Step |
Hyp |
Ref |
Expression |
1 |
|
atoml.1 |
|- A e. CH |
2 |
|
atelch |
|- ( B e. HAtoms -> B e. CH ) |
3 |
1
|
choccli |
|- ( _|_ ` A ) e. CH |
4 |
|
chincl |
|- ( ( ( _|_ ` A ) e. CH /\ B e. CH ) -> ( ( _|_ ` A ) i^i B ) e. CH ) |
5 |
3 4
|
mpan |
|- ( B e. CH -> ( ( _|_ ` A ) i^i B ) e. CH ) |
6 |
|
chj0 |
|- ( ( ( _|_ ` A ) i^i B ) e. CH -> ( ( ( _|_ ` A ) i^i B ) vH 0H ) = ( ( _|_ ` A ) i^i B ) ) |
7 |
5 6
|
syl |
|- ( B e. CH -> ( ( ( _|_ ` A ) i^i B ) vH 0H ) = ( ( _|_ ` A ) i^i B ) ) |
8 |
|
incom |
|- ( ( _|_ ` A ) i^i B ) = ( B i^i ( _|_ ` A ) ) |
9 |
7 8
|
eqtrdi |
|- ( B e. CH -> ( ( ( _|_ ` A ) i^i B ) vH 0H ) = ( B i^i ( _|_ ` A ) ) ) |
10 |
|
h0elch |
|- 0H e. CH |
11 |
|
chjcom |
|- ( ( ( ( _|_ ` A ) i^i B ) e. CH /\ 0H e. CH ) -> ( ( ( _|_ ` A ) i^i B ) vH 0H ) = ( 0H vH ( ( _|_ ` A ) i^i B ) ) ) |
12 |
5 10 11
|
sylancl |
|- ( B e. CH -> ( ( ( _|_ ` A ) i^i B ) vH 0H ) = ( 0H vH ( ( _|_ ` A ) i^i B ) ) ) |
13 |
9 12
|
eqtr3d |
|- ( B e. CH -> ( B i^i ( _|_ ` A ) ) = ( 0H vH ( ( _|_ ` A ) i^i B ) ) ) |
14 |
|
incom |
|- ( ( _|_ ` A ) i^i A ) = ( A i^i ( _|_ ` A ) ) |
15 |
1
|
chocini |
|- ( A i^i ( _|_ ` A ) ) = 0H |
16 |
14 15
|
eqtri |
|- ( ( _|_ ` A ) i^i A ) = 0H |
17 |
16
|
oveq1i |
|- ( ( ( _|_ ` A ) i^i A ) vH ( ( _|_ ` A ) i^i B ) ) = ( 0H vH ( ( _|_ ` A ) i^i B ) ) |
18 |
13 17
|
eqtr4di |
|- ( B e. CH -> ( B i^i ( _|_ ` A ) ) = ( ( ( _|_ ` A ) i^i A ) vH ( ( _|_ ` A ) i^i B ) ) ) |
19 |
18
|
adantr |
|- ( ( B e. CH /\ A C_H B ) -> ( B i^i ( _|_ ` A ) ) = ( ( ( _|_ ` A ) i^i A ) vH ( ( _|_ ` A ) i^i B ) ) ) |
20 |
1
|
cmidi |
|- A C_H A |
21 |
1 1 20
|
cmcm2ii |
|- A C_H ( _|_ ` A ) |
22 |
|
fh2 |
|- ( ( ( ( _|_ ` A ) e. CH /\ A e. CH /\ B e. CH ) /\ ( A C_H ( _|_ ` A ) /\ A C_H B ) ) -> ( ( _|_ ` A ) i^i ( A vH B ) ) = ( ( ( _|_ ` A ) i^i A ) vH ( ( _|_ ` A ) i^i B ) ) ) |
23 |
21 22
|
mpanr1 |
|- ( ( ( ( _|_ ` A ) e. CH /\ A e. CH /\ B e. CH ) /\ A C_H B ) -> ( ( _|_ ` A ) i^i ( A vH B ) ) = ( ( ( _|_ ` A ) i^i A ) vH ( ( _|_ ` A ) i^i B ) ) ) |
24 |
1 23
|
mp3anl2 |
|- ( ( ( ( _|_ ` A ) e. CH /\ B e. CH ) /\ A C_H B ) -> ( ( _|_ ` A ) i^i ( A vH B ) ) = ( ( ( _|_ ` A ) i^i A ) vH ( ( _|_ ` A ) i^i B ) ) ) |
25 |
3 24
|
mpanl1 |
|- ( ( B e. CH /\ A C_H B ) -> ( ( _|_ ` A ) i^i ( A vH B ) ) = ( ( ( _|_ ` A ) i^i A ) vH ( ( _|_ ` A ) i^i B ) ) ) |
26 |
19 25
|
eqtr4d |
|- ( ( B e. CH /\ A C_H B ) -> ( B i^i ( _|_ ` A ) ) = ( ( _|_ ` A ) i^i ( A vH B ) ) ) |
27 |
2 26
|
sylan |
|- ( ( B e. HAtoms /\ A C_H B ) -> ( B i^i ( _|_ ` A ) ) = ( ( _|_ ` A ) i^i ( A vH B ) ) ) |
28 |
|
incom |
|- ( ( _|_ ` A ) i^i ( A vH B ) ) = ( ( A vH B ) i^i ( _|_ ` A ) ) |
29 |
27 28
|
eqtrdi |
|- ( ( B e. HAtoms /\ A C_H B ) -> ( B i^i ( _|_ ` A ) ) = ( ( A vH B ) i^i ( _|_ ` A ) ) ) |
30 |
29
|
adantr |
|- ( ( ( B e. HAtoms /\ A C_H B ) /\ -. B C_ A ) -> ( B i^i ( _|_ ` A ) ) = ( ( A vH B ) i^i ( _|_ ` A ) ) ) |
31 |
1
|
atoml2i |
|- ( ( B e. HAtoms /\ -. B C_ A ) -> ( ( A vH B ) i^i ( _|_ ` A ) ) e. HAtoms ) |
32 |
31
|
adantlr |
|- ( ( ( B e. HAtoms /\ A C_H B ) /\ -. B C_ A ) -> ( ( A vH B ) i^i ( _|_ ` A ) ) e. HAtoms ) |
33 |
30 32
|
eqeltrd |
|- ( ( ( B e. HAtoms /\ A C_H B ) /\ -. B C_ A ) -> ( B i^i ( _|_ ` A ) ) e. HAtoms ) |
34 |
|
atssma |
|- ( ( B e. HAtoms /\ ( _|_ ` A ) e. CH ) -> ( B C_ ( _|_ ` A ) <-> ( B i^i ( _|_ ` A ) ) e. HAtoms ) ) |
35 |
3 34
|
mpan2 |
|- ( B e. HAtoms -> ( B C_ ( _|_ ` A ) <-> ( B i^i ( _|_ ` A ) ) e. HAtoms ) ) |
36 |
35
|
ad2antrr |
|- ( ( ( B e. HAtoms /\ A C_H B ) /\ -. B C_ A ) -> ( B C_ ( _|_ ` A ) <-> ( B i^i ( _|_ ` A ) ) e. HAtoms ) ) |
37 |
33 36
|
mpbird |
|- ( ( ( B e. HAtoms /\ A C_H B ) /\ -. B C_ A ) -> B C_ ( _|_ ` A ) ) |
38 |
37
|
ex |
|- ( ( B e. HAtoms /\ A C_H B ) -> ( -. B C_ A -> B C_ ( _|_ ` A ) ) ) |
39 |
38
|
orrd |
|- ( ( B e. HAtoms /\ A C_H B ) -> ( B C_ A \/ B C_ ( _|_ ` A ) ) ) |