Theorem unitgrp 14425

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 14354	. . 3 |- ( R e. Ring -> R e. SRing )
6	2, 4, 5	unitgrpbasd 14424	. 2 |- ( R e. Ring -> U = ( Base ` G ) )
7		eqid 2238	. . . 4 |- (mulGrp ` R ) = (mulGrp ` R )
8		eqid 2238	. . . 4 |- ( .r ` R ) = ( .r ` R )
9	7, 8	mgpplusgg 14223	. . 3 |- ( R e. Ring -> ( .r ` R ) = ( +g ` (mulGrp ` R ) ) )
10		basfn 13413	. . . . 5 |- Base Fn _V
11		elex 2833	. . . . 5 |- ( R e. Ring -> R e. _V )
12		funfvex 5712	. . . . . 6 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
13	12	funfni 5483	. . . . 5 |- ( ( Base Fn _V /\ R e. _V ) -> ( Base ` R ) e. _V )
14	10, 11, 13	sylancr 418	. . . 4 |- ( R e. Ring -> ( Base ` R ) e. _V )
15		eqidd 2239	. . . . 5 |- ( R e. Ring -> ( Base ` R ) = ( Base ` R ) )
16	15, 2, 5	unitssd 14418	. . . 4 |- ( R e. Ring -> U C_ ( Base ` R ) )
17	14, 16	ssexd 4273	. . 3 |- ( R e. Ring -> U e. _V )
18	7	mgpex 14225	. . 3 |- ( R e. Ring -> (mulGrp ` R ) e. _V )
19	4, 9, 17, 18	ressplusgd 13485	. 2 |- ( R e. Ring -> ( .r ` R ) = ( +g ` G ) )
20	1, 8	unitmulcl 14422	. 2 |- ( ( R e. Ring /\ x e. U /\ y e. U ) -> ( x ( .r ` R ) y ) e. U )
21		eqidd 2239	. . . . 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 1034	. . . . 5 |- ( ( R e. Ring /\ ( x e. U /\ y e. U /\ z e. U ) ) -> x e. U )
25	21, 22, 23, 24	unitcld 14417	. . . 4 |- ( ( R e. Ring /\ ( x e. U /\ y e. U /\ z e. U ) ) -> x e. ( Base ` R ) )
26		simpr2 1035	. . . . 5 |- ( ( R e. Ring /\ ( x e. U /\ y e. U /\ z e. U ) ) -> y e. U )
27	21, 22, 23, 26	unitcld 14417	. . . 4 |- ( ( R e. Ring /\ ( x e. U /\ y e. U /\ z e. U ) ) -> y e. ( Base ` R ) )
28		simpr3 1036	. . . . 5 |- ( ( R e. Ring /\ ( x e. U /\ y e. U /\ z e. U ) ) -> z e. U )
29	21, 22, 23, 28	unitcld 14417	. . . 4 |- ( ( R e. Ring /\ ( x e. U /\ y e. U /\ z e. U ) ) -> z e. ( Base ` R ) )
30	25, 27, 29	3jca 1208	. . 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 2238	. . . 4 |- ( Base ` R ) = ( Base ` R )
32	31, 8	ringass 14322	. . 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 2238	. . 3 |- ( 1r ` R ) = ( 1r ` R )
35	1, 34	1unit 14416	. 2 |- ( R e. Ring -> ( 1r ` R ) e. U )
36		eqidd 2239	. . . 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 14417	. . 3 |- ( ( R e. Ring /\ x e. U ) -> x e. ( Base ` R ) )
41	31, 8, 34	ringlidm 14330	. . 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 2239	. . . . 5 |- ( ( R e. Ring /\ x e. U ) -> ( 1r ` R ) = ( 1r ` R ) )
44		eqidd 2239	. . . . 5 |- ( ( R e. Ring /\ x e. U ) -> ( ||r ` R ) = ( ||r ` R ) )
45		eqidd 2239	. . . . 5 |- ( ( R e. Ring /\ x e. U ) -> (oppr ` R ) = (oppr ` R ) )
46		eqidd 2239	. . . . 5 |- ( ( R e. Ring /\ x e. U ) -> ( ||r ` (oppr ` R ) ) = ( ||r ` (oppr ` R ) ) )
47	37, 43, 44, 45, 46, 38	isunitd 14415	. . . 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 2239	. . . . . 6 |- ( ( R e. Ring /\ x e. U ) -> ( .r ` R ) = ( .r ` R ) )
50	36, 44, 38, 49, 40	dvdsr2d 14404	. . . . 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 2238	. . . . . . . 8 |- (oppr ` R ) = (oppr ` R )
52	51, 31	opprbasg 14382	. . . . . . 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 14386	. . . . . . . 8 |- ( R e. Ring -> (oppr ` R ) e. Ring )
55		ringsrg 14354	. . . . . . . 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 2239	. . . . . 6 |- ( ( R e. Ring /\ x e. U ) -> ( .r ` (oppr ` R ) ) = ( .r ` (oppr ` R ) ) )
59	53, 46, 57, 58, 40	dvdsr2d 14404	. . . . 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 477	. . . 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 2721	. . . . 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 2239	. . . . . . . . . . . 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 2239	. . . . . . . . . . . 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 492	. . . . . . . . . . . 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 2239	. . . . . . . . . . . 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 535	. . . . . . . . . . . 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 492	. . . . . . . . . . . 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 14405	. . . . . . . . . . 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 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 ) ) ) ) -> R e. Ring )
70		simplr 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 ) ) ) ) -> y e. ( Base ` R ) )
71	31, 8	ringass 14322	. . . . . . . . . . . . . 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 1280	. . . . . . . . . . . . 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 545	. . . . . . . . . . . . . 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 6100	. . . . . . . . . . . . 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 492	. . . . . . . . . . . . . . . 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 2238	. . . . . . . . . . . . . . . . 17 |- ( .r ` (oppr ` R ) ) = ( .r ` (oppr ` R ) )
77	31, 8, 51, 76	opprmulg 14378	. . . . . . . . . . . . . . . 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 1278	. . . . . . . . . . . . . . 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 546	. . . . . . . . . . . . . . 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 2273	. . . . . . . . . . . . . 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 6101	. . . . . . . . . . . . 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 2279	. . . . . . . . . . . 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 14330	. . . . . . . . . . . . 13 |- ( ( R e. Ring /\ m e. ( Base ` R ) ) -> ( ( 1r ` R ) ( .r ` R ) m ) = m )
84	69, 66, 83	syl2anc 415	. . . . . . . . . . . 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 14331	. . . . . . . . . . . . 13 |- ( ( R e. Ring /\ y e. ( Base ` R ) ) -> ( y ( .r ` R ) ( 1r ` R ) ) = y )
86	69, 70, 85	syl2anc 415	. . . . . . . . . . . 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 2279	. . . . . . . . . . 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 4161	. . . . . . . . . 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 2239	. . . . . . . . . . . 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 2239	. . . . . . . . . . . 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 14405	. . . . . . . . . . 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 14378	. . . . . . . . . . . . 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 1278	. . . . . . . . . . . 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 2271	. . . . . . . . . . 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 4156	. . . . . . . . . 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 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 ) ) ) ) -> ( 1r ` R ) = ( 1r ` R ) )
100		eqidd 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 ) ) ) ) -> (oppr ` R ) = (oppr ` R ) )
101	98, 99, 63, 100, 90, 64	isunitd 14415	. . . . . . . . . 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 957	. . . . . . . . 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 2669	. . . . . . 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 2649	. . . . 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 13829	1 |- ( R e. Ring -> G e. Grp )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   E.wrex 2529   _Vcvv 2821   class class class wbr 4130   Fn wfn 5372   ` cfv 5377  (class class class)co 6085   Basecbs 13354   ↾_s cress 13355   .rcmulr 13434   Grpcgrp 13807  mulGrpcmgp 14219   1rcur 14264  SRingcsrg 14269   Ringcrg 14302  opp_rcoppr 14374   ||_rcdsr 14394  Unitcui 14395

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-pre-ltirr 8291  ax-pre-lttrn 8293  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-tpos 6516  df-pnf 8362  df-mnf 8363  df-ltxr 8365  df-inn 9306  df-2 9364  df-3 9365  df-ndx 13357  df-slot 13358  df-base 13360  df-sets 13361  df-iress 13362  df-plusg 13446  df-mulr 13447  df-0g 13614  df-mgm 13678  df-sgrp 13719  df-mnd 13732  df-grp 13810  df-minusg 13811  df-cmn 14091  df-abl 14092  df-mgp 14220  df-ur 14265  df-srg 14270  df-ring 14304  df-oppr 14375  df-dvdsr 14397  df-unit 14398

This theorem is used by:  unitabl  14426  unitsubm  14428  invrfvald  14431  unitinvcl  14432  unitinvinv  14433  unitlinv  14435  unitrinv  14436  rdivmuldivd  14453  rhmunitinv  14487  subrgugrp  14550  expghmap  14944