Theorem unitgrp 13928

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 13859	. . 3 |- ( R e. Ring -> R e. SRing )
6	2, 4, 5	unitgrpbasd 13927	. 2 |- ( R e. Ring -> U = ( Base ` G ) )
7		eqid 2206	. . . 4 |- (mulGrp ` R ) = (mulGrp ` R )
8		eqid 2206	. . . 4 |- ( .r ` R ) = ( .r ` R )
9	7, 8	mgpplusgg 13736	. . 3 |- ( R e. Ring -> ( .r ` R ) = ( +g ` (mulGrp ` R ) ) )
10		basfn 12940	. . . . 5 |- Base Fn _V
11		elex 2785	. . . . 5 |- ( R e. Ring -> R e. _V )
12		funfvex 5603	. . . . . 6 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
13	12	funfni 5382	. . . . 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 2207	. . . . 5 |- ( R e. Ring -> ( Base ` R ) = ( Base ` R ) )
16	15, 2, 5	unitssd 13921	. . . 4 |- ( R e. Ring -> U C_ ( Base ` R ) )
17	14, 16	ssexd 4189	. . 3 |- ( R e. Ring -> U e. _V )
18	7	mgpex 13737	. . 3 |- ( R e. Ring -> (mulGrp ` R ) e. _V )
19	4, 9, 17, 18	ressplusgd 13011	. 2 |- ( R e. Ring -> ( .r ` R ) = ( +g ` G ) )
20	1, 8	unitmulcl 13925	. 2 |- ( ( R e. Ring /\ x e. U /\ y e. U ) -> ( x ( .r ` R ) y ) e. U )
21		eqidd 2207	. . . . 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 1006	. . . . 5 |- ( ( R e. Ring /\ ( x e. U /\ y e. U /\ z e. U ) ) -> x e. U )
25	21, 22, 23, 24	unitcld 13920	. . . 4 |- ( ( R e. Ring /\ ( x e. U /\ y e. U /\ z e. U ) ) -> x e. ( Base ` R ) )
26		simpr2 1007	. . . . 5 |- ( ( R e. Ring /\ ( x e. U /\ y e. U /\ z e. U ) ) -> y e. U )
27	21, 22, 23, 26	unitcld 13920	. . . 4 |- ( ( R e. Ring /\ ( x e. U /\ y e. U /\ z e. U ) ) -> y e. ( Base ` R ) )
28		simpr3 1008	. . . . 5 |- ( ( R e. Ring /\ ( x e. U /\ y e. U /\ z e. U ) ) -> z e. U )
29	21, 22, 23, 28	unitcld 13920	. . . 4 |- ( ( R e. Ring /\ ( x e. U /\ y e. U /\ z e. U ) ) -> z e. ( Base ` R ) )
30	25, 27, 29	3jca 1180	. . 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 2206	. . . 4 |- ( Base ` R ) = ( Base ` R )
32	31, 8	ringass 13828	. . 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 2206	. . 3 |- ( 1r ` R ) = ( 1r ` R )
35	1, 34	1unit 13919	. 2 |- ( R e. Ring -> ( 1r ` R ) e. U )
36		eqidd 2207	. . . 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 13920	. . 3 |- ( ( R e. Ring /\ x e. U ) -> x e. ( Base ` R ) )
41	31, 8, 34	ringlidm 13835	. . 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 2207	. . . . 5 |- ( ( R e. Ring /\ x e. U ) -> ( 1r ` R ) = ( 1r ` R ) )
44		eqidd 2207	. . . . 5 |- ( ( R e. Ring /\ x e. U ) -> ( ||r ` R ) = ( ||r ` R ) )
45		eqidd 2207	. . . . 5 |- ( ( R e. Ring /\ x e. U ) -> (oppr ` R ) = (oppr ` R ) )
46		eqidd 2207	. . . . 5 |- ( ( R e. Ring /\ x e. U ) -> ( ||r ` (oppr ` R ) ) = ( ||r ` (oppr ` R ) ) )
47	37, 43, 44, 45, 46, 38	isunitd 13918	. . . 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 2207	. . . . . 6 |- ( ( R e. Ring /\ x e. U ) -> ( .r ` R ) = ( .r ` R ) )
50	36, 44, 38, 49, 40	dvdsr2d 13907	. . . . 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 2206	. . . . . . . 8 |- (oppr ` R ) = (oppr ` R )
52	51, 31	opprbasg 13887	. . . . . . 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 13891	. . . . . . . 8 |- ( R e. Ring -> (oppr ` R ) e. Ring )
55		ringsrg 13859	. . . . . . . 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 2207	. . . . . 6 |- ( ( R e. Ring /\ x e. U ) -> ( .r ` (oppr ` R ) ) = ( .r ` (oppr ` R ) ) )
59	53, 46, 57, 58, 40	dvdsr2d 13907	. . . . 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 2677	. . . . 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 2207	. . . . . . . . . . . 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 2207	. . . . . . . . . . . 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 2207	. . . . . . . . . . . 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 13908	. . . . . . . . . . 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 13828	. . . . . . . . . . . . . 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 1252	. . . . . . . . . . . . 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 5969	. . . . . . . . . . . . 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 2206	. . . . . . . . . . . . . . . . 17 |- ( .r ` (oppr ` R ) ) = ( .r ` (oppr ` R ) )
77	31, 8, 51, 76	opprmulg 13883	. . . . . . . . . . . . . . . 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 1250	. . . . . . . . . . . . . . 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 2241	. . . . . . . . . . . . . 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 5970	. . . . . . . . . . . . 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 2247	. . . . . . . . . . . 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 13835	. . . . . . . . . . . . 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 13836	. . . . . . . . . . . . 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 2247	. . . . . . . . . . 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 4079	. . . . . . . . . 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 2207	. . . . . . . . . . . 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 2207	. . . . . . . . . . . 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 13908	. . . . . . . . . . 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 13883	. . . . . . . . . . . . 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 1250	. . . . . . . . . . . 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 2239	. . . . . . . . . . 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 4074	. . . . . . . . . 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 2207	. . . . . . . . . . 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 2207	. . . . . . . . . . 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 13918	. . . . . . . . . 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 947	. . . . . . . . 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 2625	. . . . . . 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 2606	. . . . 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 13404	1 |- ( R e. Ring -> G e. Grp )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 981   = wceq 1373   e. wcel 2177   E.wrex 2486   _Vcvv 2773   class class class wbr 4048   Fn wfn 5272   ` cfv 5277  (class class class)co 5954   Basecbs 12882   ↾_s cress 12883   .rcmulr 12960   Grpcgrp 13382  mulGrpcmgp 13732   1rcur 13771  SRingcsrg 13775   Ringcrg 13808  opp_rcoppr 13879   ||_rcdsr 13898  Unitcui 13899

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 2179  ax-14 2180  ax-ext 2188  ax-coll 4164  ax-sep 4167  ax-nul 4175  ax-pow 4223  ax-pr 4258  ax-un 4485  ax-setind 4590  ax-cnex 8029  ax-resscn 8030  ax-1cn 8031  ax-1re 8032  ax-icn 8033  ax-addcl 8034  ax-addrcl 8035  ax-mulcl 8036  ax-addcom 8038  ax-addass 8040  ax-i2m1 8043  ax-0lt1 8044  ax-0id 8046  ax-rnegex 8047  ax-pre-ltirr 8050  ax-pre-lttrn 8052  ax-pre-ltadd 8054

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 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3001  df-csb 3096  df-dif 3170  df-un 3172  df-in 3174  df-ss 3181  df-nul 3463  df-pw 3620  df-sn 3641  df-pr 3642  df-op 3644  df-uni 3854  df-int 3889  df-iun 3932  df-br 4049  df-opab 4111  df-mpt 4112  df-id 4345  df-xp 4686  df-rel 4687  df-cnv 4688  df-co 4689  df-dm 4690  df-rn 4691  df-res 4692  df-ima 4693  df-iota 5238  df-fun 5279  df-fn 5280  df-f 5281  df-f1 5282  df-fo 5283  df-f1o 5284  df-fv 5285  df-riota 5909  df-ov 5957  df-oprab 5958  df-mpo 5959  df-tpos 6341  df-pnf 8122  df-mnf 8123  df-ltxr 8125  df-inn 9050  df-2 9108  df-3 9109  df-ndx 12885  df-slot 12886  df-base 12888  df-sets 12889  df-iress 12890  df-plusg 12972  df-mulr 12973  df-0g 13140  df-mgm 13238  df-sgrp 13284  df-mnd 13299  df-grp 13385  df-minusg 13386  df-cmn 13672  df-abl 13673  df-mgp 13733  df-ur 13772  df-srg 13776  df-ring 13810  df-oppr 13880  df-dvdsr 13901  df-unit 13902

This theorem is referenced by:  unitabl  13929  unitsubm  13931  invrfvald  13934  unitinvcl  13935  unitinvinv  13936  unitlinv  13938  unitrinv  13939  rdivmuldivd  13956  rhmunitinv  13990  subrgugrp  14052  expghmap  14419