Theorem opprunitd 13987

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 2208	. . . . . 6 |- ( ph -> ( 1r ` R ) = ( 1r ` R ) )
3		eqidd 2208	. . . . . 6 |- ( ph -> ( ||r ` R ) = ( ||r ` R ) )
4		opprunitd.2	. . . . . 6 |- ( ph -> S = (oppr ` R ) )
5		eqidd 2208	. . . . . 6 |- ( ph -> ( ||r ` S ) = ( ||r ` S ) )
6		opprunitd.r	. . . . . . 7 |- ( ph -> R e. Ring )
7		ringsrg 13924	. . . . . . 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 13983	. . . . 5 |- ( ph -> ( x e. U <-> ( x ( ||r ` R ) ( 1r ` R ) /\ x ( ||r ` S ) ( 1r ` R ) ) ) )
10		eqid 2207	. . . . . . . . . . . . . . 15 |- (oppr ` R ) = (oppr ` R )
11	10	opprring 13956	. . . . . . . . . . . . . 14 |- ( R e. Ring -> (oppr ` R ) e. Ring )
12	6, 11	syl 14	. . . . . . . . . . . . 13 |- ( ph -> (oppr ` R ) e. Ring )
13	4, 12	eqeltrd 2284	. . . . . . . . . . . 12 |- ( ph -> S e. Ring )
14		vex 2779	. . . . . . . . . . . . 13 |- y e. _V
15	14	a1i 9	. . . . . . . . . . . 12 |- ( ph -> y e. _V )
16		vex 2779	. . . . . . . . . . . . 13 |- x e. _V
17	16	a1i 9	. . . . . . . . . . . 12 |- ( ph -> x e. _V )
18		eqid 2207	. . . . . . . . . . . . 13 |- ( Base ` S ) = ( Base ` S )
19		eqid 2207	. . . . . . . . . . . . 13 |- ( .r ` S ) = ( .r ` S )
20		eqid 2207	. . . . . . . . . . . . 13 |- (oppr ` S ) = (oppr ` S )
21		eqid 2207	. . . . . . . . . . . . 13 |- ( .r ` (oppr ` S ) ) = ( .r ` (oppr ` S ) )
22	18, 19, 20, 21	opprmulg 13948	. . . . . . . . . . . 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 1250	. . . . . . . . . . 11 |- ( ph -> ( y ( .r ` (oppr ` S ) ) x ) = ( x ( .r ` S ) y ) )
24	4	fveq2d 5603	. . . . . . . . . . . 12 |- ( ph -> ( .r ` S ) = ( .r ` (oppr ` R ) ) )
25	24	oveqd 5984	. . . . . . . . . . 11 |- ( ph -> ( x ( .r ` S ) y ) = ( x ( .r ` (oppr ` R ) ) y ) )
26		eqid 2207	. . . . . . . . . . . . 13 |- ( Base ` R ) = ( Base ` R )
27		eqid 2207	. . . . . . . . . . . . 13 |- ( .r ` R ) = ( .r ` R )
28		eqid 2207	. . . . . . . . . . . . 13 |- ( .r ` (oppr ` R ) ) = ( .r ` (oppr ` R ) )
29	26, 27, 10, 28	opprmulg 13948	. . . . . . . . . . . 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 1250	. . . . . . . . . . 11 |- ( ph -> ( x ( .r ` (oppr ` R ) ) y ) = ( y ( .r ` R ) x ) )
31	23, 25, 30	3eqtrrd 2245	. . . . . . . . . 10 |- ( ph -> ( y ( .r ` R ) x ) = ( y ( .r ` (oppr ` S ) ) x ) )
32	31	eqeq1d 2216	. . . . . . . . 9 |- ( ph -> ( ( y ( .r ` R ) x ) = ( 1r ` R ) <-> ( y ( .r ` (oppr ` S ) ) x ) = ( 1r ` R ) ) )
33	32	rexbidv 2509	. . . . . . . 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 2208	. . . . . . . 8 |- ( ph -> ( Base ` R ) = ( Base ` R ) )
36		eqidd 2208	. . . . . . . 8 |- ( ph -> ( .r ` R ) = ( .r ` R ) )
37	35, 3, 8, 36	dvdsrd 13971	. . . . . . 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 13952	. . . . . . . . . 10 |- ( R e. SRing -> ( Base ` R ) = ( Base ` (oppr ` R ) ) )
39	8, 38	syl 14	. . . . . . . . 9 |- ( ph -> ( Base ` R ) = ( Base ` (oppr ` R ) ) )
40	4	fveq2d 5603	. . . . . . . . 9 |- ( ph -> ( Base ` S ) = ( Base ` (oppr ` R ) ) )
41	20, 18	opprbasg 13952	. . . . . . . . . 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 2246	. . . . . . . 8 |- ( ph -> ( Base ` R ) = ( Base ` (oppr ` S ) ) )
44		eqidd 2208	. . . . . . . 8 |- ( ph -> ( ||r ` (oppr ` S ) ) = ( ||r ` (oppr ` S ) ) )
45	20	opprring 13956	. . . . . . . . . 10 |- ( S e. Ring -> (oppr ` S ) e. Ring )
46	13, 45	syl 14	. . . . . . . . 9 |- ( ph -> (oppr ` S ) e. Ring )
47		ringsrg 13924	. . . . . . . . 9 |- ( (oppr ` S ) e. Ring -> (oppr ` S ) e. SRing )
48	46, 47	syl 14	. . . . . . . 8 |- ( ph -> (oppr ` S ) e. SRing )
49		eqidd 2208	. . . . . . . 8 |- ( ph -> ( .r ` (oppr ` S ) ) = ( .r ` (oppr ` S ) ) )
50	43, 44, 48, 49	dvdsrd 13971	. . . . . . 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 2208	. . . 4 |- ( ph -> (Unit ` S ) = (Unit ` S ) )
56		eqid 2207	. . . . . . 7 |- ( 1r ` R ) = ( 1r ` R )
57	10, 56	oppr1g 13959	. . . . . 6 |- ( R e. Ring -> ( 1r ` R ) = ( 1r ` (oppr ` R ) ) )
58	6, 57	syl 14	. . . . 5 |- ( ph -> ( 1r ` R ) = ( 1r ` (oppr ` R ) ) )
59	4	fveq2d 5603	. . . . 5 |- ( ph -> ( 1r ` S ) = ( 1r ` (oppr ` R ) ) )
60	58, 59	eqtr4d 2243	. . . 4 |- ( ph -> ( 1r ` R ) = ( 1r ` S ) )
61		eqidd 2208	. . . 4 |- ( ph -> (oppr ` S ) = (oppr ` S ) )
62		ringsrg 13924	. . . . 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 13983	. . 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 2205	1 |- ( ph -> U = (Unit ` S ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2178   E.wrex 2487   _Vcvv 2776   class class class wbr 4059   ` cfv 5290  (class class class)co 5967   Basecbs 12947   .rcmulr 13025   1rcur 13836  SRingcsrg 13840   Ringcrg 13873  opp_rcoppr 13944   ||_rcdsr 13963  Unitcui 13964

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-addass 8062  ax-i2m1 8065  ax-0lt1 8066  ax-0id 8068  ax-rnegex 8069  ax-pre-ltirr 8072  ax-pre-lttrn 8074  ax-pre-ltadd 8076

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-tpos 6354  df-pnf 8144  df-mnf 8145  df-ltxr 8147  df-inn 9072  df-2 9130  df-3 9131  df-ndx 12950  df-slot 12951  df-base 12953  df-sets 12954  df-plusg 13037  df-mulr 13038  df-0g 13205  df-mgm 13303  df-sgrp 13349  df-mnd 13364  df-grp 13450  df-minusg 13451  df-cmn 13737  df-abl 13738  df-mgp 13798  df-ur 13837  df-srg 13841  df-ring 13875  df-oppr 13945  df-dvdsr 13966  df-unit 13967

This theorem is referenced by: (None)