Theorem opprunitd 14205

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 2232	. . . . . 6 |- ( ph -> ( 1r ` R ) = ( 1r ` R ) )
3		eqidd 2232	. . . . . 6 |- ( ph -> ( ||r ` R ) = ( ||r ` R ) )
4		opprunitd.2	. . . . . 6 |- ( ph -> S = (oppr ` R ) )
5		eqidd 2232	. . . . . 6 |- ( ph -> ( ||r ` S ) = ( ||r ` S ) )
6		opprunitd.r	. . . . . . 7 |- ( ph -> R e. Ring )
7		ringsrg 14141	. . . . . . 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 14201	. . . . 5 |- ( ph -> ( x e. U <-> ( x ( ||r ` R ) ( 1r ` R ) /\ x ( ||r ` S ) ( 1r ` R ) ) ) )
10		eqid 2231	. . . . . . . . . . . . . . 15 |- (oppr ` R ) = (oppr ` R )
11	10	opprring 14173	. . . . . . . . . . . . . 14 |- ( R e. Ring -> (oppr ` R ) e. Ring )
12	6, 11	syl 14	. . . . . . . . . . . . 13 |- ( ph -> (oppr ` R ) e. Ring )
13	4, 12	eqeltrd 2308	. . . . . . . . . . . 12 |- ( ph -> S e. Ring )
14		vex 2806	. . . . . . . . . . . . 13 |- y e. _V
15	14	a1i 9	. . . . . . . . . . . 12 |- ( ph -> y e. _V )
16		vex 2806	. . . . . . . . . . . . 13 |- x e. _V
17	16	a1i 9	. . . . . . . . . . . 12 |- ( ph -> x e. _V )
18		eqid 2231	. . . . . . . . . . . . 13 |- ( Base ` S ) = ( Base ` S )
19		eqid 2231	. . . . . . . . . . . . 13 |- ( .r ` S ) = ( .r ` S )
20		eqid 2231	. . . . . . . . . . . . 13 |- (oppr ` S ) = (oppr ` S )
21		eqid 2231	. . . . . . . . . . . . 13 |- ( .r ` (oppr ` S ) ) = ( .r ` (oppr ` S ) )
22	18, 19, 20, 21	opprmulg 14165	. . . . . . . . . . . 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 1274	. . . . . . . . . . 11 |- ( ph -> ( y ( .r ` (oppr ` S ) ) x ) = ( x ( .r ` S ) y ) )
24	4	fveq2d 5652	. . . . . . . . . . . 12 |- ( ph -> ( .r ` S ) = ( .r ` (oppr ` R ) ) )
25	24	oveqd 6045	. . . . . . . . . . 11 |- ( ph -> ( x ( .r ` S ) y ) = ( x ( .r ` (oppr ` R ) ) y ) )
26		eqid 2231	. . . . . . . . . . . . 13 |- ( Base ` R ) = ( Base ` R )
27		eqid 2231	. . . . . . . . . . . . 13 |- ( .r ` R ) = ( .r ` R )
28		eqid 2231	. . . . . . . . . . . . 13 |- ( .r ` (oppr ` R ) ) = ( .r ` (oppr ` R ) )
29	26, 27, 10, 28	opprmulg 14165	. . . . . . . . . . . 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 1274	. . . . . . . . . . 11 |- ( ph -> ( x ( .r ` (oppr ` R ) ) y ) = ( y ( .r ` R ) x ) )
31	23, 25, 30	3eqtrrd 2269	. . . . . . . . . 10 |- ( ph -> ( y ( .r ` R ) x ) = ( y ( .r ` (oppr ` S ) ) x ) )
32	31	eqeq1d 2240	. . . . . . . . 9 |- ( ph -> ( ( y ( .r ` R ) x ) = ( 1r ` R ) <-> ( y ( .r ` (oppr ` S ) ) x ) = ( 1r ` R ) ) )
33	32	rexbidv 2534	. . . . . . . 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 2232	. . . . . . . 8 |- ( ph -> ( Base ` R ) = ( Base ` R ) )
36		eqidd 2232	. . . . . . . 8 |- ( ph -> ( .r ` R ) = ( .r ` R ) )
37	35, 3, 8, 36	dvdsrd 14189	. . . . . . 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 14169	. . . . . . . . . 10 |- ( R e. SRing -> ( Base ` R ) = ( Base ` (oppr ` R ) ) )
39	8, 38	syl 14	. . . . . . . . 9 |- ( ph -> ( Base ` R ) = ( Base ` (oppr ` R ) ) )
40	4	fveq2d 5652	. . . . . . . . 9 |- ( ph -> ( Base ` S ) = ( Base ` (oppr ` R ) ) )
41	20, 18	opprbasg 14169	. . . . . . . . . 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 2270	. . . . . . . 8 |- ( ph -> ( Base ` R ) = ( Base ` (oppr ` S ) ) )
44		eqidd 2232	. . . . . . . 8 |- ( ph -> ( ||r ` (oppr ` S ) ) = ( ||r ` (oppr ` S ) ) )
45	20	opprring 14173	. . . . . . . . . 10 |- ( S e. Ring -> (oppr ` S ) e. Ring )
46	13, 45	syl 14	. . . . . . . . 9 |- ( ph -> (oppr ` S ) e. Ring )
47		ringsrg 14141	. . . . . . . . 9 |- ( (oppr ` S ) e. Ring -> (oppr ` S ) e. SRing )
48	46, 47	syl 14	. . . . . . . 8 |- ( ph -> (oppr ` S ) e. SRing )
49		eqidd 2232	. . . . . . . 8 |- ( ph -> ( .r ` (oppr ` S ) ) = ( .r ` (oppr ` S ) ) )
50	43, 44, 48, 49	dvdsrd 14189	. . . . . . 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 2232	. . . 4 |- ( ph -> (Unit ` S ) = (Unit ` S ) )
56		eqid 2231	. . . . . . 7 |- ( 1r ` R ) = ( 1r ` R )
57	10, 56	oppr1g 14176	. . . . . 6 |- ( R e. Ring -> ( 1r ` R ) = ( 1r ` (oppr ` R ) ) )
58	6, 57	syl 14	. . . . 5 |- ( ph -> ( 1r ` R ) = ( 1r ` (oppr ` R ) ) )
59	4	fveq2d 5652	. . . . 5 |- ( ph -> ( 1r ` S ) = ( 1r ` (oppr ` R ) ) )
60	58, 59	eqtr4d 2267	. . . 4 |- ( ph -> ( 1r ` R ) = ( 1r ` S ) )
61		eqidd 2232	. . . 4 |- ( ph -> (oppr ` S ) = (oppr ` S ) )
62		ringsrg 14141	. . . . 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 14201	. . 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 2229	1 |- ( ph -> U = (Unit ` S ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202   E.wrex 2512   _Vcvv 2803   class class class wbr 4093   ` cfv 5333  (class class class)co 6028   Basecbs 13162   .rcmulr 13241   1rcur 14053  SRingcsrg 14057   Ringcrg 14090  opp_rcoppr 14161   ||_rcdsr 14180  Unitcui 14181

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-addcom 8192  ax-addass 8194  ax-i2m1 8197  ax-0lt1 8198  ax-0id 8200  ax-rnegex 8201  ax-pre-ltirr 8204  ax-pre-lttrn 8206  ax-pre-ltadd 8208

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-tpos 6454  df-pnf 8275  df-mnf 8276  df-ltxr 8278  df-inn 9203  df-2 9261  df-3 9262  df-ndx 13165  df-slot 13166  df-base 13168  df-sets 13169  df-plusg 13253  df-mulr 13254  df-0g 13421  df-mgm 13519  df-sgrp 13565  df-mnd 13580  df-grp 13666  df-minusg 13667  df-cmn 13953  df-abl 13954  df-mgp 14015  df-ur 14054  df-srg 14058  df-ring 14092  df-oppr 14162  df-dvdsr 14183  df-unit 14184

This theorem is referenced by: (None)