Theorem unitgrp 13612

Description: The group of units is a group under multiplication. (Contributed by Mario Carneiro, 2-Dec-2014.)

Hypotheses

Ref	Expression
unitgrp.1	|- U = (Unit ` R )
unitgrp.2	|- G = ( (mulGrp ` R ) ↾s U )

Assertion

Ref	Expression
unitgrp	|- ( R e. Ring -> G e. Grp )

Proof of Theorem unitgrp

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

Step	Hyp	Ref	Expression
1		unitgrp.1	. . . 4 |- U = (Unit ` R )
2	1	a1i 9	. . 3 |- ( R e. Ring -> U = (Unit ` R ) )
3		unitgrp.2	. . . 4 |- G = ( (mulGrp ` R ) ↾s U )
4	3	a1i 9	. . 3 |- ( R e. Ring -> G = ( (mulGrp ` R ) ↾s U ) )
5		ringsrg 13543	. . 3 |- ( R e. Ring -> R e. SRing )
6	2, 4, 5	unitgrpbasd 13611	. 2 |- ( R e. Ring -> U = ( Base ` G ) )
7		eqid 2193	. . . 4 |- (mulGrp ` R ) = (mulGrp ` R )
8		eqid 2193	. . . 4 |- ( .r ` R ) = ( .r ` R )
9	7, 8	mgpplusgg 13420	. . 3 |- ( R e. Ring -> ( .r ` R ) = ( +g ` (mulGrp ` R ) ) )
10		basfn 12676	. . . . 5 |- Base Fn _V
11		elex 2771	. . . . 5 |- ( R e. Ring -> R e. _V )
12		funfvex 5571	. . . . . 6 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
13	12	funfni 5354	. . . . 5 |- ( ( Base Fn _V /\ R e. _V ) -> ( Base ` R ) e. _V )
14	10, 11, 13	sylancr 414	. . . 4 |- ( R e. Ring -> ( Base ` R ) e. _V )
15		eqidd 2194	. . . . 5 |- ( R e. Ring -> ( Base ` R ) = ( Base ` R ) )
16	15, 2, 5	unitssd 13605	. . . 4 |- ( R e. Ring -> U C_ ( Base ` R ) )
17	14, 16	ssexd 4169	. . 3 |- ( R e. Ring -> U e. _V )
18	7	mgpex 13421	. . 3 |- ( R e. Ring -> (mulGrp ` R ) e. _V )
19	4, 9, 17, 18	ressplusgd 12746	. 2 |- ( R e. Ring -> ( .r ` R ) = ( +g ` G ) )
20	1, 8	unitmulcl 13609	. 2 |- ( ( R e. Ring /\ x e. U /\ y e. U ) -> ( x ( .r ` R ) y ) e. U )
21		eqidd 2194	. . . . 5 |- ( ( R e. Ring /\ ( x e. U /\ y e. U /\ z e. U ) ) -> ( Base ` R ) = ( Base ` R ) )
22	1	a1i 9	. . . . 5 |- ( ( R e. Ring /\ ( x e. U /\ y e. U /\ z e. U ) ) -> U = (Unit ` R ) )
23	5	adantr 276	. . . . 5 |- ( ( R e. Ring /\ ( x e. U /\ y e. U /\ z e. U ) ) -> R e. SRing )
24		simpr1 1005	. . . . 5 |- ( ( R e. Ring /\ ( x e. U /\ y e. U /\ z e. U ) ) -> x e. U )
25	21, 22, 23, 24	unitcld 13604	. . . 4 |- ( ( R e. Ring /\ ( x e. U /\ y e. U /\ z e. U ) ) -> x e. ( Base ` R ) )
26		simpr2 1006	. . . . 5 |- ( ( R e. Ring /\ ( x e. U /\ y e. U /\ z e. U ) ) -> y e. U )
27	21, 22, 23, 26	unitcld 13604	. . . 4 |- ( ( R e. Ring /\ ( x e. U /\ y e. U /\ z e. U ) ) -> y e. ( Base ` R ) )
28		simpr3 1007	. . . . 5 |- ( ( R e. Ring /\ ( x e. U /\ y e. U /\ z e. U ) ) -> z e. U )
29	21, 22, 23, 28	unitcld 13604	. . . 4 |- ( ( R e. Ring /\ ( x e. U /\ y e. U /\ z e. U ) ) -> z e. ( Base ` R ) )
30	25, 27, 29	3jca 1179	. . 3 |- ( ( R e. Ring /\ ( x e. U /\ y e. U /\ z e. U ) ) -> ( x e. ( Base ` R ) /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) )
31		eqid 2193	. . . 4 |- ( Base ` R ) = ( Base ` R )
32	31, 8	ringass 13512	. . 3 |- ( ( R e. Ring /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) ) -> ( ( x ( .r ` R ) y ) ( .r ` R ) z ) = ( x ( .r ` R ) ( y ( .r ` R ) z ) ) )
33	30, 32	syldan 282	. 2 |- ( ( R e. Ring /\ ( x e. U /\ y e. U /\ z e. U ) ) -> ( ( x ( .r ` R ) y ) ( .r ` R ) z ) = ( x ( .r ` R ) ( y ( .r ` R ) z ) ) )
34		eqid 2193	. . 3 |- ( 1r ` R ) = ( 1r ` R )
35	1, 34	1unit 13603	. 2 |- ( R e. Ring -> ( 1r ` R ) e. U )
36		eqidd 2194	. . . 4 |- ( ( R e. Ring /\ x e. U ) -> ( Base ` R ) = ( Base ` R ) )
37	1	a1i 9	. . . 4 |- ( ( R e. Ring /\ x e. U ) -> U = (Unit ` R ) )
38	5	adantr 276	. . . 4 |- ( ( R e. Ring /\ x e. U ) -> R e. SRing )
39		simpr 110	. . . 4 |- ( ( R e. Ring /\ x e. U ) -> x e. U )
40	36, 37, 38, 39	unitcld 13604	. . 3 |- ( ( R e. Ring /\ x e. U ) -> x e. ( Base ` R ) )
41	31, 8, 34	ringlidm 13519	. . 3 |- ( ( R e. Ring /\ x e. ( Base ` R ) ) -> ( ( 1r ` R ) ( .r ` R ) x ) = x )
42	40, 41	syldan 282	. 2 |- ( ( R e. Ring /\ x e. U ) -> ( ( 1r ` R ) ( .r ` R ) x ) = x )
43		eqidd 2194	. . . . 5 |- ( ( R e. Ring /\ x e. U ) -> ( 1r ` R ) = ( 1r ` R ) )
44		eqidd 2194	. . . . 5 |- ( ( R e. Ring /\ x e. U ) -> ( ||r ` R ) = ( ||r ` R ) )
45		eqidd 2194	. . . . 5 |- ( ( R e. Ring /\ x e. U ) -> (oppr ` R ) = (oppr ` R ) )
46		eqidd 2194	. . . . 5 |- ( ( R e. Ring /\ x e. U ) -> ( ||r ` (oppr ` R ) ) = ( ||r ` (oppr ` R ) ) )
47	37, 43, 44, 45, 46, 38	isunitd 13602	. . . 4 |- ( ( R e. Ring /\ x e. U ) -> ( x e. U <-> ( x ( ||r ` R ) ( 1r ` R ) /\ x ( ||r ` (oppr ` R ) ) ( 1r ` R ) ) ) )
48	39, 47	mpbid 147	. . 3 |- ( ( R e. Ring /\ x e. U ) -> ( x ( ||r ` R ) ( 1r ` R ) /\ x ( ||r ` (oppr ` R ) ) ( 1r ` R ) ) )
49		eqidd 2194	. . . . . 6 |- ( ( R e. Ring /\ x e. U ) -> ( .r ` R ) = ( .r ` R ) )
50	36, 44, 38, 49, 40	dvdsr2d 13591	. . . . 5 |- ( ( R e. Ring /\ x e. U ) -> ( x ( ||r ` R ) ( 1r ` R ) <-> E. y e. ( Base ` R ) ( y ( .r ` R ) x ) = ( 1r ` R ) ) )
51		eqid 2193	. . . . . . . 8 |- (oppr ` R ) = (oppr ` R )
52	51, 31	opprbasg 13571	. . . . . . 7 |- ( R e. Ring -> ( Base ` R ) = ( Base ` (oppr ` R ) ) )
53	52	adantr 276	. . . . . 6 |- ( ( R e. Ring /\ x e. U ) -> ( Base ` R ) = ( Base ` (oppr ` R ) ) )
54	51	opprring 13575	. . . . . . . 8 |- ( R e. Ring -> (oppr ` R ) e. Ring )
55		ringsrg 13543	. . . . . . . 8 |- ( (oppr ` R ) e. Ring -> (oppr ` R ) e. SRing )
56	54, 55	syl 14	. . . . . . 7 |- ( R e. Ring -> (oppr ` R ) e. SRing )
57	56	adantr 276	. . . . . 6 |- ( ( R e. Ring /\ x e. U ) -> (oppr ` R ) e. SRing )
58		eqidd 2194	. . . . . 6 |- ( ( R e. Ring /\ x e. U ) -> ( .r ` (oppr ` R ) ) = ( .r ` (oppr ` R ) ) )
59	53, 46, 57, 58, 40	dvdsr2d 13591	. . . . 5 |- ( ( R e. Ring /\ x e. U ) -> ( x ( ||r ` (oppr ` R ) ) ( 1r ` R ) <-> E. m e. ( Base ` R ) ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) )
60	50, 59	anbi12d 473	. . . 4 |- ( ( R e. Ring /\ x e. U ) -> ( ( x ( ||r ` R ) ( 1r ` R ) /\ x ( ||r ` (oppr ` R ) ) ( 1r ` R ) ) <-> ( E. y e. ( Base ` R ) ( y ( .r ` R ) x ) = ( 1r ` R ) /\ E. m e. ( Base ` R ) ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) ) )
61		reeanv 2664	. . . . 5 |- ( E. y e. ( Base ` R ) E. m e. ( Base ` R ) ( ( y ( .r ` R ) x ) = ( 1r ` R ) /\ ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) <-> ( E. y e. ( Base ` R ) ( y ( .r ` R ) x ) = ( 1r ` R ) /\ E. m e. ( Base ` R ) ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) )
62		eqidd 2194	. . . . . . . . . . . 12 |- ( ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) /\ ( m e. ( Base ` R ) /\ ( ( y ( .r ` R ) x ) = ( 1r ` R ) /\ ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) ) ) -> ( Base ` R ) = ( Base ` R ) )
63		eqidd 2194	. . . . . . . . . . . 12 |- ( ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) /\ ( m e. ( Base ` R ) /\ ( ( y ( .r ` R ) x ) = ( 1r ` R ) /\ ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) ) ) -> ( ||r ` R ) = ( ||r ` R ) )
64	38	ad2antrr 488	. . . . . . . . . . . 12 |- ( ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) /\ ( m e. ( Base ` R ) /\ ( ( y ( .r ` R ) x ) = ( 1r ` R ) /\ ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) ) ) -> R e. SRing )
65		eqidd 2194	. . . . . . . . . . . 12 |- ( ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) /\ ( m e. ( Base ` R ) /\ ( ( y ( .r ` R ) x ) = ( 1r ` R ) /\ ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) ) ) -> ( .r ` R ) = ( .r ` R ) )
66		simprl 529	. . . . . . . . . . . 12 |- ( ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) /\ ( m e. ( Base ` R ) /\ ( ( y ( .r ` R ) x ) = ( 1r ` R ) /\ ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) ) ) -> m e. ( Base ` R ) )
67	40	ad2antrr 488	. . . . . . . . . . . 12 |- ( ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) /\ ( m e. ( Base ` R ) /\ ( ( y ( .r ` R ) x ) = ( 1r ` R ) /\ ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) ) ) -> x e. ( Base ` R ) )
68	62, 63, 64, 65, 66, 67	dvdsrmuld 13592	. . . . . . . . . . 11 |- ( ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) /\ ( m e. ( Base ` R ) /\ ( ( y ( .r ` R ) x ) = ( 1r ` R ) /\ ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) ) ) -> m ( ||r ` R ) ( x ( .r ` R ) m ) )
69		simplll 533	. . . . . . . . . . . . . 14 |- ( ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) /\ ( m e. ( Base ` R ) /\ ( ( y ( .r ` R ) x ) = ( 1r ` R ) /\ ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) ) ) -> R e. Ring )
70		simplr 528	. . . . . . . . . . . . . 14 |- ( ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) /\ ( m e. ( Base ` R ) /\ ( ( y ( .r ` R ) x ) = ( 1r ` R ) /\ ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) ) ) -> y e. ( Base ` R ) )
71	31, 8	ringass 13512	. . . . . . . . . . . . . 14 |- ( ( R e. Ring /\ ( y e. ( Base ` R ) /\ x e. ( Base ` R ) /\ m e. ( Base ` R ) ) ) -> ( ( y ( .r ` R ) x ) ( .r ` R ) m ) = ( y ( .r ` R ) ( x ( .r ` R ) m ) ) )
72	69, 70, 67, 66, 71	syl13anc 1251	. . . . . . . . . . . . 13 |- ( ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) /\ ( m e. ( Base ` R ) /\ ( ( y ( .r ` R ) x ) = ( 1r ` R ) /\ ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) ) ) -> ( ( y ( .r ` R ) x ) ( .r ` R ) m ) = ( y ( .r ` R ) ( x ( .r ` R ) m ) ) )
73		simprrl 539	. . . . . . . . . . . . . 14 |- ( ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) /\ ( m e. ( Base ` R ) /\ ( ( y ( .r ` R ) x ) = ( 1r ` R ) /\ ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) ) ) -> ( y ( .r ` R ) x ) = ( 1r ` R ) )
74	73	oveq1d 5933	. . . . . . . . . . . . 13 |- ( ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) /\ ( m e. ( Base ` R ) /\ ( ( y ( .r ` R ) x ) = ( 1r ` R ) /\ ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) ) ) -> ( ( y ( .r ` R ) x ) ( .r ` R ) m ) = ( ( 1r ` R ) ( .r ` R ) m ) )
75	39	ad2antrr 488	. . . . . . . . . . . . . . . 16 |- ( ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) /\ ( m e. ( Base ` R ) /\ ( ( y ( .r ` R ) x ) = ( 1r ` R ) /\ ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) ) ) -> x e. U )
76		eqid 2193	. . . . . . . . . . . . . . . . 17 |- ( .r ` (oppr ` R ) ) = ( .r ` (oppr ` R ) )
77	31, 8, 51, 76	opprmulg 13567	. . . . . . . . . . . . . . . 16 |- ( ( R e. Ring /\ m e. ( Base ` R ) /\ x e. U ) -> ( m ( .r ` (oppr ` R ) ) x ) = ( x ( .r ` R ) m ) )
78	69, 66, 75, 77	syl3anc 1249	. . . . . . . . . . . . . . 15 |- ( ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) /\ ( m e. ( Base ` R ) /\ ( ( y ( .r ` R ) x ) = ( 1r ` R ) /\ ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) ) ) -> ( m ( .r ` (oppr ` R ) ) x ) = ( x ( .r ` R ) m ) )
79		simprrr 540	. . . . . . . . . . . . . . 15 |- ( ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) /\ ( m e. ( Base ` R ) /\ ( ( y ( .r ` R ) x ) = ( 1r ` R ) /\ ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) ) ) -> ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) )
80	78, 79	eqtr3d 2228	. . . . . . . . . . . . . 14 |- ( ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) /\ ( m e. ( Base ` R ) /\ ( ( y ( .r ` R ) x ) = ( 1r ` R ) /\ ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) ) ) -> ( x ( .r ` R ) m ) = ( 1r ` R ) )
81	80	oveq2d 5934	. . . . . . . . . . . . 13 |- ( ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) /\ ( m e. ( Base ` R ) /\ ( ( y ( .r ` R ) x ) = ( 1r ` R ) /\ ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) ) ) -> ( y ( .r ` R ) ( x ( .r ` R ) m ) ) = ( y ( .r ` R ) ( 1r ` R ) ) )
82	72, 74, 81	3eqtr3d 2234	. . . . . . . . . . . 12 |- ( ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) /\ ( m e. ( Base ` R ) /\ ( ( y ( .r ` R ) x ) = ( 1r ` R ) /\ ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) ) ) -> ( ( 1r ` R ) ( .r ` R ) m ) = ( y ( .r ` R ) ( 1r ` R ) ) )
83	31, 8, 34	ringlidm 13519	. . . . . . . . . . . . 13 |- ( ( R e. Ring /\ m e. ( Base ` R ) ) -> ( ( 1r ` R ) ( .r ` R ) m ) = m )
84	69, 66, 83	syl2anc 411	. . . . . . . . . . . 12 |- ( ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) /\ ( m e. ( Base ` R ) /\ ( ( y ( .r ` R ) x ) = ( 1r ` R ) /\ ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) ) ) -> ( ( 1r ` R ) ( .r ` R ) m ) = m )
85	31, 8, 34	ringridm 13520	. . . . . . . . . . . . 13 |- ( ( R e. Ring /\ y e. ( Base ` R ) ) -> ( y ( .r ` R ) ( 1r ` R ) ) = y )
86	69, 70, 85	syl2anc 411	. . . . . . . . . . . 12 |- ( ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) /\ ( m e. ( Base ` R ) /\ ( ( y ( .r ` R ) x ) = ( 1r ` R ) /\ ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) ) ) -> ( y ( .r ` R ) ( 1r ` R ) ) = y )
87	82, 84, 86	3eqtr3d 2234	. . . . . . . . . . 11 |- ( ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) /\ ( m e. ( Base ` R ) /\ ( ( y ( .r ` R ) x ) = ( 1r ` R ) /\ ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) ) ) -> m = y )
88	68, 87, 80	3brtr3d 4060	. . . . . . . . . 10 |- ( ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) /\ ( m e. ( Base ` R ) /\ ( ( y ( .r ` R ) x ) = ( 1r ` R ) /\ ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) ) ) -> y ( ||r ` R ) ( 1r ` R ) )
89	69, 52	syl 14	. . . . . . . . . . . 12 |- ( ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) /\ ( m e. ( Base ` R ) /\ ( ( y ( .r ` R ) x ) = ( 1r ` R ) /\ ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) ) ) -> ( Base ` R ) = ( Base ` (oppr ` R ) ) )
90		eqidd 2194	. . . . . . . . . . . 12 |- ( ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) /\ ( m e. ( Base ` R ) /\ ( ( y ( .r ` R ) x ) = ( 1r ` R ) /\ ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) ) ) -> ( ||r ` (oppr ` R ) ) = ( ||r ` (oppr ` R ) ) )
91	69, 56	syl 14	. . . . . . . . . . . 12 |- ( ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) /\ ( m e. ( Base ` R ) /\ ( ( y ( .r ` R ) x ) = ( 1r ` R ) /\ ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) ) ) -> (oppr ` R ) e. SRing )
92		eqidd 2194	. . . . . . . . . . . 12 |- ( ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) /\ ( m e. ( Base ` R ) /\ ( ( y ( .r ` R ) x ) = ( 1r ` R ) /\ ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) ) ) -> ( .r ` (oppr ` R ) ) = ( .r ` (oppr ` R ) ) )
93	89, 90, 91, 92, 70, 67	dvdsrmuld 13592	. . . . . . . . . . 11 |- ( ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) /\ ( m e. ( Base ` R ) /\ ( ( y ( .r ` R ) x ) = ( 1r ` R ) /\ ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) ) ) -> y ( ||r ` (oppr ` R ) ) ( x ( .r ` (oppr ` R ) ) y ) )
94	31, 8, 51, 76	opprmulg 13567	. . . . . . . . . . . . 13 |- ( ( R e. Ring /\ x e. U /\ y e. ( Base ` R ) ) -> ( x ( .r ` (oppr ` R ) ) y ) = ( y ( .r ` R ) x ) )
95	69, 75, 70, 94	syl3anc 1249	. . . . . . . . . . . 12 |- ( ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) /\ ( m e. ( Base ` R ) /\ ( ( y ( .r ` R ) x ) = ( 1r ` R ) /\ ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) ) ) -> ( x ( .r ` (oppr ` R ) ) y ) = ( y ( .r ` R ) x ) )
96	95, 73	eqtrd 2226	. . . . . . . . . . 11 |- ( ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) /\ ( m e. ( Base ` R ) /\ ( ( y ( .r ` R ) x ) = ( 1r ` R ) /\ ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) ) ) -> ( x ( .r ` (oppr ` R ) ) y ) = ( 1r ` R ) )
97	93, 96	breqtrd 4055	. . . . . . . . . 10 |- ( ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) /\ ( m e. ( Base ` R ) /\ ( ( y ( .r ` R ) x ) = ( 1r ` R ) /\ ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) ) ) -> y ( ||r ` (oppr ` R ) ) ( 1r ` R ) )
98	1	a1i 9	. . . . . . . . . . 11 |- ( ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) /\ ( m e. ( Base ` R ) /\ ( ( y ( .r ` R ) x ) = ( 1r ` R ) /\ ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) ) ) -> U = (Unit ` R ) )
99		eqidd 2194	. . . . . . . . . . 11 |- ( ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) /\ ( m e. ( Base ` R ) /\ ( ( y ( .r ` R ) x ) = ( 1r ` R ) /\ ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) ) ) -> ( 1r ` R ) = ( 1r ` R ) )
100		eqidd 2194	. . . . . . . . . . 11 |- ( ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) /\ ( m e. ( Base ` R ) /\ ( ( y ( .r ` R ) x ) = ( 1r ` R ) /\ ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) ) ) -> (oppr ` R ) = (oppr ` R ) )
101	98, 99, 63, 100, 90, 64	isunitd 13602	. . . . . . . . . 10 |- ( ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) /\ ( m e. ( Base ` R ) /\ ( ( y ( .r ` R ) x ) = ( 1r ` R ) /\ ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) ) ) -> ( y e. U <-> ( y ( ||r ` R ) ( 1r ` R ) /\ y ( ||r ` (oppr ` R ) ) ( 1r ` R ) ) ) )
102	88, 97, 101	mpbir2and 946	. . . . . . . . 9 |- ( ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) /\ ( m e. ( Base ` R ) /\ ( ( y ( .r ` R ) x ) = ( 1r ` R ) /\ ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) ) ) -> y e. U )
103	102, 73	jca 306	. . . . . . . 8 |- ( ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) /\ ( m e. ( Base ` R ) /\ ( ( y ( .r ` R ) x ) = ( 1r ` R ) /\ ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) ) ) -> ( y e. U /\ ( y ( .r ` R ) x ) = ( 1r ` R ) ) )
104	103	rexlimdvaa 2612	. . . . . . 7 |- ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) -> ( E. m e. ( Base ` R ) ( ( y ( .r ` R ) x ) = ( 1r ` R ) /\ ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) -> ( y e. U /\ ( y ( .r ` R ) x ) = ( 1r ` R ) ) ) )
105	104	expimpd 363	. . . . . 6 |- ( ( R e. Ring /\ x e. U ) -> ( ( y e. ( Base ` R ) /\ E. m e. ( Base ` R ) ( ( y ( .r ` R ) x ) = ( 1r ` R ) /\ ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) ) -> ( y e. U /\ ( y ( .r ` R ) x ) = ( 1r ` R ) ) ) )
106	105	reximdv2 2593	. . . . 5 |- ( ( R e. Ring /\ x e. U ) -> ( E. y e. ( Base ` R ) E. m e. ( Base ` R ) ( ( y ( .r ` R ) x ) = ( 1r ` R ) /\ ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) -> E. y e. U ( y ( .r ` R ) x ) = ( 1r ` R ) ) )
107	61, 106	biimtrrid 153	. . . 4 |- ( ( R e. Ring /\ x e. U ) -> ( ( E. y e. ( Base ` R ) ( y ( .r ` R ) x ) = ( 1r ` R ) /\ E. m e. ( Base ` R ) ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) -> E. y e. U ( y ( .r ` R ) x ) = ( 1r ` R ) ) )
108	60, 107	sylbid 150	. . 3 |- ( ( R e. Ring /\ x e. U ) -> ( ( x ( ||r ` R ) ( 1r ` R ) /\ x ( ||r ` (oppr ` R ) ) ( 1r ` R ) ) -> E. y e. U ( y ( .r ` R ) x ) = ( 1r ` R ) ) )
109	48, 108	mpd 13	. 2 |- ( ( R e. Ring /\ x e. U ) -> E. y e. U ( y ( .r ` R ) x ) = ( 1r ` R ) )
110	6, 19, 20, 33, 35, 42, 109	isgrpde 13094	1 |- ( R e. Ring -> G e. Grp )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   e. wcel 2164   E.wrex 2473   _Vcvv 2760   class class class wbr 4029   Fn wfn 5249   ` cfv 5254  (class class class)co 5918   Basecbs 12618   ↾_s cress 12619   .rcmulr 12696   Grpcgrp 13072  mulGrpcmgp 13416   1rcur 13455  SRingcsrg 13459   Ringcrg 13492  opp_rcoppr 13563   ||_rcdsr 13582  Unitcui 13583

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 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-pre-ltirr 7984  ax-pre-lttrn 7986  ax-pre-ltadd 7988

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 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-tpos 6298  df-pnf 8056  df-mnf 8057  df-ltxr 8059  df-inn 8983  df-2 9041  df-3 9042  df-ndx 12621  df-slot 12622  df-base 12624  df-sets 12625  df-iress 12626  df-plusg 12708  df-mulr 12709  df-0g 12869  df-mgm 12939  df-sgrp 12985  df-mnd 12998  df-grp 13075  df-minusg 13076  df-cmn 13356  df-abl 13357  df-mgp 13417  df-ur 13456  df-srg 13460  df-ring 13494  df-oppr 13564  df-dvdsr 13585  df-unit 13586

This theorem is referenced by:  unitabl  13613  unitsubm  13615  invrfvald  13618  unitinvcl  13619  unitinvinv  13620  unitlinv  13622  unitrinv  13623  rdivmuldivd  13640  rhmunitinv  13674  subrgugrp  13736  expghmap  14095