Step |
Hyp |
Ref |
Expression |
1 |
|
isdrng2.b |
⊢ 𝐵 = ( Base ‘ 𝑅 ) |
2 |
|
isdrng2.z |
⊢ 0 = ( 0g ‘ 𝑅 ) |
3 |
|
isdrng2.g |
⊢ 𝐺 = ( ( mulGrp ‘ 𝑅 ) ↾s ( 𝐵 ∖ { 0 } ) ) |
4 |
|
eqid |
⊢ ( Unit ‘ 𝑅 ) = ( Unit ‘ 𝑅 ) |
5 |
1 4 2
|
isdrng |
⊢ ( 𝑅 ∈ DivRing ↔ ( 𝑅 ∈ Ring ∧ ( Unit ‘ 𝑅 ) = ( 𝐵 ∖ { 0 } ) ) ) |
6 |
|
oveq2 |
⊢ ( ( Unit ‘ 𝑅 ) = ( 𝐵 ∖ { 0 } ) → ( ( mulGrp ‘ 𝑅 ) ↾s ( Unit ‘ 𝑅 ) ) = ( ( mulGrp ‘ 𝑅 ) ↾s ( 𝐵 ∖ { 0 } ) ) ) |
7 |
6
|
adantl |
⊢ ( ( 𝑅 ∈ Ring ∧ ( Unit ‘ 𝑅 ) = ( 𝐵 ∖ { 0 } ) ) → ( ( mulGrp ‘ 𝑅 ) ↾s ( Unit ‘ 𝑅 ) ) = ( ( mulGrp ‘ 𝑅 ) ↾s ( 𝐵 ∖ { 0 } ) ) ) |
8 |
7 3
|
eqtr4di |
⊢ ( ( 𝑅 ∈ Ring ∧ ( Unit ‘ 𝑅 ) = ( 𝐵 ∖ { 0 } ) ) → ( ( mulGrp ‘ 𝑅 ) ↾s ( Unit ‘ 𝑅 ) ) = 𝐺 ) |
9 |
|
eqid |
⊢ ( ( mulGrp ‘ 𝑅 ) ↾s ( Unit ‘ 𝑅 ) ) = ( ( mulGrp ‘ 𝑅 ) ↾s ( Unit ‘ 𝑅 ) ) |
10 |
4 9
|
unitgrp |
⊢ ( 𝑅 ∈ Ring → ( ( mulGrp ‘ 𝑅 ) ↾s ( Unit ‘ 𝑅 ) ) ∈ Grp ) |
11 |
10
|
adantr |
⊢ ( ( 𝑅 ∈ Ring ∧ ( Unit ‘ 𝑅 ) = ( 𝐵 ∖ { 0 } ) ) → ( ( mulGrp ‘ 𝑅 ) ↾s ( Unit ‘ 𝑅 ) ) ∈ Grp ) |
12 |
8 11
|
eqeltrrd |
⊢ ( ( 𝑅 ∈ Ring ∧ ( Unit ‘ 𝑅 ) = ( 𝐵 ∖ { 0 } ) ) → 𝐺 ∈ Grp ) |
13 |
1 4
|
unitcl |
⊢ ( 𝑥 ∈ ( Unit ‘ 𝑅 ) → 𝑥 ∈ 𝐵 ) |
14 |
13
|
adantl |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝐺 ∈ Grp ) ∧ 𝑥 ∈ ( Unit ‘ 𝑅 ) ) → 𝑥 ∈ 𝐵 ) |
15 |
|
difss |
⊢ ( 𝐵 ∖ { 0 } ) ⊆ 𝐵 |
16 |
|
eqid |
⊢ ( mulGrp ‘ 𝑅 ) = ( mulGrp ‘ 𝑅 ) |
17 |
16 1
|
mgpbas |
⊢ 𝐵 = ( Base ‘ ( mulGrp ‘ 𝑅 ) ) |
18 |
3 17
|
ressbas2 |
⊢ ( ( 𝐵 ∖ { 0 } ) ⊆ 𝐵 → ( 𝐵 ∖ { 0 } ) = ( Base ‘ 𝐺 ) ) |
19 |
15 18
|
ax-mp |
⊢ ( 𝐵 ∖ { 0 } ) = ( Base ‘ 𝐺 ) |
20 |
|
eqid |
⊢ ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐺 ) |
21 |
19 20
|
grpidcl |
⊢ ( 𝐺 ∈ Grp → ( 0g ‘ 𝐺 ) ∈ ( 𝐵 ∖ { 0 } ) ) |
22 |
21
|
ad2antlr |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝐺 ∈ Grp ) ∧ 𝑥 ∈ ( Unit ‘ 𝑅 ) ) → ( 0g ‘ 𝐺 ) ∈ ( 𝐵 ∖ { 0 } ) ) |
23 |
|
eldifsn |
⊢ ( ( 0g ‘ 𝐺 ) ∈ ( 𝐵 ∖ { 0 } ) ↔ ( ( 0g ‘ 𝐺 ) ∈ 𝐵 ∧ ( 0g ‘ 𝐺 ) ≠ 0 ) ) |
24 |
22 23
|
sylib |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝐺 ∈ Grp ) ∧ 𝑥 ∈ ( Unit ‘ 𝑅 ) ) → ( ( 0g ‘ 𝐺 ) ∈ 𝐵 ∧ ( 0g ‘ 𝐺 ) ≠ 0 ) ) |
25 |
24
|
simprd |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝐺 ∈ Grp ) ∧ 𝑥 ∈ ( Unit ‘ 𝑅 ) ) → ( 0g ‘ 𝐺 ) ≠ 0 ) |
26 |
|
simpll |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝐺 ∈ Grp ) ∧ 𝑥 ∈ ( Unit ‘ 𝑅 ) ) → 𝑅 ∈ Ring ) |
27 |
22
|
eldifad |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝐺 ∈ Grp ) ∧ 𝑥 ∈ ( Unit ‘ 𝑅 ) ) → ( 0g ‘ 𝐺 ) ∈ 𝐵 ) |
28 |
|
simpr |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝐺 ∈ Grp ) ∧ 𝑥 ∈ ( Unit ‘ 𝑅 ) ) → 𝑥 ∈ ( Unit ‘ 𝑅 ) ) |
29 |
|
eqid |
⊢ ( /r ‘ 𝑅 ) = ( /r ‘ 𝑅 ) |
30 |
|
eqid |
⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 ) |
31 |
1 4 29 30
|
dvrcan1 |
⊢ ( ( 𝑅 ∈ Ring ∧ ( 0g ‘ 𝐺 ) ∈ 𝐵 ∧ 𝑥 ∈ ( Unit ‘ 𝑅 ) ) → ( ( ( 0g ‘ 𝐺 ) ( /r ‘ 𝑅 ) 𝑥 ) ( .r ‘ 𝑅 ) 𝑥 ) = ( 0g ‘ 𝐺 ) ) |
32 |
26 27 28 31
|
syl3anc |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝐺 ∈ Grp ) ∧ 𝑥 ∈ ( Unit ‘ 𝑅 ) ) → ( ( ( 0g ‘ 𝐺 ) ( /r ‘ 𝑅 ) 𝑥 ) ( .r ‘ 𝑅 ) 𝑥 ) = ( 0g ‘ 𝐺 ) ) |
33 |
1 4 29
|
dvrcl |
⊢ ( ( 𝑅 ∈ Ring ∧ ( 0g ‘ 𝐺 ) ∈ 𝐵 ∧ 𝑥 ∈ ( Unit ‘ 𝑅 ) ) → ( ( 0g ‘ 𝐺 ) ( /r ‘ 𝑅 ) 𝑥 ) ∈ 𝐵 ) |
34 |
26 27 28 33
|
syl3anc |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝐺 ∈ Grp ) ∧ 𝑥 ∈ ( Unit ‘ 𝑅 ) ) → ( ( 0g ‘ 𝐺 ) ( /r ‘ 𝑅 ) 𝑥 ) ∈ 𝐵 ) |
35 |
1 30 2
|
ringrz |
⊢ ( ( 𝑅 ∈ Ring ∧ ( ( 0g ‘ 𝐺 ) ( /r ‘ 𝑅 ) 𝑥 ) ∈ 𝐵 ) → ( ( ( 0g ‘ 𝐺 ) ( /r ‘ 𝑅 ) 𝑥 ) ( .r ‘ 𝑅 ) 0 ) = 0 ) |
36 |
26 34 35
|
syl2anc |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝐺 ∈ Grp ) ∧ 𝑥 ∈ ( Unit ‘ 𝑅 ) ) → ( ( ( 0g ‘ 𝐺 ) ( /r ‘ 𝑅 ) 𝑥 ) ( .r ‘ 𝑅 ) 0 ) = 0 ) |
37 |
25 32 36
|
3netr4d |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝐺 ∈ Grp ) ∧ 𝑥 ∈ ( Unit ‘ 𝑅 ) ) → ( ( ( 0g ‘ 𝐺 ) ( /r ‘ 𝑅 ) 𝑥 ) ( .r ‘ 𝑅 ) 𝑥 ) ≠ ( ( ( 0g ‘ 𝐺 ) ( /r ‘ 𝑅 ) 𝑥 ) ( .r ‘ 𝑅 ) 0 ) ) |
38 |
|
oveq2 |
⊢ ( 𝑥 = 0 → ( ( ( 0g ‘ 𝐺 ) ( /r ‘ 𝑅 ) 𝑥 ) ( .r ‘ 𝑅 ) 𝑥 ) = ( ( ( 0g ‘ 𝐺 ) ( /r ‘ 𝑅 ) 𝑥 ) ( .r ‘ 𝑅 ) 0 ) ) |
39 |
38
|
necon3i |
⊢ ( ( ( ( 0g ‘ 𝐺 ) ( /r ‘ 𝑅 ) 𝑥 ) ( .r ‘ 𝑅 ) 𝑥 ) ≠ ( ( ( 0g ‘ 𝐺 ) ( /r ‘ 𝑅 ) 𝑥 ) ( .r ‘ 𝑅 ) 0 ) → 𝑥 ≠ 0 ) |
40 |
37 39
|
syl |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝐺 ∈ Grp ) ∧ 𝑥 ∈ ( Unit ‘ 𝑅 ) ) → 𝑥 ≠ 0 ) |
41 |
|
eldifsn |
⊢ ( 𝑥 ∈ ( 𝐵 ∖ { 0 } ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 ) ) |
42 |
14 40 41
|
sylanbrc |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝐺 ∈ Grp ) ∧ 𝑥 ∈ ( Unit ‘ 𝑅 ) ) → 𝑥 ∈ ( 𝐵 ∖ { 0 } ) ) |
43 |
42
|
ex |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐺 ∈ Grp ) → ( 𝑥 ∈ ( Unit ‘ 𝑅 ) → 𝑥 ∈ ( 𝐵 ∖ { 0 } ) ) ) |
44 |
43
|
ssrdv |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐺 ∈ Grp ) → ( Unit ‘ 𝑅 ) ⊆ ( 𝐵 ∖ { 0 } ) ) |
45 |
|
eldifi |
⊢ ( 𝑥 ∈ ( 𝐵 ∖ { 0 } ) → 𝑥 ∈ 𝐵 ) |
46 |
45
|
adantl |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝐺 ∈ Grp ) ∧ 𝑥 ∈ ( 𝐵 ∖ { 0 } ) ) → 𝑥 ∈ 𝐵 ) |
47 |
|
eqid |
⊢ ( invg ‘ 𝐺 ) = ( invg ‘ 𝐺 ) |
48 |
19 47
|
grpinvcl |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑥 ∈ ( 𝐵 ∖ { 0 } ) ) → ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ∈ ( 𝐵 ∖ { 0 } ) ) |
49 |
48
|
adantll |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝐺 ∈ Grp ) ∧ 𝑥 ∈ ( 𝐵 ∖ { 0 } ) ) → ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ∈ ( 𝐵 ∖ { 0 } ) ) |
50 |
49
|
eldifad |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝐺 ∈ Grp ) ∧ 𝑥 ∈ ( 𝐵 ∖ { 0 } ) ) → ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ∈ 𝐵 ) |
51 |
|
eqid |
⊢ ( ∥r ‘ 𝑅 ) = ( ∥r ‘ 𝑅 ) |
52 |
1 51 30
|
dvdsrmul |
⊢ ( ( 𝑥 ∈ 𝐵 ∧ ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ∈ 𝐵 ) → 𝑥 ( ∥r ‘ 𝑅 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ( .r ‘ 𝑅 ) 𝑥 ) ) |
53 |
46 50 52
|
syl2anc |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝐺 ∈ Grp ) ∧ 𝑥 ∈ ( 𝐵 ∖ { 0 } ) ) → 𝑥 ( ∥r ‘ 𝑅 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ( .r ‘ 𝑅 ) 𝑥 ) ) |
54 |
1
|
fvexi |
⊢ 𝐵 ∈ V |
55 |
|
difexg |
⊢ ( 𝐵 ∈ V → ( 𝐵 ∖ { 0 } ) ∈ V ) |
56 |
16 30
|
mgpplusg |
⊢ ( .r ‘ 𝑅 ) = ( +g ‘ ( mulGrp ‘ 𝑅 ) ) |
57 |
3 56
|
ressplusg |
⊢ ( ( 𝐵 ∖ { 0 } ) ∈ V → ( .r ‘ 𝑅 ) = ( +g ‘ 𝐺 ) ) |
58 |
54 55 57
|
mp2b |
⊢ ( .r ‘ 𝑅 ) = ( +g ‘ 𝐺 ) |
59 |
19 58 20 47
|
grplinv |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑥 ∈ ( 𝐵 ∖ { 0 } ) ) → ( ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ( .r ‘ 𝑅 ) 𝑥 ) = ( 0g ‘ 𝐺 ) ) |
60 |
59
|
adantll |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝐺 ∈ Grp ) ∧ 𝑥 ∈ ( 𝐵 ∖ { 0 } ) ) → ( ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ( .r ‘ 𝑅 ) 𝑥 ) = ( 0g ‘ 𝐺 ) ) |
61 |
|
eqid |
⊢ ( 1r ‘ 𝑅 ) = ( 1r ‘ 𝑅 ) |
62 |
1 61
|
ringidcl |
⊢ ( 𝑅 ∈ Ring → ( 1r ‘ 𝑅 ) ∈ 𝐵 ) |
63 |
1 30 61
|
ringlidm |
⊢ ( ( 𝑅 ∈ Ring ∧ ( 1r ‘ 𝑅 ) ∈ 𝐵 ) → ( ( 1r ‘ 𝑅 ) ( .r ‘ 𝑅 ) ( 1r ‘ 𝑅 ) ) = ( 1r ‘ 𝑅 ) ) |
64 |
62 63
|
mpdan |
⊢ ( 𝑅 ∈ Ring → ( ( 1r ‘ 𝑅 ) ( .r ‘ 𝑅 ) ( 1r ‘ 𝑅 ) ) = ( 1r ‘ 𝑅 ) ) |
65 |
64
|
adantr |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐺 ∈ Grp ) → ( ( 1r ‘ 𝑅 ) ( .r ‘ 𝑅 ) ( 1r ‘ 𝑅 ) ) = ( 1r ‘ 𝑅 ) ) |
66 |
|
simpr |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐺 ∈ Grp ) → 𝐺 ∈ Grp ) |
67 |
4 61
|
1unit |
⊢ ( 𝑅 ∈ Ring → ( 1r ‘ 𝑅 ) ∈ ( Unit ‘ 𝑅 ) ) |
68 |
67
|
adantr |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐺 ∈ Grp ) → ( 1r ‘ 𝑅 ) ∈ ( Unit ‘ 𝑅 ) ) |
69 |
44 68
|
sseldd |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐺 ∈ Grp ) → ( 1r ‘ 𝑅 ) ∈ ( 𝐵 ∖ { 0 } ) ) |
70 |
19 58 20
|
grpid |
⊢ ( ( 𝐺 ∈ Grp ∧ ( 1r ‘ 𝑅 ) ∈ ( 𝐵 ∖ { 0 } ) ) → ( ( ( 1r ‘ 𝑅 ) ( .r ‘ 𝑅 ) ( 1r ‘ 𝑅 ) ) = ( 1r ‘ 𝑅 ) ↔ ( 0g ‘ 𝐺 ) = ( 1r ‘ 𝑅 ) ) ) |
71 |
66 69 70
|
syl2anc |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐺 ∈ Grp ) → ( ( ( 1r ‘ 𝑅 ) ( .r ‘ 𝑅 ) ( 1r ‘ 𝑅 ) ) = ( 1r ‘ 𝑅 ) ↔ ( 0g ‘ 𝐺 ) = ( 1r ‘ 𝑅 ) ) ) |
72 |
65 71
|
mpbid |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐺 ∈ Grp ) → ( 0g ‘ 𝐺 ) = ( 1r ‘ 𝑅 ) ) |
73 |
72
|
adantr |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝐺 ∈ Grp ) ∧ 𝑥 ∈ ( 𝐵 ∖ { 0 } ) ) → ( 0g ‘ 𝐺 ) = ( 1r ‘ 𝑅 ) ) |
74 |
60 73
|
eqtrd |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝐺 ∈ Grp ) ∧ 𝑥 ∈ ( 𝐵 ∖ { 0 } ) ) → ( ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ( .r ‘ 𝑅 ) 𝑥 ) = ( 1r ‘ 𝑅 ) ) |
75 |
53 74
|
breqtrd |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝐺 ∈ Grp ) ∧ 𝑥 ∈ ( 𝐵 ∖ { 0 } ) ) → 𝑥 ( ∥r ‘ 𝑅 ) ( 1r ‘ 𝑅 ) ) |
76 |
|
eqid |
⊢ ( oppr ‘ 𝑅 ) = ( oppr ‘ 𝑅 ) |
77 |
76 1
|
opprbas |
⊢ 𝐵 = ( Base ‘ ( oppr ‘ 𝑅 ) ) |
78 |
|
eqid |
⊢ ( ∥r ‘ ( oppr ‘ 𝑅 ) ) = ( ∥r ‘ ( oppr ‘ 𝑅 ) ) |
79 |
|
eqid |
⊢ ( .r ‘ ( oppr ‘ 𝑅 ) ) = ( .r ‘ ( oppr ‘ 𝑅 ) ) |
80 |
77 78 79
|
dvdsrmul |
⊢ ( ( 𝑥 ∈ 𝐵 ∧ ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ∈ 𝐵 ) → 𝑥 ( ∥r ‘ ( oppr ‘ 𝑅 ) ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ( .r ‘ ( oppr ‘ 𝑅 ) ) 𝑥 ) ) |
81 |
46 50 80
|
syl2anc |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝐺 ∈ Grp ) ∧ 𝑥 ∈ ( 𝐵 ∖ { 0 } ) ) → 𝑥 ( ∥r ‘ ( oppr ‘ 𝑅 ) ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ( .r ‘ ( oppr ‘ 𝑅 ) ) 𝑥 ) ) |
82 |
1 30 76 79
|
opprmul |
⊢ ( ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ( .r ‘ ( oppr ‘ 𝑅 ) ) 𝑥 ) = ( 𝑥 ( .r ‘ 𝑅 ) ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ) |
83 |
19 58 20 47
|
grprinv |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑥 ∈ ( 𝐵 ∖ { 0 } ) ) → ( 𝑥 ( .r ‘ 𝑅 ) ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ) = ( 0g ‘ 𝐺 ) ) |
84 |
83
|
adantll |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝐺 ∈ Grp ) ∧ 𝑥 ∈ ( 𝐵 ∖ { 0 } ) ) → ( 𝑥 ( .r ‘ 𝑅 ) ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ) = ( 0g ‘ 𝐺 ) ) |
85 |
84 73
|
eqtrd |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝐺 ∈ Grp ) ∧ 𝑥 ∈ ( 𝐵 ∖ { 0 } ) ) → ( 𝑥 ( .r ‘ 𝑅 ) ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ) = ( 1r ‘ 𝑅 ) ) |
86 |
82 85
|
eqtrid |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝐺 ∈ Grp ) ∧ 𝑥 ∈ ( 𝐵 ∖ { 0 } ) ) → ( ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ( .r ‘ ( oppr ‘ 𝑅 ) ) 𝑥 ) = ( 1r ‘ 𝑅 ) ) |
87 |
81 86
|
breqtrd |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝐺 ∈ Grp ) ∧ 𝑥 ∈ ( 𝐵 ∖ { 0 } ) ) → 𝑥 ( ∥r ‘ ( oppr ‘ 𝑅 ) ) ( 1r ‘ 𝑅 ) ) |
88 |
4 61 51 76 78
|
isunit |
⊢ ( 𝑥 ∈ ( Unit ‘ 𝑅 ) ↔ ( 𝑥 ( ∥r ‘ 𝑅 ) ( 1r ‘ 𝑅 ) ∧ 𝑥 ( ∥r ‘ ( oppr ‘ 𝑅 ) ) ( 1r ‘ 𝑅 ) ) ) |
89 |
75 87 88
|
sylanbrc |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝐺 ∈ Grp ) ∧ 𝑥 ∈ ( 𝐵 ∖ { 0 } ) ) → 𝑥 ∈ ( Unit ‘ 𝑅 ) ) |
90 |
44 89
|
eqelssd |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐺 ∈ Grp ) → ( Unit ‘ 𝑅 ) = ( 𝐵 ∖ { 0 } ) ) |
91 |
12 90
|
impbida |
⊢ ( 𝑅 ∈ Ring → ( ( Unit ‘ 𝑅 ) = ( 𝐵 ∖ { 0 } ) ↔ 𝐺 ∈ Grp ) ) |
92 |
91
|
pm5.32i |
⊢ ( ( 𝑅 ∈ Ring ∧ ( Unit ‘ 𝑅 ) = ( 𝐵 ∖ { 0 } ) ) ↔ ( 𝑅 ∈ Ring ∧ 𝐺 ∈ Grp ) ) |
93 |
5 92
|
bitri |
⊢ ( 𝑅 ∈ DivRing ↔ ( 𝑅 ∈ Ring ∧ 𝐺 ∈ Grp ) ) |