Theorem opprunitd 13609

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 2194	. . . . . 6 |- ( ph -> ( 1r ` R ) = ( 1r ` R ) )
3		eqidd 2194	. . . . . 6 |- ( ph -> ( ||r ` R ) = ( ||r ` R ) )
4		opprunitd.2	. . . . . 6 |- ( ph -> S = (oppr ` R ) )
5		eqidd 2194	. . . . . 6 |- ( ph -> ( ||r ` S ) = ( ||r ` S ) )
6		opprunitd.r	. . . . . . 7 |- ( ph -> R e. Ring )
7		ringsrg 13546	. . . . . . 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 13605	. . . . 5 |- ( ph -> ( x e. U <-> ( x ( ||r ` R ) ( 1r ` R ) /\ x ( ||r ` S ) ( 1r ` R ) ) ) )
10		eqid 2193	. . . . . . . . . . . . . . 15 |- (oppr ` R ) = (oppr ` R )
11	10	opprring 13578	. . . . . . . . . . . . . 14 |- ( R e. Ring -> (oppr ` R ) e. Ring )
12	6, 11	syl 14	. . . . . . . . . . . . 13 |- ( ph -> (oppr ` R ) e. Ring )
13	4, 12	eqeltrd 2270	. . . . . . . . . . . 12 |- ( ph -> S e. Ring )
14		vex 2763	. . . . . . . . . . . . 13 |- y e. _V
15	14	a1i 9	. . . . . . . . . . . 12 |- ( ph -> y e. _V )
16		vex 2763	. . . . . . . . . . . . 13 |- x e. _V
17	16	a1i 9	. . . . . . . . . . . 12 |- ( ph -> x e. _V )
18		eqid 2193	. . . . . . . . . . . . 13 |- ( Base ` S ) = ( Base ` S )
19		eqid 2193	. . . . . . . . . . . . 13 |- ( .r ` S ) = ( .r ` S )
20		eqid 2193	. . . . . . . . . . . . 13 |- (oppr ` S ) = (oppr ` S )
21		eqid 2193	. . . . . . . . . . . . 13 |- ( .r ` (oppr ` S ) ) = ( .r ` (oppr ` S ) )
22	18, 19, 20, 21	opprmulg 13570	. . . . . . . . . . . 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 1249	. . . . . . . . . . 11 |- ( ph -> ( y ( .r ` (oppr ` S ) ) x ) = ( x ( .r ` S ) y ) )
24	4	fveq2d 5559	. . . . . . . . . . . 12 |- ( ph -> ( .r ` S ) = ( .r ` (oppr ` R ) ) )
25	24	oveqd 5936	. . . . . . . . . . 11 |- ( ph -> ( x ( .r ` S ) y ) = ( x ( .r ` (oppr ` R ) ) y ) )
26		eqid 2193	. . . . . . . . . . . . 13 |- ( Base ` R ) = ( Base ` R )
27		eqid 2193	. . . . . . . . . . . . 13 |- ( .r ` R ) = ( .r ` R )
28		eqid 2193	. . . . . . . . . . . . 13 |- ( .r ` (oppr ` R ) ) = ( .r ` (oppr ` R ) )
29	26, 27, 10, 28	opprmulg 13570	. . . . . . . . . . . 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 1249	. . . . . . . . . . 11 |- ( ph -> ( x ( .r ` (oppr ` R ) ) y ) = ( y ( .r ` R ) x ) )
31	23, 25, 30	3eqtrrd 2231	. . . . . . . . . 10 |- ( ph -> ( y ( .r ` R ) x ) = ( y ( .r ` (oppr ` S ) ) x ) )
32	31	eqeq1d 2202	. . . . . . . . 9 |- ( ph -> ( ( y ( .r ` R ) x ) = ( 1r ` R ) <-> ( y ( .r ` (oppr ` S ) ) x ) = ( 1r ` R ) ) )
33	32	rexbidv 2495	. . . . . . . 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 2194	. . . . . . . 8 |- ( ph -> ( Base ` R ) = ( Base ` R ) )
36		eqidd 2194	. . . . . . . 8 |- ( ph -> ( .r ` R ) = ( .r ` R ) )
37	35, 3, 8, 36	dvdsrd 13593	. . . . . . 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 13574	. . . . . . . . . 10 |- ( R e. SRing -> ( Base ` R ) = ( Base ` (oppr ` R ) ) )
39	8, 38	syl 14	. . . . . . . . 9 |- ( ph -> ( Base ` R ) = ( Base ` (oppr ` R ) ) )
40	4	fveq2d 5559	. . . . . . . . 9 |- ( ph -> ( Base ` S ) = ( Base ` (oppr ` R ) ) )
41	20, 18	opprbasg 13574	. . . . . . . . . 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 2232	. . . . . . . 8 |- ( ph -> ( Base ` R ) = ( Base ` (oppr ` S ) ) )
44		eqidd 2194	. . . . . . . 8 |- ( ph -> ( ||r ` (oppr ` S ) ) = ( ||r ` (oppr ` S ) ) )
45	20	opprring 13578	. . . . . . . . . 10 |- ( S e. Ring -> (oppr ` S ) e. Ring )
46	13, 45	syl 14	. . . . . . . . 9 |- ( ph -> (oppr ` S ) e. Ring )
47		ringsrg 13546	. . . . . . . . 9 |- ( (oppr ` S ) e. Ring -> (oppr ` S ) e. SRing )
48	46, 47	syl 14	. . . . . . . 8 |- ( ph -> (oppr ` S ) e. SRing )
49		eqidd 2194	. . . . . . . 8 |- ( ph -> ( .r ` (oppr ` S ) ) = ( .r ` (oppr ` S ) ) )
50	43, 44, 48, 49	dvdsrd 13593	. . . . . . 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 2194	. . . 4 |- ( ph -> (Unit ` S ) = (Unit ` S ) )
56		eqid 2193	. . . . . . 7 |- ( 1r ` R ) = ( 1r ` R )
57	10, 56	oppr1g 13581	. . . . . 6 |- ( R e. Ring -> ( 1r ` R ) = ( 1r ` (oppr ` R ) ) )
58	6, 57	syl 14	. . . . 5 |- ( ph -> ( 1r ` R ) = ( 1r ` (oppr ` R ) ) )
59	4	fveq2d 5559	. . . . 5 |- ( ph -> ( 1r ` S ) = ( 1r ` (oppr ` R ) ) )
60	58, 59	eqtr4d 2229	. . . 4 |- ( ph -> ( 1r ` R ) = ( 1r ` S ) )
61		eqidd 2194	. . . 4 |- ( ph -> (oppr ` S ) = (oppr ` S ) )
62		ringsrg 13546	. . . . 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 13605	. . 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 2191	1 |- ( ph -> U = (Unit ` S ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   E.wrex 2473   _Vcvv 2760   class class class wbr 4030   ` cfv 5255  (class class class)co 5919   Basecbs 12621   .rcmulr 12699   1rcur 13458  SRingcsrg 13462   Ringcrg 13495  opp_rcoppr 13566   ||_rcdsr 13585  Unitcui 13586

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-addcom 7974  ax-addass 7976  ax-i2m1 7979  ax-0lt1 7980  ax-0id 7982  ax-rnegex 7983  ax-pre-ltirr 7986  ax-pre-lttrn 7988  ax-pre-ltadd 7990

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-tpos 6300  df-pnf 8058  df-mnf 8059  df-ltxr 8061  df-inn 8985  df-2 9043  df-3 9044  df-ndx 12624  df-slot 12625  df-base 12627  df-sets 12628  df-plusg 12711  df-mulr 12712  df-0g 12872  df-mgm 12942  df-sgrp 12988  df-mnd 13001  df-grp 13078  df-minusg 13079  df-cmn 13359  df-abl 13360  df-mgp 13420  df-ur 13459  df-srg 13463  df-ring 13497  df-oppr 13567  df-dvdsr 13588  df-unit 13589

This theorem is referenced by: (None)