Theorem opprunitd 13277

Description: Being a unit is a symmetric property, so it transfers to the opposite ring. (Contributed by Mario Carneiro, 4-Dec-2014.)

Hypotheses

Ref	Expression
opprunitd.1	|- ( ph -> U = (Unit ` R ) )
opprunitd.2	|- ( ph -> S = (oppr ` R ) )
opprunitd.r	|- ( ph -> R e. Ring )

Assertion

Ref	Expression
opprunitd	|- ( ph -> U = (Unit ` S ) )

Proof of Theorem opprunitd

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

Step	Hyp	Ref	Expression
1		opprunitd.1	. . . . . 6 |- ( ph -> U = (Unit ` R ) )
2		eqidd 2178	. . . . . 6 |- ( ph -> ( 1r ` R ) = ( 1r ` R ) )
3		eqidd 2178	. . . . . 6 |- ( ph -> ( ||r ` R ) = ( ||r ` R ) )
4		opprunitd.2	. . . . . 6 |- ( ph -> S = (oppr ` R ) )
5		eqidd 2178	. . . . . 6 |- ( ph -> ( ||r ` S ) = ( ||r ` S ) )
6		opprunitd.r	. . . . . . 7 |- ( ph -> R e. Ring )
7		ringsrg 13222	. . . . . . 7 |- ( R e. Ring -> R e. SRing )
8	6, 7	syl 14	. . . . . 6 |- ( ph -> R e. SRing )
9	1, 2, 3, 4, 5, 8	isunitd 13273	. . . . 5 |- ( ph -> ( x e. U <-> ( x ( ||r ` R ) ( 1r ` R ) /\ x ( ||r ` S ) ( 1r ` R ) ) ) )
10		eqid 2177	. . . . . . . . . . . . . . 15 |- (oppr ` R ) = (oppr ` R )
11	10	opprring 13247	. . . . . . . . . . . . . 14 |- ( R e. Ring -> (oppr ` R ) e. Ring )
12	6, 11	syl 14	. . . . . . . . . . . . 13 |- ( ph -> (oppr ` R ) e. Ring )
13	4, 12	eqeltrd 2254	. . . . . . . . . . . 12 |- ( ph -> S e. Ring )
14		vex 2740	. . . . . . . . . . . . 13 |- y e. _V
15	14	a1i 9	. . . . . . . . . . . 12 |- ( ph -> y e. _V )
16		vex 2740	. . . . . . . . . . . . 13 |- x e. _V
17	16	a1i 9	. . . . . . . . . . . 12 |- ( ph -> x e. _V )
18		eqid 2177	. . . . . . . . . . . . 13 |- ( Base ` S ) = ( Base ` S )
19		eqid 2177	. . . . . . . . . . . . 13 |- ( .r ` S ) = ( .r ` S )
20		eqid 2177	. . . . . . . . . . . . 13 |- (oppr ` S ) = (oppr ` S )
21		eqid 2177	. . . . . . . . . . . . 13 |- ( .r ` (oppr ` S ) ) = ( .r ` (oppr ` S ) )
22	18, 19, 20, 21	opprmulg 13241	. . . . . . . . . . . 12 |- ( ( S e. Ring /\ y e. _V /\ x e. _V ) -> ( y ( .r ` (oppr ` S ) ) x ) = ( x ( .r ` S ) y ) )
23	13, 15, 17, 22	syl3anc 1238	. . . . . . . . . . 11 |- ( ph -> ( y ( .r ` (oppr ` S ) ) x ) = ( x ( .r ` S ) y ) )
24	4	fveq2d 5519	. . . . . . . . . . . 12 |- ( ph -> ( .r ` S ) = ( .r ` (oppr ` R ) ) )
25	24	oveqd 5891	. . . . . . . . . . 11 |- ( ph -> ( x ( .r ` S ) y ) = ( x ( .r ` (oppr ` R ) ) y ) )
26		eqid 2177	. . . . . . . . . . . . 13 |- ( Base ` R ) = ( Base ` R )
27		eqid 2177	. . . . . . . . . . . . 13 |- ( .r ` R ) = ( .r ` R )
28		eqid 2177	. . . . . . . . . . . . 13 |- ( .r ` (oppr ` R ) ) = ( .r ` (oppr ` R ) )
29	26, 27, 10, 28	opprmulg 13241	. . . . . . . . . . . 12 |- ( ( R e. Ring /\ x e. _V /\ y e. _V ) -> ( x ( .r ` (oppr ` R ) ) y ) = ( y ( .r ` R ) x ) )
30	6, 17, 15, 29	syl3anc 1238	. . . . . . . . . . 11 |- ( ph -> ( x ( .r ` (oppr ` R ) ) y ) = ( y ( .r ` R ) x ) )
31	23, 25, 30	3eqtrrd 2215	. . . . . . . . . 10 |- ( ph -> ( y ( .r ` R ) x ) = ( y ( .r ` (oppr ` S ) ) x ) )
32	31	eqeq1d 2186	. . . . . . . . 9 |- ( ph -> ( ( y ( .r ` R ) x ) = ( 1r ` R ) <-> ( y ( .r ` (oppr ` S ) ) x ) = ( 1r ` R ) ) )
33	32	rexbidv 2478	. . . . . . . 8 |- ( ph -> ( E. y e. ( Base ` R ) ( y ( .r ` R ) x ) = ( 1r ` R ) <-> E. y e. ( Base ` R ) ( y ( .r ` (oppr ` S ) ) x ) = ( 1r ` R ) ) )
34	33	anbi2d 464	. . . . . . 7 |- ( ph -> ( ( x e. ( Base ` R ) /\ E. y e. ( Base ` R ) ( y ( .r ` R ) x ) = ( 1r ` R ) ) <-> ( x e. ( Base ` R ) /\ E. y e. ( Base ` R ) ( y ( .r ` (oppr ` S ) ) x ) = ( 1r ` R ) ) ) )
35		eqidd 2178	. . . . . . . 8 |- ( ph -> ( Base ` R ) = ( Base ` R ) )
36		eqidd 2178	. . . . . . . 8 |- ( ph -> ( .r ` R ) = ( .r ` R ) )
37	35, 3, 8, 36	dvdsrd 13261	. . . . . . 7 |- ( ph -> ( x ( ||r ` R ) ( 1r ` R ) <-> ( x e. ( Base ` R ) /\ E. y e. ( Base ` R ) ( y ( .r ` R ) x ) = ( 1r ` R ) ) ) )
38	10, 26	opprbasg 13245	. . . . . . . . . 10 |- ( R e. SRing -> ( Base ` R ) = ( Base ` (oppr ` R ) ) )
39	8, 38	syl 14	. . . . . . . . 9 |- ( ph -> ( Base ` R ) = ( Base ` (oppr ` R ) ) )
40	4	fveq2d 5519	. . . . . . . . 9 |- ( ph -> ( Base ` S ) = ( Base ` (oppr ` R ) ) )
41	20, 18	opprbasg 13245	. . . . . . . . . 10 |- ( S e. Ring -> ( Base ` S ) = ( Base ` (oppr ` S ) ) )
42	13, 41	syl 14	. . . . . . . . 9 |- ( ph -> ( Base ` S ) = ( Base ` (oppr ` S ) ) )
43	39, 40, 42	3eqtr2d 2216	. . . . . . . 8 |- ( ph -> ( Base ` R ) = ( Base ` (oppr ` S ) ) )
44		eqidd 2178	. . . . . . . 8 |- ( ph -> ( ||r ` (oppr ` S ) ) = ( ||r ` (oppr ` S ) ) )
45	20	opprring 13247	. . . . . . . . . 10 |- ( S e. Ring -> (oppr ` S ) e. Ring )
46	13, 45	syl 14	. . . . . . . . 9 |- ( ph -> (oppr ` S ) e. Ring )
47		ringsrg 13222	. . . . . . . . 9 |- ( (oppr ` S ) e. Ring -> (oppr ` S ) e. SRing )
48	46, 47	syl 14	. . . . . . . 8 |- ( ph -> (oppr ` S ) e. SRing )
49		eqidd 2178	. . . . . . . 8 |- ( ph -> ( .r ` (oppr ` S ) ) = ( .r ` (oppr ` S ) ) )
50	43, 44, 48, 49	dvdsrd 13261	. . . . . . 7 |- ( ph -> ( x ( ||r ` (oppr ` S ) ) ( 1r ` R ) <-> ( x e. ( Base ` R ) /\ E. y e. ( Base ` R ) ( y ( .r ` (oppr ` S ) ) x ) = ( 1r ` R ) ) ) )
51	34, 37, 50	3bitr4d 220	. . . . . 6 |- ( ph -> ( x ( ||r ` R ) ( 1r ` R ) <-> x ( ||r ` (oppr ` S ) ) ( 1r ` R ) ) )
52	51	anbi1d 465	. . . . 5 |- ( ph -> ( ( x ( ||r ` R ) ( 1r ` R ) /\ x ( ||r ` S ) ( 1r ` R ) ) <-> ( x ( ||r ` (oppr ` S ) ) ( 1r ` R ) /\ x ( ||r ` S ) ( 1r ` R ) ) ) )
53	9, 52	bitrd 188	. . . 4 |- ( ph -> ( x e. U <-> ( x ( ||r ` (oppr ` S ) ) ( 1r ` R ) /\ x ( ||r ` S ) ( 1r ` R ) ) ) )
54	53	biancomd 271	. . 3 |- ( ph -> ( x e. U <-> ( x ( ||r ` S ) ( 1r ` R ) /\ x ( ||r ` (oppr ` S ) ) ( 1r ` R ) ) ) )
55		eqidd 2178	. . . 4 |- ( ph -> (Unit ` S ) = (Unit ` S ) )
56		eqid 2177	. . . . . . 7 |- ( 1r ` R ) = ( 1r ` R )
57	10, 56	oppr1g 13250	. . . . . 6 |- ( R e. Ring -> ( 1r ` R ) = ( 1r ` (oppr ` R ) ) )
58	6, 57	syl 14	. . . . 5 |- ( ph -> ( 1r ` R ) = ( 1r ` (oppr ` R ) ) )
59	4	fveq2d 5519	. . . . 5 |- ( ph -> ( 1r ` S ) = ( 1r ` (oppr ` R ) ) )
60	58, 59	eqtr4d 2213	. . . 4 |- ( ph -> ( 1r ` R ) = ( 1r ` S ) )
61		eqidd 2178	. . . 4 |- ( ph -> (oppr ` S ) = (oppr ` S ) )
62		ringsrg 13222	. . . . 5 |- ( S e. Ring -> S e. SRing )
63	13, 62	syl 14	. . . 4 |- ( ph -> S e. SRing )
64	55, 60, 5, 61, 44, 63	isunitd 13273	. . 3 |- ( ph -> ( x e. (Unit ` S ) <-> ( x ( ||r ` S ) ( 1r ` R ) /\ x ( ||r ` (oppr ` S ) ) ( 1r ` R ) ) ) )
65	54, 64	bitr4d 191	. 2 |- ( ph -> ( x e. U <-> x e. (Unit ` S ) ) )
66	65	eqrdv 2175	1 |- ( ph -> U = (Unit ` S ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   E.wrex 2456   _Vcvv 2737   class class class wbr 4003   ` cfv 5216  (class class class)co 5874   Basecbs 12461   .rcmulr 12536   1rcur 13140  SRingcsrg 13144   Ringcrg 13177  opp_rcoppr 13237   ||_rcdsr 13253  Unitcui 13254

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-addcom 7910  ax-addass 7912  ax-i2m1 7915  ax-0lt1 7916  ax-0id 7918  ax-rnegex 7919  ax-pre-ltirr 7922  ax-pre-lttrn 7924  ax-pre-ltadd 7926

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-tpos 6245  df-pnf 7993  df-mnf 7994  df-ltxr 7996  df-inn 8919  df-2 8977  df-3 8978  df-ndx 12464  df-slot 12465  df-base 12467  df-sets 12468  df-plusg 12548  df-mulr 12549  df-0g 12706  df-mgm 12774  df-sgrp 12807  df-mnd 12817  df-grp 12879  df-minusg 12880  df-cmn 13088  df-abl 13089  df-mgp 13129  df-ur 13141  df-srg 13145  df-ring 13179  df-oppr 13238  df-dvdsr 13256  df-unit 13257

This theorem is referenced by: (None)