Step |
Hyp |
Ref |
Expression |
1 |
|
lmodvsmmulgdi.v |
|
2 |
|
lmodvsmmulgdi.f |
|
3 |
|
lmodvsmmulgdi.s |
|
4 |
|
lmodvsmmulgdi.k |
|
5 |
|
lmodvsmmulgdi.p |
|
6 |
|
lmodvsmmulgdi.e |
|
7 |
|
oveq1 |
|
8 |
|
oveq1 |
|
9 |
8
|
oveq1d |
|
10 |
7 9
|
eqeq12d |
|
11 |
10
|
imbi2d |
|
12 |
|
oveq1 |
|
13 |
|
oveq1 |
|
14 |
13
|
oveq1d |
|
15 |
12 14
|
eqeq12d |
|
16 |
15
|
imbi2d |
|
17 |
|
oveq1 |
|
18 |
|
oveq1 |
|
19 |
18
|
oveq1d |
|
20 |
17 19
|
eqeq12d |
|
21 |
20
|
imbi2d |
|
22 |
|
oveq1 |
|
23 |
|
oveq1 |
|
24 |
23
|
oveq1d |
|
25 |
22 24
|
eqeq12d |
|
26 |
25
|
imbi2d |
|
27 |
|
simpr |
|
28 |
|
simpr |
|
29 |
28
|
adantr |
|
30 |
|
eqid |
|
31 |
|
eqid |
|
32 |
1 2 3 30 31
|
lmod0vs |
|
33 |
27 29 32
|
syl2anc |
|
34 |
|
simpl |
|
35 |
34
|
adantr |
|
36 |
4 30 6
|
mulg0 |
|
37 |
35 36
|
syl |
|
38 |
37
|
oveq1d |
|
39 |
1 2 3 4
|
lmodvscl |
|
40 |
27 35 29 39
|
syl3anc |
|
41 |
1 31 5
|
mulg0 |
|
42 |
40 41
|
syl |
|
43 |
33 38 42
|
3eqtr4rd |
|
44 |
|
lmodgrp |
|
45 |
44
|
grpmndd |
|
46 |
45
|
ad2antll |
|
47 |
|
simpl |
|
48 |
40
|
adantl |
|
49 |
|
eqid |
|
50 |
1 5 49
|
mulgnn0p1 |
|
51 |
46 47 48 50
|
syl3anc |
|
52 |
51
|
adantr |
|
53 |
|
oveq1 |
|
54 |
27
|
adantl |
|
55 |
2
|
lmodring |
|
56 |
|
ringmnd |
|
57 |
55 56
|
syl |
|
58 |
57
|
ad2antll |
|
59 |
|
simprll |
|
60 |
4 6
|
mulgnn0cl |
|
61 |
58 47 59 60
|
syl3anc |
|
62 |
29
|
adantl |
|
63 |
|
eqid |
|
64 |
1 49 2 3 4 63
|
lmodvsdir |
|
65 |
54 61 59 62 64
|
syl13anc |
|
66 |
4 6 63
|
mulgnn0p1 |
|
67 |
58 47 59 66
|
syl3anc |
|
68 |
67
|
eqcomd |
|
69 |
68
|
oveq1d |
|
70 |
65 69
|
eqtr3d |
|
71 |
53 70
|
sylan9eqr |
|
72 |
52 71
|
eqtrd |
|
73 |
72
|
exp31 |
|
74 |
73
|
a2d |
|
75 |
11 16 21 26 43 74
|
nn0ind |
|
76 |
75
|
exp4c |
|
77 |
76
|
3imp21 |
|
78 |
77
|
impcom |
|