Theorem opprring 18631

Description: An opposite ring is a ring. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Aug-2015.)

Hypothesis

Ref	Expression
opprbas.1	|- O = (oppr ` R )

Assertion

Ref	Expression
opprring	|- ( R e. Ring -> O e. Ring )

Proof of Theorem opprring

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		opprbas.1	. . . 4 |- O = (oppr ` R )
2		eqid 2622	. . . 4 |- ( Base ` R ) = ( Base ` R )
3	1, 2	opprbas 18629	. . 3 |- ( Base ` R ) = ( Base ` O )
4	3	a1i 11	. 2 |- ( R e. Ring -> ( Base ` R ) = ( Base ` O ) )
5		eqid 2622	. . . 4 |- ( +g ` R ) = ( +g ` R )
6	1, 5	oppradd 18630	. . 3 |- ( +g ` R ) = ( +g ` O )
7	6	a1i 11	. 2 |- ( R e. Ring -> ( +g ` R ) = ( +g ` O ) )
8		eqidd 2623	. 2 |- ( R e. Ring -> ( .r ` O ) = ( .r ` O ) )
9		ringgrp 18552	. . 3 |- ( R e. Ring -> R e. Grp )
10	3, 6	grpprop 17438	. . 3 |- ( R e. Grp <-> O e. Grp )
11	9, 10	sylib 208	. 2 |- ( R e. Ring -> O e. Grp )
12		eqid 2622	. . . 4 |- ( .r ` R ) = ( .r ` R )
13		eqid 2622	. . . 4 |- ( .r ` O ) = ( .r ` O )
14	2, 12, 1, 13	opprmul 18626	. . 3 |- ( x ( .r ` O ) y ) = ( y ( .r ` R ) x )
15	2, 12	ringcl 18561	. . . 4 |- ( ( R e. Ring /\ y e. ( Base ` R ) /\ x e. ( Base ` R ) ) -> ( y ( .r ` R ) x ) e. ( Base ` R ) )
16	15	3com23 1271	. . 3 |- ( ( R e. Ring /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> ( y ( .r ` R ) x ) e. ( Base ` R ) )
17	14, 16	syl5eqel 2705	. 2 |- ( ( R e. Ring /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> ( x ( .r ` O ) y ) e. ( Base ` R ) )
18		simpl 473	. . . . 5 |- ( ( R e. Ring /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) ) -> R e. Ring )
19		simpr3 1069	. . . . 5 |- ( ( R e. Ring /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) ) -> z e. ( Base ` R ) )
20		simpr2 1068	. . . . 5 |- ( ( R e. Ring /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) ) -> y e. ( Base ` R ) )
21		simpr1 1067	. . . . 5 |- ( ( R e. Ring /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) ) -> x e. ( Base ` R ) )
22	2, 12	ringass 18564	. . . . 5 |- ( ( R e. Ring /\ ( z e. ( Base ` R ) /\ y e. ( Base ` R ) /\ x e. ( Base ` R ) ) ) -> ( ( z ( .r ` R ) y ) ( .r ` R ) x ) = ( z ( .r ` R ) ( y ( .r ` R ) x ) ) )
23	18, 19, 20, 21, 22	syl13anc 1328	. . . 4 |- ( ( R e. Ring /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) ) -> ( ( z ( .r ` R ) y ) ( .r ` R ) x ) = ( z ( .r ` R ) ( y ( .r ` R ) x ) ) )
24	23	eqcomd 2628	. . 3 |- ( ( R e. Ring /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) ) -> ( z ( .r ` R ) ( y ( .r ` R ) x ) ) = ( ( z ( .r ` R ) y ) ( .r ` R ) x ) )
25	14	oveq1i 6660	. . . 4 |- ( ( x ( .r ` O ) y ) ( .r ` O ) z ) = ( ( y ( .r ` R ) x ) ( .r ` O ) z )
26	2, 12, 1, 13	opprmul 18626	. . . 4 |- ( ( y ( .r ` R ) x ) ( .r ` O ) z ) = ( z ( .r ` R ) ( y ( .r ` R ) x ) )
27	25, 26	eqtri 2644	. . 3 |- ( ( x ( .r ` O ) y ) ( .r ` O ) z ) = ( z ( .r ` R ) ( y ( .r ` R ) x ) )
28	2, 12, 1, 13	opprmul 18626	. . . . 5 |- ( y ( .r ` O ) z ) = ( z ( .r ` R ) y )
29	28	oveq2i 6661	. . . 4 |- ( x ( .r ` O ) ( y ( .r ` O ) z ) ) = ( x ( .r ` O ) ( z ( .r ` R ) y ) )
30	2, 12, 1, 13	opprmul 18626	. . . 4 |- ( x ( .r ` O ) ( z ( .r ` R ) y ) ) = ( ( z ( .r ` R ) y ) ( .r ` R ) x )
31	29, 30	eqtri 2644	. . 3 |- ( x ( .r ` O ) ( y ( .r ` O ) z ) ) = ( ( z ( .r ` R ) y ) ( .r ` R ) x )
32	24, 27, 31	3eqtr4g 2681	. 2 |- ( ( R e. Ring /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) ) -> ( ( x ( .r ` O ) y ) ( .r ` O ) z ) = ( x ( .r ` O ) ( y ( .r ` O ) z ) ) )
33	2, 5, 12	ringdir 18567	. . . 4 |- ( ( R e. Ring /\ ( y e. ( Base ` R ) /\ z e. ( Base ` R ) /\ x e. ( Base ` R ) ) ) -> ( ( y ( +g ` R ) z ) ( .r ` R ) x ) = ( ( y ( .r ` R ) x ) ( +g ` R ) ( z ( .r ` R ) x ) ) )
34	18, 20, 19, 21, 33	syl13anc 1328	. . 3 |- ( ( R e. Ring /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) ) -> ( ( y ( +g ` R ) z ) ( .r ` R ) x ) = ( ( y ( .r ` R ) x ) ( +g ` R ) ( z ( .r ` R ) x ) ) )
35	2, 12, 1, 13	opprmul 18626	. . 3 |- ( x ( .r ` O ) ( y ( +g ` R ) z ) ) = ( ( y ( +g ` R ) z ) ( .r ` R ) x )
36	2, 12, 1, 13	opprmul 18626	. . . 4 |- ( x ( .r ` O ) z ) = ( z ( .r ` R ) x )
37	14, 36	oveq12i 6662	. . 3 |- ( ( x ( .r ` O ) y ) ( +g ` R ) ( x ( .r ` O ) z ) ) = ( ( y ( .r ` R ) x ) ( +g ` R ) ( z ( .r ` R ) x ) )
38	34, 35, 37	3eqtr4g 2681	. 2 |- ( ( R e. Ring /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) ) -> ( x ( .r ` O ) ( y ( +g ` R ) z ) ) = ( ( x ( .r ` O ) y ) ( +g ` R ) ( x ( .r ` O ) z ) ) )
39	2, 5, 12	ringdi 18566	. . . 4 |- ( ( R e. Ring /\ ( z e. ( Base ` R ) /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) -> ( z ( .r ` R ) ( x ( +g ` R ) y ) ) = ( ( z ( .r ` R ) x ) ( +g ` R ) ( z ( .r ` R ) y ) ) )
40	18, 19, 21, 20, 39	syl13anc 1328	. . 3 |- ( ( R e. Ring /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) ) -> ( z ( .r ` R ) ( x ( +g ` R ) y ) ) = ( ( z ( .r ` R ) x ) ( +g ` R ) ( z ( .r ` R ) y ) ) )
41	2, 12, 1, 13	opprmul 18626	. . 3 |- ( ( x ( +g ` R ) y ) ( .r ` O ) z ) = ( z ( .r ` R ) ( x ( +g ` R ) y ) )
42	36, 28	oveq12i 6662	. . 3 |- ( ( x ( .r ` O ) z ) ( +g ` R ) ( y ( .r ` O ) z ) ) = ( ( z ( .r ` R ) x ) ( +g ` R ) ( z ( .r ` R ) y ) )
43	40, 41, 42	3eqtr4g 2681	. 2 |- ( ( R e. Ring /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) ) -> ( ( x ( +g ` R ) y ) ( .r ` O ) z ) = ( ( x ( .r ` O ) z ) ( +g ` R ) ( y ( .r ` O ) z ) ) )
44		eqid 2622	. . 3 |- ( 1r ` R ) = ( 1r ` R )
45	2, 44	ringidcl 18568	. 2 |- ( R e. Ring -> ( 1r ` R ) e. ( Base ` R ) )
46	2, 12, 1, 13	opprmul 18626	. . 3 |- ( ( 1r ` R ) ( .r ` O ) x ) = ( x ( .r ` R ) ( 1r ` R ) )
47	2, 12, 44	ringridm 18572	. . 3 |- ( ( R e. Ring /\ x e. ( Base ` R ) ) -> ( x ( .r ` R ) ( 1r ` R ) ) = x )
48	46, 47	syl5eq 2668	. 2 |- ( ( R e. Ring /\ x e. ( Base ` R ) ) -> ( ( 1r ` R ) ( .r ` O ) x ) = x )
49	2, 12, 1, 13	opprmul 18626	. . 3 |- ( x ( .r ` O ) ( 1r ` R ) ) = ( ( 1r ` R ) ( .r ` R ) x )
50	2, 12, 44	ringlidm 18571	. . 3 |- ( ( R e. Ring /\ x e. ( Base ` R ) ) -> ( ( 1r ` R ) ( .r ` R ) x ) = x )
51	49, 50	syl5eq 2668	. 2 |- ( ( R e. Ring /\ x e. ( Base ` R ) ) -> ( x ( .r ` O ) ( 1r ` R ) ) = x )
52	4, 7, 8, 11, 17, 32, 38, 43, 45, 48, 51	isringd 18585	1 |- ( R e. Ring -> O e. Ring )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   .rcmulr 15942   Grpcgrp 17422   1rcur 18501   Ringcrg 18547  opp_rcoppr 18622

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  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-nel 2898  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-om 7066  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-plusg 15954  df-mulr 15955  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-mgp 18490  df-ur 18502  df-ring 18549  df-oppr 18623

This theorem is referenced by:  opprringb  18632  mulgass3  18637  1unit  18658  unitmulcl  18664  unitnegcl  18681  irredlmul  18708  isdrngrd  18773  issrngd  18861  2idlcpbl  19234  opprnzr  19265  ply1divalg2  23898  lduallmodlem  34439