Step |
Hyp |
Ref |
Expression |
1 |
|
ocss |
|- ( A C_ ~H -> ( _|_ ` A ) C_ ~H ) |
2 |
|
sshjval |
|- ( ( A C_ ~H /\ ( _|_ ` A ) C_ ~H ) -> ( A vH ( _|_ ` A ) ) = ( _|_ ` ( _|_ ` ( A u. ( _|_ ` A ) ) ) ) ) |
3 |
1 2
|
mpdan |
|- ( A C_ ~H -> ( A vH ( _|_ ` A ) ) = ( _|_ ` ( _|_ ` ( A u. ( _|_ ` A ) ) ) ) ) |
4 |
|
ssun1 |
|- A C_ ( A u. ( _|_ ` A ) ) |
5 |
1
|
ancli |
|- ( A C_ ~H -> ( A C_ ~H /\ ( _|_ ` A ) C_ ~H ) ) |
6 |
|
unss |
|- ( ( A C_ ~H /\ ( _|_ ` A ) C_ ~H ) <-> ( A u. ( _|_ ` A ) ) C_ ~H ) |
7 |
5 6
|
sylib |
|- ( A C_ ~H -> ( A u. ( _|_ ` A ) ) C_ ~H ) |
8 |
|
occon |
|- ( ( A C_ ~H /\ ( A u. ( _|_ ` A ) ) C_ ~H ) -> ( A C_ ( A u. ( _|_ ` A ) ) -> ( _|_ ` ( A u. ( _|_ ` A ) ) ) C_ ( _|_ ` A ) ) ) |
9 |
7 8
|
mpdan |
|- ( A C_ ~H -> ( A C_ ( A u. ( _|_ ` A ) ) -> ( _|_ ` ( A u. ( _|_ ` A ) ) ) C_ ( _|_ ` A ) ) ) |
10 |
4 9
|
mpi |
|- ( A C_ ~H -> ( _|_ ` ( A u. ( _|_ ` A ) ) ) C_ ( _|_ ` A ) ) |
11 |
|
ssun2 |
|- ( _|_ ` A ) C_ ( A u. ( _|_ ` A ) ) |
12 |
|
occon |
|- ( ( ( _|_ ` A ) C_ ~H /\ ( A u. ( _|_ ` A ) ) C_ ~H ) -> ( ( _|_ ` A ) C_ ( A u. ( _|_ ` A ) ) -> ( _|_ ` ( A u. ( _|_ ` A ) ) ) C_ ( _|_ ` ( _|_ ` A ) ) ) ) |
13 |
1 7 12
|
syl2anc |
|- ( A C_ ~H -> ( ( _|_ ` A ) C_ ( A u. ( _|_ ` A ) ) -> ( _|_ ` ( A u. ( _|_ ` A ) ) ) C_ ( _|_ ` ( _|_ ` A ) ) ) ) |
14 |
11 13
|
mpi |
|- ( A C_ ~H -> ( _|_ ` ( A u. ( _|_ ` A ) ) ) C_ ( _|_ ` ( _|_ ` A ) ) ) |
15 |
10 14
|
ssind |
|- ( A C_ ~H -> ( _|_ ` ( A u. ( _|_ ` A ) ) ) C_ ( ( _|_ ` A ) i^i ( _|_ ` ( _|_ ` A ) ) ) ) |
16 |
|
ocsh |
|- ( A C_ ~H -> ( _|_ ` A ) e. SH ) |
17 |
|
ocin |
|- ( ( _|_ ` A ) e. SH -> ( ( _|_ ` A ) i^i ( _|_ ` ( _|_ ` A ) ) ) = 0H ) |
18 |
16 17
|
syl |
|- ( A C_ ~H -> ( ( _|_ ` A ) i^i ( _|_ ` ( _|_ ` A ) ) ) = 0H ) |
19 |
15 18
|
sseqtrd |
|- ( A C_ ~H -> ( _|_ ` ( A u. ( _|_ ` A ) ) ) C_ 0H ) |
20 |
|
ocsh |
|- ( ( A u. ( _|_ ` A ) ) C_ ~H -> ( _|_ ` ( A u. ( _|_ ` A ) ) ) e. SH ) |
21 |
|
sh0le |
|- ( ( _|_ ` ( A u. ( _|_ ` A ) ) ) e. SH -> 0H C_ ( _|_ ` ( A u. ( _|_ ` A ) ) ) ) |
22 |
7 20 21
|
3syl |
|- ( A C_ ~H -> 0H C_ ( _|_ ` ( A u. ( _|_ ` A ) ) ) ) |
23 |
19 22
|
eqssd |
|- ( A C_ ~H -> ( _|_ ` ( A u. ( _|_ ` A ) ) ) = 0H ) |
24 |
23
|
fveq2d |
|- ( A C_ ~H -> ( _|_ ` ( _|_ ` ( A u. ( _|_ ` A ) ) ) ) = ( _|_ ` 0H ) ) |
25 |
|
choc0 |
|- ( _|_ ` 0H ) = ~H |
26 |
24 25
|
eqtrdi |
|- ( A C_ ~H -> ( _|_ ` ( _|_ ` ( A u. ( _|_ ` A ) ) ) ) = ~H ) |
27 |
3 26
|
eqtrd |
|- ( A C_ ~H -> ( A vH ( _|_ ` A ) ) = ~H ) |