Theorem riscer 16142

Description: Ring isomorphism is an equivalence relation.

Assertion

Ref	Expression
riscer	|- Er ~=r

Proof of Theorem riscer

Step	Hyp	Ref	Expression
1		visset 2295	. . . 4 |- r e. _V
2		visset 2295	. . . 4 |- s e. _V
3	1, 2	isrisc 16139	. . 3 |- (r~=rs <-> ((r e. Ring /\ s e. Ring) /\ E.f f e. (r RngIso s)))
4		rngisocnv 16135	. . . . . . 7 |- ((r e. Ring /\ s e. Ring /\ f e. (r RngIso s)) -> `'f e. (s RngIso r))
5	4	3expia 1069	. . . . . 6 |- ((r e. Ring /\ s e. Ring) -> (f e. (r RngIso s) -> `'f e. (s RngIso r)))
6		risci 16141	. . . . . . . 8 |- ((s e. Ring /\ r e. Ring /\ `'f e. (s RngIso r)) -> s~=rr)
7	6	3expia 1069	. . . . . . 7 |- ((s e. Ring /\ r e. Ring) -> (`'f e. (s RngIso r) -> s~=rr))
8	7	ancoms 484	. . . . . 6 |- ((r e. Ring /\ s e. Ring) -> (`'f e. (s RngIso r) -> s~=rr))
9	5, 8	syld 30	. . . . 5 |- ((r e. Ring /\ s e. Ring) -> (f e. (r RngIso s) -> s~=rr))
10	9	19.23adv 1584	. . . 4 |- ((r e. Ring /\ s e. Ring) -> (E.f f e. (r RngIso s) -> s~=rr))
11	10	imp 377	. . 3 |- (((r e. Ring /\ s e. Ring) /\ E.f f e. (r RngIso s)) -> s~=rr)
12	3, 11	sylbi 216	. 2 |- (r~=rs -> s~=rr)
13		rngisoco 16136	. . . . . . . . . . 11 |- (((r e. Ring /\ s e. Ring /\ t e. Ring) /\ (f e. (r RngIso s) /\ g e. (s RngIso t))) -> (g o. f) e. (r RngIso t))
14	13	ex 402	. . . . . . . . . 10 |- ((r e. Ring /\ s e. Ring /\ t e. Ring) -> ((f e. (r RngIso s) /\ g e. (s RngIso t)) -> (g o. f) e. (r RngIso t)))
15		risci 16141	. . . . . . . . . . . 12 |- ((r e. Ring /\ t e. Ring /\ (g o. f) e. (r RngIso t)) -> r~=rt)
16	15	3expia 1069	. . . . . . . . . . 11 |- ((r e. Ring /\ t e. Ring) -> ((g o. f) e. (r RngIso t) -> r~=rt))
17	16	3adant2 895	. . . . . . . . . 10 |- ((r e. Ring /\ s e. Ring /\ t e. Ring) -> ((g o. f) e. (r RngIso t) -> r~=rt))
18	14, 17	syld 30	. . . . . . . . 9 |- ((r e. Ring /\ s e. Ring /\ t e. Ring) -> ((f e. (r RngIso s) /\ g e. (s RngIso t)) -> r~=rt))
19	18	19.23advv 1676	. . . . . . . 8 |- ((r e. Ring /\ s e. Ring /\ t e. Ring) -> (E.fE.g(f e. (r RngIso s) /\ g e. (s RngIso t)) -> r~=rt))
20		eeanv 1707	. . . . . . . 8 |- (E.fE.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)))
21	19, 20	syl5ibr 224	. . . . . . 7 |- ((r e. Ring /\ s e. Ring /\ t e. Ring) -> ((E.f f e. (r RngIso s) /\ E.g g e. (s RngIso t)) -> r~=rt))
22	21	3expb 1068	. . . . . 6 |- ((r e. Ring /\ (s e. Ring /\ t e. Ring)) -> ((E.f f e. (r RngIso s) /\ E.g g e. (s RngIso t)) -> r~=rt))
23	22	adantlr 429	. . . . 5 |- (((r e. Ring /\ s e. Ring) /\ (s e. Ring /\ t e. Ring)) -> ((E.f f e. (r RngIso s) /\ E.g g e. (s RngIso t)) -> r~=rt))
24	23	imp 377	. . . 4 |- ((((r e. Ring /\ s e. Ring) /\ (s e. Ring /\ t e. Ring)) /\ (E.f f e. (r RngIso s) /\ E.g g e. (s RngIso t))) -> r~=rt)
25	24	an4s 566	. . 3 |- ((((r e. Ring /\ s e. Ring) /\ E.f f e. (r RngIso s)) /\ ((s e. Ring /\ t e. Ring) /\ E.g g e. (s RngIso t))) -> r~=rt)
26		visset 2295	. . . 4 |- t e. _V
27	2, 26	isrisc 16139	. . 3 |- (s~=rt <-> ((s e. Ring /\ t e. Ring) /\ E.g g e. (s RngIso t)))
28	25, 3, 27	syl2anb 504	. 2 |- ((r~=rs /\ s~=rt) -> r~=rt)
29	12, 28	ster 5325	1 |- Er ~=r

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   e. wcel 1300  E.wex 1326   class class class wbr 3338  `'ccnv 3985   o. ccom 3990  (class class class)co 4884  Er wer 5315  Ringcring 9463   RngIso crngiso 16115  ~=rcrisc 16116

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-1st 5020  df-2nd 5021  df-er 5318  df-grp 9316  df-gid 9317  df-abl 9408  df-ring 9464  df-ass 10360  df-exid 10362  df-mgm 10366  df-sgr 10378  df-mnd 10385  df-rnghom 16117  df-rngiso 16130  df-risc 16137

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