Step |
Hyp |
Ref |
Expression |
1 |
|
chincl |
|- ( ( A e. CH /\ B e. CH ) -> ( A i^i B ) e. CH ) |
2 |
|
chincl |
|- ( ( A e. CH /\ C e. CH ) -> ( A i^i C ) e. CH ) |
3 |
|
chjcl |
|- ( ( ( A i^i B ) e. CH /\ ( A i^i C ) e. CH ) -> ( ( A i^i B ) vH ( A i^i C ) ) e. CH ) |
4 |
1 2 3
|
syl2an |
|- ( ( ( A e. CH /\ B e. CH ) /\ ( A e. CH /\ C e. CH ) ) -> ( ( A i^i B ) vH ( A i^i C ) ) e. CH ) |
5 |
4
|
anandis |
|- ( ( A e. CH /\ ( B e. CH /\ C e. CH ) ) -> ( ( A i^i B ) vH ( A i^i C ) ) e. CH ) |
6 |
|
chjcl |
|- ( ( B e. CH /\ C e. CH ) -> ( B vH C ) e. CH ) |
7 |
|
chincl |
|- ( ( A e. CH /\ ( B vH C ) e. CH ) -> ( A i^i ( B vH C ) ) e. CH ) |
8 |
6 7
|
sylan2 |
|- ( ( A e. CH /\ ( B e. CH /\ C e. CH ) ) -> ( A i^i ( B vH C ) ) e. CH ) |
9 |
|
chsh |
|- ( ( A i^i ( B vH C ) ) e. CH -> ( A i^i ( B vH C ) ) e. SH ) |
10 |
8 9
|
syl |
|- ( ( A e. CH /\ ( B e. CH /\ C e. CH ) ) -> ( A i^i ( B vH C ) ) e. SH ) |
11 |
5 10
|
jca |
|- ( ( A e. CH /\ ( B e. CH /\ C e. CH ) ) -> ( ( ( A i^i B ) vH ( A i^i C ) ) e. CH /\ ( A i^i ( B vH C ) ) e. SH ) ) |
12 |
11
|
3impb |
|- ( ( A e. CH /\ B e. CH /\ C e. CH ) -> ( ( ( A i^i B ) vH ( A i^i C ) ) e. CH /\ ( A i^i ( B vH C ) ) e. SH ) ) |
13 |
12
|
adantr |
|- ( ( ( A e. CH /\ B e. CH /\ C e. CH ) /\ ( B C_H A /\ B C_H C ) ) -> ( ( ( A i^i B ) vH ( A i^i C ) ) e. CH /\ ( A i^i ( B vH C ) ) e. SH ) ) |
14 |
|
ledi |
|- ( ( A e. CH /\ B e. CH /\ C e. CH ) -> ( ( A i^i B ) vH ( A i^i C ) ) C_ ( A i^i ( B vH C ) ) ) |
15 |
14
|
adantr |
|- ( ( ( A e. CH /\ B e. CH /\ C e. CH ) /\ ( B C_H A /\ B C_H C ) ) -> ( ( A i^i B ) vH ( A i^i C ) ) C_ ( A i^i ( B vH C ) ) ) |
16 |
|
chdmj1 |
|- ( ( ( A i^i B ) e. CH /\ ( A i^i C ) e. CH ) -> ( _|_ ` ( ( A i^i B ) vH ( A i^i C ) ) ) = ( ( _|_ ` ( A i^i B ) ) i^i ( _|_ ` ( A i^i C ) ) ) ) |
17 |
1 2 16
|
syl2an |
|- ( ( ( A e. CH /\ B e. CH ) /\ ( A e. CH /\ C e. CH ) ) -> ( _|_ ` ( ( A i^i B ) vH ( A i^i C ) ) ) = ( ( _|_ ` ( A i^i B ) ) i^i ( _|_ ` ( A i^i C ) ) ) ) |
18 |
|
chdmm1 |
|- ( ( A e. CH /\ B e. CH ) -> ( _|_ ` ( A i^i B ) ) = ( ( _|_ ` A ) vH ( _|_ ` B ) ) ) |
19 |
18
|
adantr |
|- ( ( ( A e. CH /\ B e. CH ) /\ ( A e. CH /\ C e. CH ) ) -> ( _|_ ` ( A i^i B ) ) = ( ( _|_ ` A ) vH ( _|_ ` B ) ) ) |
20 |
19
|
ineq1d |
|- ( ( ( A e. CH /\ B e. CH ) /\ ( A e. CH /\ C e. CH ) ) -> ( ( _|_ ` ( A i^i B ) ) i^i ( _|_ ` ( A i^i C ) ) ) = ( ( ( _|_ ` A ) vH ( _|_ ` B ) ) i^i ( _|_ ` ( A i^i C ) ) ) ) |
21 |
17 20
|
eqtrd |
|- ( ( ( A e. CH /\ B e. CH ) /\ ( A e. CH /\ C e. CH ) ) -> ( _|_ ` ( ( A i^i B ) vH ( A i^i C ) ) ) = ( ( ( _|_ ` A ) vH ( _|_ ` B ) ) i^i ( _|_ ` ( A i^i C ) ) ) ) |
22 |
21
|
3impdi |
|- ( ( A e. CH /\ B e. CH /\ C e. CH ) -> ( _|_ ` ( ( A i^i B ) vH ( A i^i C ) ) ) = ( ( ( _|_ ` A ) vH ( _|_ ` B ) ) i^i ( _|_ ` ( A i^i C ) ) ) ) |
23 |
22
|
ineq2d |
|- ( ( A e. CH /\ B e. CH /\ C e. CH ) -> ( ( A i^i ( B vH C ) ) i^i ( _|_ ` ( ( A i^i B ) vH ( A i^i C ) ) ) ) = ( ( A i^i ( B vH C ) ) i^i ( ( ( _|_ ` A ) vH ( _|_ ` B ) ) i^i ( _|_ ` ( A i^i C ) ) ) ) ) |
24 |
23
|
adantr |
|- ( ( ( A e. CH /\ B e. CH /\ C e. CH ) /\ ( B C_H A /\ B C_H C ) ) -> ( ( A i^i ( B vH C ) ) i^i ( _|_ ` ( ( A i^i B ) vH ( A i^i C ) ) ) ) = ( ( A i^i ( B vH C ) ) i^i ( ( ( _|_ ` A ) vH ( _|_ ` B ) ) i^i ( _|_ ` ( A i^i C ) ) ) ) ) |
25 |
|
in4 |
|- ( ( A i^i ( B vH C ) ) i^i ( ( ( _|_ ` A ) vH ( _|_ ` B ) ) i^i ( _|_ ` ( A i^i C ) ) ) ) = ( ( A i^i ( ( _|_ ` A ) vH ( _|_ ` B ) ) ) i^i ( ( B vH C ) i^i ( _|_ ` ( A i^i C ) ) ) ) |
26 |
|
cmcm2 |
|- ( ( A e. CH /\ B e. CH ) -> ( A C_H B <-> A C_H ( _|_ ` B ) ) ) |
27 |
|
cmcm |
|- ( ( A e. CH /\ B e. CH ) -> ( A C_H B <-> B C_H A ) ) |
28 |
|
choccl |
|- ( B e. CH -> ( _|_ ` B ) e. CH ) |
29 |
|
cmbr3 |
|- ( ( A e. CH /\ ( _|_ ` B ) e. CH ) -> ( A C_H ( _|_ ` B ) <-> ( A i^i ( ( _|_ ` A ) vH ( _|_ ` B ) ) ) = ( A i^i ( _|_ ` B ) ) ) ) |
30 |
28 29
|
sylan2 |
|- ( ( A e. CH /\ B e. CH ) -> ( A C_H ( _|_ ` B ) <-> ( A i^i ( ( _|_ ` A ) vH ( _|_ ` B ) ) ) = ( A i^i ( _|_ ` B ) ) ) ) |
31 |
26 27 30
|
3bitr3d |
|- ( ( A e. CH /\ B e. CH ) -> ( B C_H A <-> ( A i^i ( ( _|_ ` A ) vH ( _|_ ` B ) ) ) = ( A i^i ( _|_ ` B ) ) ) ) |
32 |
31
|
biimpa |
|- ( ( ( A e. CH /\ B e. CH ) /\ B C_H A ) -> ( A i^i ( ( _|_ ` A ) vH ( _|_ ` B ) ) ) = ( A i^i ( _|_ ` B ) ) ) |
33 |
|
incom |
|- ( A i^i ( _|_ ` B ) ) = ( ( _|_ ` B ) i^i A ) |
34 |
32 33
|
eqtrdi |
|- ( ( ( A e. CH /\ B e. CH ) /\ B C_H A ) -> ( A i^i ( ( _|_ ` A ) vH ( _|_ ` B ) ) ) = ( ( _|_ ` B ) i^i A ) ) |
35 |
34
|
3adantl3 |
|- ( ( ( A e. CH /\ B e. CH /\ C e. CH ) /\ B C_H A ) -> ( A i^i ( ( _|_ ` A ) vH ( _|_ ` B ) ) ) = ( ( _|_ ` B ) i^i A ) ) |
36 |
35
|
adantrr |
|- ( ( ( A e. CH /\ B e. CH /\ C e. CH ) /\ ( B C_H A /\ B C_H C ) ) -> ( A i^i ( ( _|_ ` A ) vH ( _|_ ` B ) ) ) = ( ( _|_ ` B ) i^i A ) ) |
37 |
36
|
ineq1d |
|- ( ( ( A e. CH /\ B e. CH /\ C e. CH ) /\ ( B C_H A /\ B C_H C ) ) -> ( ( A i^i ( ( _|_ ` A ) vH ( _|_ ` B ) ) ) i^i ( ( B vH C ) i^i ( _|_ ` ( A i^i C ) ) ) ) = ( ( ( _|_ ` B ) i^i A ) i^i ( ( B vH C ) i^i ( _|_ ` ( A i^i C ) ) ) ) ) |
38 |
25 37
|
syl5eq |
|- ( ( ( A e. CH /\ B e. CH /\ C e. CH ) /\ ( B C_H A /\ B C_H C ) ) -> ( ( A i^i ( B vH C ) ) i^i ( ( ( _|_ ` A ) vH ( _|_ ` B ) ) i^i ( _|_ ` ( A i^i C ) ) ) ) = ( ( ( _|_ ` B ) i^i A ) i^i ( ( B vH C ) i^i ( _|_ ` ( A i^i C ) ) ) ) ) |
39 |
24 38
|
eqtrd |
|- ( ( ( A e. CH /\ B e. CH /\ C e. CH ) /\ ( B C_H A /\ B C_H C ) ) -> ( ( A i^i ( B vH C ) ) i^i ( _|_ ` ( ( A i^i B ) vH ( A i^i C ) ) ) ) = ( ( ( _|_ ` B ) i^i A ) i^i ( ( B vH C ) i^i ( _|_ ` ( A i^i C ) ) ) ) ) |
40 |
|
in4 |
|- ( ( ( _|_ ` B ) i^i A ) i^i ( ( B vH C ) i^i ( _|_ ` ( A i^i C ) ) ) ) = ( ( ( _|_ ` B ) i^i ( B vH C ) ) i^i ( A i^i ( _|_ ` ( A i^i C ) ) ) ) |
41 |
39 40
|
eqtrdi |
|- ( ( ( A e. CH /\ B e. CH /\ C e. CH ) /\ ( B C_H A /\ B C_H C ) ) -> ( ( A i^i ( B vH C ) ) i^i ( _|_ ` ( ( A i^i B ) vH ( A i^i C ) ) ) ) = ( ( ( _|_ ` B ) i^i ( B vH C ) ) i^i ( A i^i ( _|_ ` ( A i^i C ) ) ) ) ) |
42 |
|
ococ |
|- ( B e. CH -> ( _|_ ` ( _|_ ` B ) ) = B ) |
43 |
42
|
oveq1d |
|- ( B e. CH -> ( ( _|_ ` ( _|_ ` B ) ) vH C ) = ( B vH C ) ) |
44 |
43
|
ineq2d |
|- ( B e. CH -> ( ( _|_ ` B ) i^i ( ( _|_ ` ( _|_ ` B ) ) vH C ) ) = ( ( _|_ ` B ) i^i ( B vH C ) ) ) |
45 |
44
|
3ad2ant2 |
|- ( ( A e. CH /\ B e. CH /\ C e. CH ) -> ( ( _|_ ` B ) i^i ( ( _|_ ` ( _|_ ` B ) ) vH C ) ) = ( ( _|_ ` B ) i^i ( B vH C ) ) ) |
46 |
45
|
adantr |
|- ( ( ( A e. CH /\ B e. CH /\ C e. CH ) /\ ( B C_H A /\ B C_H C ) ) -> ( ( _|_ ` B ) i^i ( ( _|_ ` ( _|_ ` B ) ) vH C ) ) = ( ( _|_ ` B ) i^i ( B vH C ) ) ) |
47 |
|
cmcm3 |
|- ( ( B e. CH /\ C e. CH ) -> ( B C_H C <-> ( _|_ ` B ) C_H C ) ) |
48 |
|
cmbr3 |
|- ( ( ( _|_ ` B ) e. CH /\ C e. CH ) -> ( ( _|_ ` B ) C_H C <-> ( ( _|_ ` B ) i^i ( ( _|_ ` ( _|_ ` B ) ) vH C ) ) = ( ( _|_ ` B ) i^i C ) ) ) |
49 |
28 48
|
sylan |
|- ( ( B e. CH /\ C e. CH ) -> ( ( _|_ ` B ) C_H C <-> ( ( _|_ ` B ) i^i ( ( _|_ ` ( _|_ ` B ) ) vH C ) ) = ( ( _|_ ` B ) i^i C ) ) ) |
50 |
47 49
|
bitrd |
|- ( ( B e. CH /\ C e. CH ) -> ( B C_H C <-> ( ( _|_ ` B ) i^i ( ( _|_ ` ( _|_ ` B ) ) vH C ) ) = ( ( _|_ ` B ) i^i C ) ) ) |
51 |
50
|
biimpa |
|- ( ( ( B e. CH /\ C e. CH ) /\ B C_H C ) -> ( ( _|_ ` B ) i^i ( ( _|_ ` ( _|_ ` B ) ) vH C ) ) = ( ( _|_ ` B ) i^i C ) ) |
52 |
51
|
3adantl1 |
|- ( ( ( A e. CH /\ B e. CH /\ C e. CH ) /\ B C_H C ) -> ( ( _|_ ` B ) i^i ( ( _|_ ` ( _|_ ` B ) ) vH C ) ) = ( ( _|_ ` B ) i^i C ) ) |
53 |
52
|
adantrl |
|- ( ( ( A e. CH /\ B e. CH /\ C e. CH ) /\ ( B C_H A /\ B C_H C ) ) -> ( ( _|_ ` B ) i^i ( ( _|_ ` ( _|_ ` B ) ) vH C ) ) = ( ( _|_ ` B ) i^i C ) ) |
54 |
46 53
|
eqtr3d |
|- ( ( ( A e. CH /\ B e. CH /\ C e. CH ) /\ ( B C_H A /\ B C_H C ) ) -> ( ( _|_ ` B ) i^i ( B vH C ) ) = ( ( _|_ ` B ) i^i C ) ) |
55 |
54
|
ineq1d |
|- ( ( ( A e. CH /\ B e. CH /\ C e. CH ) /\ ( B C_H A /\ B C_H C ) ) -> ( ( ( _|_ ` B ) i^i ( B vH C ) ) i^i ( A i^i ( _|_ ` ( A i^i C ) ) ) ) = ( ( ( _|_ ` B ) i^i C ) i^i ( A i^i ( _|_ ` ( A i^i C ) ) ) ) ) |
56 |
|
inass |
|- ( ( ( _|_ ` B ) i^i C ) i^i ( A i^i ( _|_ ` ( A i^i C ) ) ) ) = ( ( _|_ ` B ) i^i ( C i^i ( A i^i ( _|_ ` ( A i^i C ) ) ) ) ) |
57 |
|
in12 |
|- ( C i^i ( A i^i ( _|_ ` ( A i^i C ) ) ) ) = ( A i^i ( C i^i ( _|_ ` ( A i^i C ) ) ) ) |
58 |
|
inass |
|- ( ( A i^i C ) i^i ( _|_ ` ( A i^i C ) ) ) = ( A i^i ( C i^i ( _|_ ` ( A i^i C ) ) ) ) |
59 |
57 58
|
eqtr4i |
|- ( C i^i ( A i^i ( _|_ ` ( A i^i C ) ) ) ) = ( ( A i^i C ) i^i ( _|_ ` ( A i^i C ) ) ) |
60 |
|
chocin |
|- ( ( A i^i C ) e. CH -> ( ( A i^i C ) i^i ( _|_ ` ( A i^i C ) ) ) = 0H ) |
61 |
2 60
|
syl |
|- ( ( A e. CH /\ C e. CH ) -> ( ( A i^i C ) i^i ( _|_ ` ( A i^i C ) ) ) = 0H ) |
62 |
59 61
|
syl5eq |
|- ( ( A e. CH /\ C e. CH ) -> ( C i^i ( A i^i ( _|_ ` ( A i^i C ) ) ) ) = 0H ) |
63 |
62
|
ineq2d |
|- ( ( A e. CH /\ C e. CH ) -> ( ( _|_ ` B ) i^i ( C i^i ( A i^i ( _|_ ` ( A i^i C ) ) ) ) ) = ( ( _|_ ` B ) i^i 0H ) ) |
64 |
56 63
|
syl5eq |
|- ( ( A e. CH /\ C e. CH ) -> ( ( ( _|_ ` B ) i^i C ) i^i ( A i^i ( _|_ ` ( A i^i C ) ) ) ) = ( ( _|_ ` B ) i^i 0H ) ) |
65 |
64
|
3adant2 |
|- ( ( A e. CH /\ B e. CH /\ C e. CH ) -> ( ( ( _|_ ` B ) i^i C ) i^i ( A i^i ( _|_ ` ( A i^i C ) ) ) ) = ( ( _|_ ` B ) i^i 0H ) ) |
66 |
|
chm0 |
|- ( ( _|_ ` B ) e. CH -> ( ( _|_ ` B ) i^i 0H ) = 0H ) |
67 |
28 66
|
syl |
|- ( B e. CH -> ( ( _|_ ` B ) i^i 0H ) = 0H ) |
68 |
67
|
3ad2ant2 |
|- ( ( A e. CH /\ B e. CH /\ C e. CH ) -> ( ( _|_ ` B ) i^i 0H ) = 0H ) |
69 |
65 68
|
eqtrd |
|- ( ( A e. CH /\ B e. CH /\ C e. CH ) -> ( ( ( _|_ ` B ) i^i C ) i^i ( A i^i ( _|_ ` ( A i^i C ) ) ) ) = 0H ) |
70 |
69
|
adantr |
|- ( ( ( A e. CH /\ B e. CH /\ C e. CH ) /\ ( B C_H A /\ B C_H C ) ) -> ( ( ( _|_ ` B ) i^i C ) i^i ( A i^i ( _|_ ` ( A i^i C ) ) ) ) = 0H ) |
71 |
55 70
|
eqtrd |
|- ( ( ( A e. CH /\ B e. CH /\ C e. CH ) /\ ( B C_H A /\ B C_H C ) ) -> ( ( ( _|_ ` B ) i^i ( B vH C ) ) i^i ( A i^i ( _|_ ` ( A i^i C ) ) ) ) = 0H ) |
72 |
41 71
|
eqtrd |
|- ( ( ( A e. CH /\ B e. CH /\ C e. CH ) /\ ( B C_H A /\ B C_H C ) ) -> ( ( A i^i ( B vH C ) ) i^i ( _|_ ` ( ( A i^i B ) vH ( A i^i C ) ) ) ) = 0H ) |
73 |
|
pjoml |
|- ( ( ( ( ( A i^i B ) vH ( A i^i C ) ) e. CH /\ ( A i^i ( B vH C ) ) e. SH ) /\ ( ( ( A i^i B ) vH ( A i^i C ) ) C_ ( A i^i ( B vH C ) ) /\ ( ( A i^i ( B vH C ) ) i^i ( _|_ ` ( ( A i^i B ) vH ( A i^i C ) ) ) ) = 0H ) ) -> ( ( A i^i B ) vH ( A i^i C ) ) = ( A i^i ( B vH C ) ) ) |
74 |
13 15 72 73
|
syl12anc |
|- ( ( ( A e. CH /\ B e. CH /\ C e. CH ) /\ ( B C_H A /\ B C_H C ) ) -> ( ( A i^i B ) vH ( A i^i C ) ) = ( A i^i ( B vH C ) ) ) |
75 |
74
|
eqcomd |
|- ( ( ( A e. CH /\ B e. CH /\ C e. CH ) /\ ( B C_H A /\ B C_H C ) ) -> ( A i^i ( B vH C ) ) = ( ( A i^i B ) vH ( A i^i C ) ) ) |