Step |
Hyp |
Ref |
Expression |
1 |
|
chirred.1 |
|- A e. CH |
2 |
|
atelch |
|- ( p e. HAtoms -> p e. CH ) |
3 |
|
chjcom |
|- ( ( p e. CH /\ q e. CH ) -> ( p vH q ) = ( q vH p ) ) |
4 |
2 3
|
sylan |
|- ( ( p e. HAtoms /\ q e. CH ) -> ( p vH q ) = ( q vH p ) ) |
5 |
4
|
ad2ant2r |
|- ( ( ( p e. HAtoms /\ p C_ A ) /\ ( q e. CH /\ q C_ ( _|_ ` A ) ) ) -> ( p vH q ) = ( q vH p ) ) |
6 |
5
|
adantr |
|- ( ( ( ( p e. HAtoms /\ p C_ A ) /\ ( q e. CH /\ q C_ ( _|_ ` A ) ) ) /\ ( ( r e. HAtoms /\ r C_ A ) /\ r C_ ( p vH q ) ) ) -> ( p vH q ) = ( q vH p ) ) |
7 |
6
|
ineq2d |
|- ( ( ( ( p e. HAtoms /\ p C_ A ) /\ ( q e. CH /\ q C_ ( _|_ ` A ) ) ) /\ ( ( r e. HAtoms /\ r C_ A ) /\ r C_ ( p vH q ) ) ) -> ( ( _|_ ` r ) i^i ( p vH q ) ) = ( ( _|_ ` r ) i^i ( q vH p ) ) ) |
8 |
|
atelch |
|- ( r e. HAtoms -> r e. CH ) |
9 |
|
choccl |
|- ( r e. CH -> ( _|_ ` r ) e. CH ) |
10 |
8 9
|
syl |
|- ( r e. HAtoms -> ( _|_ ` r ) e. CH ) |
11 |
|
id |
|- ( q e. CH -> q e. CH ) |
12 |
10 11 2
|
3anim123i |
|- ( ( r e. HAtoms /\ q e. CH /\ p e. HAtoms ) -> ( ( _|_ ` r ) e. CH /\ q e. CH /\ p e. CH ) ) |
13 |
12
|
3com13 |
|- ( ( p e. HAtoms /\ q e. CH /\ r e. HAtoms ) -> ( ( _|_ ` r ) e. CH /\ q e. CH /\ p e. CH ) ) |
14 |
13
|
3expa |
|- ( ( ( p e. HAtoms /\ q e. CH ) /\ r e. HAtoms ) -> ( ( _|_ ` r ) e. CH /\ q e. CH /\ p e. CH ) ) |
15 |
14
|
adantllr |
|- ( ( ( ( p e. HAtoms /\ p C_ A ) /\ q e. CH ) /\ r e. HAtoms ) -> ( ( _|_ ` r ) e. CH /\ q e. CH /\ p e. CH ) ) |
16 |
15
|
adantlrr |
|- ( ( ( ( p e. HAtoms /\ p C_ A ) /\ ( q e. CH /\ q C_ ( _|_ ` A ) ) ) /\ r e. HAtoms ) -> ( ( _|_ ` r ) e. CH /\ q e. CH /\ p e. CH ) ) |
17 |
16
|
adantrr |
|- ( ( ( ( p e. HAtoms /\ p C_ A ) /\ ( q e. CH /\ q C_ ( _|_ ` A ) ) ) /\ ( r e. HAtoms /\ r C_ A ) ) -> ( ( _|_ ` r ) e. CH /\ q e. CH /\ p e. CH ) ) |
18 |
17
|
adantrr |
|- ( ( ( ( p e. HAtoms /\ p C_ A ) /\ ( q e. CH /\ q C_ ( _|_ ` A ) ) ) /\ ( ( r e. HAtoms /\ r C_ A ) /\ r C_ ( p vH q ) ) ) -> ( ( _|_ ` r ) e. CH /\ q e. CH /\ p e. CH ) ) |
19 |
|
simpll |
|- ( ( ( q e. CH /\ q C_ ( _|_ ` A ) ) /\ ( r e. HAtoms /\ r C_ A ) ) -> q e. CH ) |
20 |
10
|
ad2antrl |
|- ( ( ( q e. CH /\ q C_ ( _|_ ` A ) ) /\ ( r e. HAtoms /\ r C_ A ) ) -> ( _|_ ` r ) e. CH ) |
21 |
|
chsscon3 |
|- ( ( r e. CH /\ A e. CH ) -> ( r C_ A <-> ( _|_ ` A ) C_ ( _|_ ` r ) ) ) |
22 |
8 1 21
|
sylancl |
|- ( r e. HAtoms -> ( r C_ A <-> ( _|_ ` A ) C_ ( _|_ ` r ) ) ) |
23 |
22
|
biimpa |
|- ( ( r e. HAtoms /\ r C_ A ) -> ( _|_ ` A ) C_ ( _|_ ` r ) ) |
24 |
|
sstr |
|- ( ( q C_ ( _|_ ` A ) /\ ( _|_ ` A ) C_ ( _|_ ` r ) ) -> q C_ ( _|_ ` r ) ) |
25 |
23 24
|
sylan2 |
|- ( ( q C_ ( _|_ ` A ) /\ ( r e. HAtoms /\ r C_ A ) ) -> q C_ ( _|_ ` r ) ) |
26 |
25
|
adantll |
|- ( ( ( q e. CH /\ q C_ ( _|_ ` A ) ) /\ ( r e. HAtoms /\ r C_ A ) ) -> q C_ ( _|_ ` r ) ) |
27 |
|
lecm |
|- ( ( q e. CH /\ ( _|_ ` r ) e. CH /\ q C_ ( _|_ ` r ) ) -> q C_H ( _|_ ` r ) ) |
28 |
19 20 26 27
|
syl3anc |
|- ( ( ( q e. CH /\ q C_ ( _|_ ` A ) ) /\ ( r e. HAtoms /\ r C_ A ) ) -> q C_H ( _|_ ` r ) ) |
29 |
28
|
ad2ant2lr |
|- ( ( ( ( p e. HAtoms /\ p C_ A ) /\ ( q e. CH /\ q C_ ( _|_ ` A ) ) ) /\ ( ( r e. HAtoms /\ r C_ A ) /\ r C_ ( p vH q ) ) ) -> q C_H ( _|_ ` r ) ) |
30 |
|
chsscon3 |
|- ( ( p e. CH /\ A e. CH ) -> ( p C_ A <-> ( _|_ ` A ) C_ ( _|_ ` p ) ) ) |
31 |
1 30
|
mpan2 |
|- ( p e. CH -> ( p C_ A <-> ( _|_ ` A ) C_ ( _|_ ` p ) ) ) |
32 |
31
|
biimpa |
|- ( ( p e. CH /\ p C_ A ) -> ( _|_ ` A ) C_ ( _|_ ` p ) ) |
33 |
|
sstr |
|- ( ( q C_ ( _|_ ` A ) /\ ( _|_ ` A ) C_ ( _|_ ` p ) ) -> q C_ ( _|_ ` p ) ) |
34 |
32 33
|
sylan2 |
|- ( ( q C_ ( _|_ ` A ) /\ ( p e. CH /\ p C_ A ) ) -> q C_ ( _|_ ` p ) ) |
35 |
34
|
an12s |
|- ( ( p e. CH /\ ( q C_ ( _|_ ` A ) /\ p C_ A ) ) -> q C_ ( _|_ ` p ) ) |
36 |
35
|
ancom2s |
|- ( ( p e. CH /\ ( p C_ A /\ q C_ ( _|_ ` A ) ) ) -> q C_ ( _|_ ` p ) ) |
37 |
36
|
adantll |
|- ( ( ( q e. CH /\ p e. CH ) /\ ( p C_ A /\ q C_ ( _|_ ` A ) ) ) -> q C_ ( _|_ ` p ) ) |
38 |
|
choccl |
|- ( p e. CH -> ( _|_ ` p ) e. CH ) |
39 |
|
lecm |
|- ( ( q e. CH /\ ( _|_ ` p ) e. CH /\ q C_ ( _|_ ` p ) ) -> q C_H ( _|_ ` p ) ) |
40 |
38 39
|
syl3an2 |
|- ( ( q e. CH /\ p e. CH /\ q C_ ( _|_ ` p ) ) -> q C_H ( _|_ ` p ) ) |
41 |
40
|
3expia |
|- ( ( q e. CH /\ p e. CH ) -> ( q C_ ( _|_ ` p ) -> q C_H ( _|_ ` p ) ) ) |
42 |
|
cmcm2 |
|- ( ( q e. CH /\ p e. CH ) -> ( q C_H p <-> q C_H ( _|_ ` p ) ) ) |
43 |
41 42
|
sylibrd |
|- ( ( q e. CH /\ p e. CH ) -> ( q C_ ( _|_ ` p ) -> q C_H p ) ) |
44 |
43
|
adantr |
|- ( ( ( q e. CH /\ p e. CH ) /\ ( p C_ A /\ q C_ ( _|_ ` A ) ) ) -> ( q C_ ( _|_ ` p ) -> q C_H p ) ) |
45 |
37 44
|
mpd |
|- ( ( ( q e. CH /\ p e. CH ) /\ ( p C_ A /\ q C_ ( _|_ ` A ) ) ) -> q C_H p ) |
46 |
2 45
|
sylanl2 |
|- ( ( ( q e. CH /\ p e. HAtoms ) /\ ( p C_ A /\ q C_ ( _|_ ` A ) ) ) -> q C_H p ) |
47 |
46
|
ancom1s |
|- ( ( ( p e. HAtoms /\ q e. CH ) /\ ( p C_ A /\ q C_ ( _|_ ` A ) ) ) -> q C_H p ) |
48 |
47
|
an4s |
|- ( ( ( p e. HAtoms /\ p C_ A ) /\ ( q e. CH /\ q C_ ( _|_ ` A ) ) ) -> q C_H p ) |
49 |
48
|
adantr |
|- ( ( ( ( p e. HAtoms /\ p C_ A ) /\ ( q e. CH /\ q C_ ( _|_ ` A ) ) ) /\ ( ( r e. HAtoms /\ r C_ A ) /\ r C_ ( p vH q ) ) ) -> q C_H p ) |
50 |
|
fh2 |
|- ( ( ( ( _|_ ` r ) e. CH /\ q e. CH /\ p e. CH ) /\ ( q C_H ( _|_ ` r ) /\ q C_H p ) ) -> ( ( _|_ ` r ) i^i ( q vH p ) ) = ( ( ( _|_ ` r ) i^i q ) vH ( ( _|_ ` r ) i^i p ) ) ) |
51 |
18 29 49 50
|
syl12anc |
|- ( ( ( ( p e. HAtoms /\ p C_ A ) /\ ( q e. CH /\ q C_ ( _|_ ` A ) ) ) /\ ( ( r e. HAtoms /\ r C_ A ) /\ r C_ ( p vH q ) ) ) -> ( ( _|_ ` r ) i^i ( q vH p ) ) = ( ( ( _|_ ` r ) i^i q ) vH ( ( _|_ ` r ) i^i p ) ) ) |
52 |
|
sseqin2 |
|- ( q C_ ( _|_ ` r ) <-> ( ( _|_ ` r ) i^i q ) = q ) |
53 |
26 52
|
sylib |
|- ( ( ( q e. CH /\ q C_ ( _|_ ` A ) ) /\ ( r e. HAtoms /\ r C_ A ) ) -> ( ( _|_ ` r ) i^i q ) = q ) |
54 |
53
|
ad2ant2lr |
|- ( ( ( ( p e. HAtoms /\ p C_ A ) /\ ( q e. CH /\ q C_ ( _|_ ` A ) ) ) /\ ( ( r e. HAtoms /\ r C_ A ) /\ r C_ ( p vH q ) ) ) -> ( ( _|_ ` r ) i^i q ) = q ) |
55 |
|
incom |
|- ( ( _|_ ` r ) i^i p ) = ( p i^i ( _|_ ` r ) ) |
56 |
1
|
chirredlem1 |
|- ( ( ( p e. HAtoms /\ ( q e. CH /\ q C_ ( _|_ ` A ) ) ) /\ ( ( r e. HAtoms /\ r C_ A ) /\ r C_ ( p vH q ) ) ) -> ( p i^i ( _|_ ` r ) ) = 0H ) |
57 |
56
|
adantllr |
|- ( ( ( ( p e. HAtoms /\ p C_ A ) /\ ( q e. CH /\ q C_ ( _|_ ` A ) ) ) /\ ( ( r e. HAtoms /\ r C_ A ) /\ r C_ ( p vH q ) ) ) -> ( p i^i ( _|_ ` r ) ) = 0H ) |
58 |
55 57
|
syl5eq |
|- ( ( ( ( p e. HAtoms /\ p C_ A ) /\ ( q e. CH /\ q C_ ( _|_ ` A ) ) ) /\ ( ( r e. HAtoms /\ r C_ A ) /\ r C_ ( p vH q ) ) ) -> ( ( _|_ ` r ) i^i p ) = 0H ) |
59 |
54 58
|
oveq12d |
|- ( ( ( ( p e. HAtoms /\ p C_ A ) /\ ( q e. CH /\ q C_ ( _|_ ` A ) ) ) /\ ( ( r e. HAtoms /\ r C_ A ) /\ r C_ ( p vH q ) ) ) -> ( ( ( _|_ ` r ) i^i q ) vH ( ( _|_ ` r ) i^i p ) ) = ( q vH 0H ) ) |
60 |
|
chj0 |
|- ( q e. CH -> ( q vH 0H ) = q ) |
61 |
60
|
adantr |
|- ( ( q e. CH /\ q C_ ( _|_ ` A ) ) -> ( q vH 0H ) = q ) |
62 |
61
|
ad2antlr |
|- ( ( ( ( p e. HAtoms /\ p C_ A ) /\ ( q e. CH /\ q C_ ( _|_ ` A ) ) ) /\ ( ( r e. HAtoms /\ r C_ A ) /\ r C_ ( p vH q ) ) ) -> ( q vH 0H ) = q ) |
63 |
59 62
|
eqtrd |
|- ( ( ( ( p e. HAtoms /\ p C_ A ) /\ ( q e. CH /\ q C_ ( _|_ ` A ) ) ) /\ ( ( r e. HAtoms /\ r C_ A ) /\ r C_ ( p vH q ) ) ) -> ( ( ( _|_ ` r ) i^i q ) vH ( ( _|_ ` r ) i^i p ) ) = q ) |
64 |
7 51 63
|
3eqtrd |
|- ( ( ( ( p e. HAtoms /\ p C_ A ) /\ ( q e. CH /\ q C_ ( _|_ ` A ) ) ) /\ ( ( r e. HAtoms /\ r C_ A ) /\ r C_ ( p vH q ) ) ) -> ( ( _|_ ` r ) i^i ( p vH q ) ) = q ) |