . . . . . . . . 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 33786	. . . . . . . . . . . . . . 15 |- ( ( r e. RingOps /\ t e. RingOps /\ ( g o. f ) e. ( r RngIso t ) ) -> r ~=R t )
22	21	3expia 1267	. . . . . . . . . . . . . 14 |- ( ( r e. RingOps /\ t e. RingOps ) -> ( ( g o. f ) e. ( r RngIso t ) -> r ~=R t ) )
23	22	3adant2 1080	. . . . . . . . . . . . 13 |- ( ( r e. RingOps /\ s e. RingOps /\ t e. RingOps ) -> ( ( g o. f ) e. ( r RngIso t ) -> r ~=R t ) )
24	20, 23	syld 47	. . . . . . . . . . . 12 |- ( ( r e. RingOps /\ s e. RingOps /\ t e. RingOps ) -> ( ( f e. ( r RngIso s ) /\ g e. ( s RngIso t ) ) -> r ~=R t ) )
25	24	exlimdvv 1862	. . . . . . . . . . 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 ) )
26	18, 25	syl5bir 233	. . . . . . . . . 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 ) )
27	26	3expb 1266	. . . . . . . . 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 ) )
28	27	adantlr 751	. . . . . . . 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 ) )
29	28	imp 445	. . . . . . 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 )
30	29	an4s 869	. . . . . 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 )
31	6, 17, 30	syl2anb 496	. . . . 5 |- ( ( r ~=R s /\ s ~=R t ) -> r ~=R t )
32	15, 31	pm3.2i 471	. . . 4 |- ( ( r ~=R s -> s ~=R r ) /\ ( ( r ~=R s /\ s ~=R t ) -> r ~=R t ) )
33	32	ax-gen 1722	. . 3 |- A. t ( ( r ~=R s -> s ~=R r ) /\ ( ( r ~=R s /\ s ~=R t ) -> r ~=R t ) )
34	33	gen2 1723	. 2 |- A. r A. s A. t ( ( r ~=R s -> s ~=R r ) /\ ( ( r ~=R s /\ s ~=R t ) -> r ~=R t ) )
35		dfer2 7743	. 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 ) ) ) )
36	2, 3, 34, 35	mpbir3an 1244	1 |- ~=R Er dom ~=R

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   class class class wbr 4653   `'ccnv 5113   dom cdm 5114   o. ccom 5118   Rel wrel 5119  (class class class)co 6650   Er wer 7739   RingOpscrngo 33693   RngIso crngiso 33760   ~=R crisc 33761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-er 7742  df-map 7859  df-grpo 27347  df-gid 27348  df-ablo 27399  df-ass 33642  df-exid 33644  df-mgmOLD 33648  df-sgrOLD 33660  df-mndo 33666  df-rngo 33694  df-rngohom 33762  df-rngoiso 33775  df-risc 33782

This theorem is referenced by: (None)