Step |
Hyp |
Ref |
Expression |
1 |
|
pjclem1.1 |
|- G e. CH |
2 |
|
pjclem1.2 |
|- H e. CH |
3 |
|
df-iop |
|- Iop = ( projh ` ~H ) |
4 |
3
|
coeq2i |
|- ( ( projh ` G ) o. Iop ) = ( ( projh ` G ) o. ( projh ` ~H ) ) |
5 |
1
|
pjfi |
|- ( projh ` G ) : ~H --> ~H |
6 |
5
|
hoid1i |
|- ( ( projh ` G ) o. Iop ) = ( projh ` G ) |
7 |
4 6
|
eqtr3i |
|- ( ( projh ` G ) o. ( projh ` ~H ) ) = ( projh ` G ) |
8 |
5
|
hoid1ri |
|- ( Iop o. ( projh ` G ) ) = ( projh ` G ) |
9 |
3
|
coeq1i |
|- ( Iop o. ( projh ` G ) ) = ( ( projh ` ~H ) o. ( projh ` G ) ) |
10 |
7 8 9
|
3eqtr2i |
|- ( ( projh ` G ) o. ( projh ` ~H ) ) = ( ( projh ` ~H ) o. ( projh ` G ) ) |
11 |
10
|
oveq1i |
|- ( ( ( projh ` G ) o. ( projh ` ~H ) ) -op ( ( projh ` G ) o. ( projh ` H ) ) ) = ( ( ( projh ` ~H ) o. ( projh ` G ) ) -op ( ( projh ` G ) o. ( projh ` H ) ) ) |
12 |
|
oveq2 |
|- ( ( ( projh ` G ) o. ( projh ` H ) ) = ( ( projh ` H ) o. ( projh ` G ) ) -> ( ( ( projh ` ~H ) o. ( projh ` G ) ) -op ( ( projh ` G ) o. ( projh ` H ) ) ) = ( ( ( projh ` ~H ) o. ( projh ` G ) ) -op ( ( projh ` H ) o. ( projh ` G ) ) ) ) |
13 |
11 12
|
syl5eq |
|- ( ( ( projh ` G ) o. ( projh ` H ) ) = ( ( projh ` H ) o. ( projh ` G ) ) -> ( ( ( projh ` G ) o. ( projh ` ~H ) ) -op ( ( projh ` G ) o. ( projh ` H ) ) ) = ( ( ( projh ` ~H ) o. ( projh ` G ) ) -op ( ( projh ` H ) o. ( projh ` G ) ) ) ) |
14 |
|
helch |
|- ~H e. CH |
15 |
14
|
pjfi |
|- ( projh ` ~H ) : ~H --> ~H |
16 |
2
|
pjfi |
|- ( projh ` H ) : ~H --> ~H |
17 |
1 15 16
|
pjddii |
|- ( ( projh ` G ) o. ( ( projh ` ~H ) -op ( projh ` H ) ) ) = ( ( ( projh ` G ) o. ( projh ` ~H ) ) -op ( ( projh ` G ) o. ( projh ` H ) ) ) |
18 |
15 16 5
|
hocsubdiri |
|- ( ( ( projh ` ~H ) -op ( projh ` H ) ) o. ( projh ` G ) ) = ( ( ( projh ` ~H ) o. ( projh ` G ) ) -op ( ( projh ` H ) o. ( projh ` G ) ) ) |
19 |
13 17 18
|
3eqtr4g |
|- ( ( ( projh ` G ) o. ( projh ` H ) ) = ( ( projh ` H ) o. ( projh ` G ) ) -> ( ( projh ` G ) o. ( ( projh ` ~H ) -op ( projh ` H ) ) ) = ( ( ( projh ` ~H ) -op ( projh ` H ) ) o. ( projh ` G ) ) ) |
20 |
2
|
pjoci |
|- ( ( projh ` ~H ) -op ( projh ` H ) ) = ( projh ` ( _|_ ` H ) ) |
21 |
20
|
coeq2i |
|- ( ( projh ` G ) o. ( ( projh ` ~H ) -op ( projh ` H ) ) ) = ( ( projh ` G ) o. ( projh ` ( _|_ ` H ) ) ) |
22 |
20
|
coeq1i |
|- ( ( ( projh ` ~H ) -op ( projh ` H ) ) o. ( projh ` G ) ) = ( ( projh ` ( _|_ ` H ) ) o. ( projh ` G ) ) |
23 |
19 21 22
|
3eqtr3g |
|- ( ( ( projh ` G ) o. ( projh ` H ) ) = ( ( projh ` H ) o. ( projh ` G ) ) -> ( ( projh ` G ) o. ( projh ` ( _|_ ` H ) ) ) = ( ( projh ` ( _|_ ` H ) ) o. ( projh ` G ) ) ) |