Step |
Hyp |
Ref |
Expression |
1 |
|
lmodvsmdi.v |
|
2 |
|
lmodvsmdi.f |
|
3 |
|
lmodvsmdi.s |
|
4 |
|
lmodvsmdi.k |
|
5 |
|
lmodvsmdi.p |
|
6 |
|
lmodvsmdi.e |
|
7 |
|
oveq1 |
|
8 |
7
|
oveq2d |
|
9 |
|
oveq1 |
|
10 |
9
|
oveq1d |
|
11 |
8 10
|
eqeq12d |
|
12 |
11
|
imbi2d |
|
13 |
|
oveq1 |
|
14 |
13
|
oveq2d |
|
15 |
|
oveq1 |
|
16 |
15
|
oveq1d |
|
17 |
14 16
|
eqeq12d |
|
18 |
17
|
imbi2d |
|
19 |
|
oveq1 |
|
20 |
19
|
oveq2d |
|
21 |
|
oveq1 |
|
22 |
21
|
oveq1d |
|
23 |
20 22
|
eqeq12d |
|
24 |
23
|
imbi2d |
|
25 |
|
oveq1 |
|
26 |
25
|
oveq2d |
|
27 |
|
oveq1 |
|
28 |
27
|
oveq1d |
|
29 |
26 28
|
eqeq12d |
|
30 |
29
|
imbi2d |
|
31 |
|
simpr |
|
32 |
31
|
adantr |
|
33 |
|
eqid |
|
34 |
1 33 5
|
mulg0 |
|
35 |
32 34
|
syl |
|
36 |
35
|
oveq2d |
|
37 |
|
simpl |
|
38 |
37
|
anim1i |
|
39 |
38
|
ancomd |
|
40 |
2 3 4 33
|
lmodvs0 |
|
41 |
39 40
|
syl |
|
42 |
31
|
anim1i |
|
43 |
42
|
ancomd |
|
44 |
|
eqid |
|
45 |
1 2 3 44 33
|
lmod0vs |
|
46 |
43 45
|
syl |
|
47 |
37
|
adantr |
|
48 |
4 44 6
|
mulg0 |
|
49 |
48
|
eqcomd |
|
50 |
47 49
|
syl |
|
51 |
50
|
oveq1d |
|
52 |
41 46 51
|
3eqtr2d |
|
53 |
36 52
|
eqtrd |
|
54 |
|
lmodgrp |
|
55 |
|
grpmnd |
|
56 |
54 55
|
syl |
|
57 |
56
|
ad2antll |
|
58 |
|
simpl |
|
59 |
32
|
adantl |
|
60 |
|
eqid |
|
61 |
1 5 60
|
mulgnn0p1 |
|
62 |
57 58 59 61
|
syl3anc |
|
63 |
62
|
oveq2d |
|
64 |
|
simpr |
|
65 |
64
|
adantl |
|
66 |
|
simprll |
|
67 |
1 5
|
mulgnn0cl |
|
68 |
57 58 59 67
|
syl3anc |
|
69 |
1 60 2 3 4
|
lmodvsdi |
|
70 |
65 66 68 59 69
|
syl13anc |
|
71 |
63 70
|
eqtrd |
|
72 |
|
oveq1 |
|
73 |
71 72
|
sylan9eq |
|
74 |
2
|
lmodfgrp |
|
75 |
|
grpmnd |
|
76 |
74 75
|
syl |
|
77 |
76
|
ad2antll |
|
78 |
4 6
|
mulgnn0cl |
|
79 |
77 58 66 78
|
syl3anc |
|
80 |
|
eqid |
|
81 |
1 60 2 3 4 80
|
lmodvsdir |
|
82 |
65 79 66 59 81
|
syl13anc |
|
83 |
4 6 80
|
mulgnn0p1 |
|
84 |
77 58 66 83
|
syl3anc |
|
85 |
84
|
eqcomd |
|
86 |
85
|
oveq1d |
|
87 |
82 86
|
eqtr3d |
|
88 |
87
|
adantr |
|
89 |
73 88
|
eqtrd |
|
90 |
89
|
exp31 |
|
91 |
90
|
a2d |
|
92 |
12 18 24 30 53 91
|
nn0ind |
|
93 |
92
|
exp4c |
|
94 |
93
|
com12 |
|
95 |
94
|
3imp |
|
96 |
95
|
impcom |
|