Step |
Hyp |
Ref |
Expression |
1 |
|
simpr |
|
2 |
|
ringgrp |
|
3 |
2
|
adantl |
|
4 |
|
orngogrp |
|
5 |
|
isogrp |
|
6 |
5
|
simprbi |
|
7 |
4 6
|
syl |
|
8 |
|
ringmnd |
|
9 |
|
submomnd |
|
10 |
7 8 9
|
syl2an |
|
11 |
|
isogrp |
|
12 |
3 10 11
|
sylanbrc |
|
13 |
|
simp-4l |
|
14 |
|
reldmress |
|
15 |
14
|
ovprc2 |
|
16 |
15
|
fveq2d |
|
17 |
16
|
adantl |
|
18 |
|
base0 |
|
19 |
17 18
|
eqtr4di |
|
20 |
|
eqid |
|
21 |
|
eqid |
|
22 |
20 21
|
ringidcl |
|
23 |
22
|
ne0d |
|
24 |
23
|
ad2antlr |
|
25 |
24
|
neneqd |
|
26 |
19 25
|
condan |
|
27 |
|
eqid |
|
28 |
|
eqid |
|
29 |
27 28
|
ressbas |
|
30 |
|
inss2 |
|
31 |
29 30
|
eqsstrrdi |
|
32 |
26 31
|
syl |
|
33 |
32
|
ad3antrrr |
|
34 |
|
simpllr |
|
35 |
33 34
|
sseldd |
|
36 |
|
simprl |
|
37 |
|
orngring |
|
38 |
|
ringgrp |
|
39 |
37 38
|
syl |
|
40 |
39
|
adantr |
|
41 |
28
|
ressinbas |
|
42 |
29
|
oveq2d |
|
43 |
41 42
|
eqtrd |
|
44 |
26 43
|
syl |
|
45 |
44 3
|
eqeltrrd |
|
46 |
28
|
issubg |
|
47 |
40 32 45 46
|
syl3anbrc |
|
48 |
|
eqid |
|
49 |
|
eqid |
|
50 |
48 49
|
subg0 |
|
51 |
47 50
|
syl |
|
52 |
44
|
fveq2d |
|
53 |
51 52
|
eqtr4d |
|
54 |
53
|
ad2antrr |
|
55 |
26
|
ad2antrr |
|
56 |
|
eqid |
|
57 |
27 56
|
ressle |
|
58 |
55 57
|
syl |
|
59 |
|
eqidd |
|
60 |
54 58 59
|
breq123d |
|
61 |
60
|
adantr |
|
62 |
36 61
|
mpbird |
|
63 |
|
simplr |
|
64 |
33 63
|
sseldd |
|
65 |
|
simprr |
|
66 |
|
eqidd |
|
67 |
54 58 66
|
breq123d |
|
68 |
67
|
adantr |
|
69 |
65 68
|
mpbird |
|
70 |
|
eqid |
|
71 |
28 56 49 70
|
orngmul |
|
72 |
13 35 62 64 69 71
|
syl122anc |
|
73 |
54
|
adantr |
|
74 |
58
|
adantr |
|
75 |
55
|
adantr |
|
76 |
27 70
|
ressmulr |
|
77 |
75 76
|
syl |
|
78 |
77
|
oveqd |
|
79 |
73 74 78
|
breq123d |
|
80 |
72 79
|
mpbid |
|
81 |
80
|
ex |
|
82 |
81
|
anasss |
|
83 |
82
|
ralrimivva |
|
84 |
|
eqid |
|
85 |
|
eqid |
|
86 |
|
eqid |
|
87 |
20 84 85 86
|
isorng |
|
88 |
1 12 83 87
|
syl3anbrc |
|