Theorem opprunitd 14074

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 2230	. . . . . 6 |- ( ph -> ( 1r ` R ) = ( 1r ` R ) )
3		eqidd 2230	. . . . . 6 |- ( ph -> ( ||r ` R ) = ( ||r ` R ) )
4		opprunitd.2	. . . . . 6 |- ( ph -> S = (oppr ` R ) )
5		eqidd 2230	. . . . . 6 |- ( ph -> ( ||r ` S ) = ( ||r ` S ) )
6		opprunitd.r	. . . . . . 7 |- ( ph -> R e. Ring )
7		ringsrg 14010	. . . . . . 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 14070	. . . . 5 |- ( ph -> ( x e. U <-> ( x ( ||r ` R ) ( 1r ` R ) /\ x ( ||r ` S ) ( 1r ` R ) ) ) )
10		eqid 2229	. . . . . . . . . . . . . . 15 |- (oppr ` R ) = (oppr ` R )
11	10	opprring 14042	. . . . . . . . . . . . . 14 |- ( R e. Ring -> (oppr ` R ) e. Ring )
12	6, 11	syl 14	. . . . . . . . . . . . 13 |- ( ph -> (oppr ` R ) e. Ring )
13	4, 12	eqeltrd 2306	. . . . . . . . . . . 12 |- ( ph -> S e. Ring )
14		vex 2802	. . . . . . . . . . . . 13 |- y e. _V
15	14	a1i 9	. . . . . . . . . . . 12 |- ( ph -> y e. _V )
16		vex 2802	. . . . . . . . . . . . 13 |- x e. _V
17	16	a1i 9	. . . . . . . . . . . 12 |- ( ph -> x e. _V )
18		eqid 2229	. . . . . . . . . . . . 13 |- ( Base ` S ) = ( Base ` S )
19		eqid 2229	. . . . . . . . . . . . 13 |- ( .r ` S ) = ( .r ` S )
20		eqid 2229	. . . . . . . . . . . . 13 |- (oppr ` S ) = (oppr ` S )
21		eqid 2229	. . . . . . . . . . . . 13 |- ( .r ` (oppr ` S ) ) = ( .r ` (oppr ` S ) )
22	18, 19, 20, 21	opprmulg 14034	. . . . . . . . . . . 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 1271	. . . . . . . . . . 11 |- ( ph -> ( y ( .r ` (oppr ` S ) ) x ) = ( x ( .r ` S ) y ) )
24	4	fveq2d 5631	. . . . . . . . . . . 12 |- ( ph -> ( .r ` S ) = ( .r ` (oppr ` R ) ) )
25	24	oveqd 6018	. . . . . . . . . . 11 |- ( ph -> ( x ( .r ` S ) y ) = ( x ( .r ` (oppr ` R ) ) y ) )
26		eqid 2229	. . . . . . . . . . . . 13 |- ( Base ` R ) = ( Base ` R )
27		eqid 2229	. . . . . . . . . . . . 13 |- ( .r ` R ) = ( .r ` R )
28		eqid 2229	. . . . . . . . . . . . 13 |- ( .r ` (oppr ` R ) ) = ( .r ` (oppr ` R ) )
29	26, 27, 10, 28	opprmulg 14034	. . . . . . . . . . . 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 1271	. . . . . . . . . . 11 |- ( ph -> ( x ( .r ` (oppr ` R ) ) y ) = ( y ( .r ` R ) x ) )
31	23, 25, 30	3eqtrrd 2267	. . . . . . . . . 10 |- ( ph -> ( y ( .r ` R ) x ) = ( y ( .r ` (oppr ` S ) ) x ) )
32	31	eqeq1d 2238	. . . . . . . . 9 |- ( ph -> ( ( y ( .r ` R ) x ) = ( 1r ` R ) <-> ( y ( .r ` (oppr ` S ) ) x ) = ( 1r ` R ) ) )
33	32	rexbidv 2531	. . . . . . . 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 2230	. . . . . . . 8 |- ( ph -> ( Base ` R ) = ( Base ` R ) )
36		eqidd 2230	. . . . . . . 8 |- ( ph -> ( .r ` R ) = ( .r ` R ) )
37	35, 3, 8, 36	dvdsrd 14058	. . . . . . 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 14038	. . . . . . . . . 10 |- ( R e. SRing -> ( Base ` R ) = ( Base ` (oppr ` R ) ) )
39	8, 38	syl 14	. . . . . . . . 9 |- ( ph -> ( Base ` R ) = ( Base ` (oppr ` R ) ) )
40	4	fveq2d 5631	. . . . . . . . 9 |- ( ph -> ( Base ` S ) = ( Base ` (oppr ` R ) ) )
41	20, 18	opprbasg 14038	. . . . . . . . . 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 2268	. . . . . . . 8 |- ( ph -> ( Base ` R ) = ( Base ` (oppr ` S ) ) )
44		eqidd 2230	. . . . . . . 8 |- ( ph -> ( ||r ` (oppr ` S ) ) = ( ||r ` (oppr ` S ) ) )
45	20	opprring 14042	. . . . . . . . . 10 |- ( S e. Ring -> (oppr ` S ) e. Ring )
46	13, 45	syl 14	. . . . . . . . 9 |- ( ph -> (oppr ` S ) e. Ring )
47		ringsrg 14010	. . . . . . . . 9 |- ( (oppr ` S ) e. Ring -> (oppr ` S ) e. SRing )
48	46, 47	syl 14	. . . . . . . 8 |- ( ph -> (oppr ` S ) e. SRing )
49		eqidd 2230	. . . . . . . 8 |- ( ph -> ( .r ` (oppr ` S ) ) = ( .r ` (oppr ` S ) ) )
50	43, 44, 48, 49	dvdsrd 14058	. . . . . . 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 2230	. . . 4 |- ( ph -> (Unit ` S ) = (Unit ` S ) )
56		eqid 2229	. . . . . . 7 |- ( 1r ` R ) = ( 1r ` R )
57	10, 56	oppr1g 14045	. . . . . 6 |- ( R e. Ring -> ( 1r ` R ) = ( 1r ` (oppr ` R ) ) )
58	6, 57	syl 14	. . . . 5 |- ( ph -> ( 1r ` R ) = ( 1r ` (oppr ` R ) ) )
59	4	fveq2d 5631	. . . . 5 |- ( ph -> ( 1r ` S ) = ( 1r ` (oppr ` R ) ) )
60	58, 59	eqtr4d 2265	. . . 4 |- ( ph -> ( 1r ` R ) = ( 1r ` S ) )
61		eqidd 2230	. . . 4 |- ( ph -> (oppr ` S ) = (oppr ` S ) )
62		ringsrg 14010	. . . . 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 14070	. . 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 2227	1 |- ( ph -> U = (Unit ` S ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   E.wrex 2509   _Vcvv 2799   class class class wbr 4083   ` cfv 5318  (class class class)co 6001   Basecbs 13032   .rcmulr 13111   1rcur 13922  SRingcsrg 13926   Ringcrg 13959  opp_rcoppr 14030   ||_rcdsr 14049  Unitcui 14050

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-addcom 8099  ax-addass 8101  ax-i2m1 8104  ax-0lt1 8105  ax-0id 8107  ax-rnegex 8108  ax-pre-ltirr 8111  ax-pre-lttrn 8113  ax-pre-ltadd 8115

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-tpos 6391  df-pnf 8183  df-mnf 8184  df-ltxr 8186  df-inn 9111  df-2 9169  df-3 9170  df-ndx 13035  df-slot 13036  df-base 13038  df-sets 13039  df-plusg 13123  df-mulr 13124  df-0g 13291  df-mgm 13389  df-sgrp 13435  df-mnd 13450  df-grp 13536  df-minusg 13537  df-cmn 13823  df-abl 13824  df-mgp 13884  df-ur 13923  df-srg 13927  df-ring 13961  df-oppr 14031  df-dvdsr 14052  df-unit 14053

This theorem is referenced by: (None)