Step |
Hyp |
Ref |
Expression |
1 |
|
goeq.1 |
⊢ 𝐴 ∈ Cℋ |
2 |
|
goeq.2 |
⊢ 𝐵 ∈ Cℋ |
3 |
|
goeq.3 |
⊢ 𝐶 ∈ Cℋ |
4 |
|
goeq.4 |
⊢ 𝐹 = ( ( ⊥ ‘ 𝐴 ) ∨ℋ ( 𝐴 ∩ 𝐵 ) ) |
5 |
|
goeq.5 |
⊢ 𝐺 = ( ( ⊥ ‘ 𝐵 ) ∨ℋ ( 𝐵 ∩ 𝐶 ) ) |
6 |
|
goeq.6 |
⊢ 𝐻 = ( ( ⊥ ‘ 𝐶 ) ∨ℋ ( 𝐶 ∩ 𝐴 ) ) |
7 |
|
goeq.7 |
⊢ 𝐷 = ( ( ⊥ ‘ 𝐵 ) ∨ℋ ( 𝐵 ∩ 𝐴 ) ) |
8 |
1
|
choccli |
⊢ ( ⊥ ‘ 𝐴 ) ∈ Cℋ |
9 |
1 2
|
chincli |
⊢ ( 𝐴 ∩ 𝐵 ) ∈ Cℋ |
10 |
8 9
|
chjcli |
⊢ ( ( ⊥ ‘ 𝐴 ) ∨ℋ ( 𝐴 ∩ 𝐵 ) ) ∈ Cℋ |
11 |
4 10
|
eqeltri |
⊢ 𝐹 ∈ Cℋ |
12 |
2
|
choccli |
⊢ ( ⊥ ‘ 𝐵 ) ∈ Cℋ |
13 |
2 3
|
chincli |
⊢ ( 𝐵 ∩ 𝐶 ) ∈ Cℋ |
14 |
12 13
|
chjcli |
⊢ ( ( ⊥ ‘ 𝐵 ) ∨ℋ ( 𝐵 ∩ 𝐶 ) ) ∈ Cℋ |
15 |
5 14
|
eqeltri |
⊢ 𝐺 ∈ Cℋ |
16 |
11 15
|
chincli |
⊢ ( 𝐹 ∩ 𝐺 ) ∈ Cℋ |
17 |
3
|
choccli |
⊢ ( ⊥ ‘ 𝐶 ) ∈ Cℋ |
18 |
3 1
|
chincli |
⊢ ( 𝐶 ∩ 𝐴 ) ∈ Cℋ |
19 |
17 18
|
chjcli |
⊢ ( ( ⊥ ‘ 𝐶 ) ∨ℋ ( 𝐶 ∩ 𝐴 ) ) ∈ Cℋ |
20 |
6 19
|
eqeltri |
⊢ 𝐻 ∈ Cℋ |
21 |
16 20
|
chincli |
⊢ ( ( 𝐹 ∩ 𝐺 ) ∩ 𝐻 ) ∈ Cℋ |
22 |
2 1
|
chincli |
⊢ ( 𝐵 ∩ 𝐴 ) ∈ Cℋ |
23 |
12 22
|
chjcli |
⊢ ( ( ⊥ ‘ 𝐵 ) ∨ℋ ( 𝐵 ∩ 𝐴 ) ) ∈ Cℋ |
24 |
7 23
|
eqeltri |
⊢ 𝐷 ∈ Cℋ |
25 |
21 24
|
stri |
⊢ ( ∀ 𝑓 ∈ States ( ( 𝑓 ‘ ( ( 𝐹 ∩ 𝐺 ) ∩ 𝐻 ) ) = 1 → ( 𝑓 ‘ 𝐷 ) = 1 ) → ( ( 𝐹 ∩ 𝐺 ) ∩ 𝐻 ) ⊆ 𝐷 ) |
26 |
|
eqid |
⊢ ( ( ⊥ ‘ 𝐶 ) ∨ℋ ( 𝐶 ∩ 𝐵 ) ) = ( ( ⊥ ‘ 𝐶 ) ∨ℋ ( 𝐶 ∩ 𝐵 ) ) |
27 |
|
eqid |
⊢ ( ( ⊥ ‘ 𝐴 ) ∨ℋ ( 𝐴 ∩ 𝐶 ) ) = ( ( ⊥ ‘ 𝐴 ) ∨ℋ ( 𝐴 ∩ 𝐶 ) ) |
28 |
1 2 3 4 5 6 7 26 27
|
golem2 |
⊢ ( 𝑓 ∈ States → ( ( 𝑓 ‘ ( ( 𝐹 ∩ 𝐺 ) ∩ 𝐻 ) ) = 1 → ( 𝑓 ‘ 𝐷 ) = 1 ) ) |
29 |
25 28
|
mprg |
⊢ ( ( 𝐹 ∩ 𝐺 ) ∩ 𝐻 ) ⊆ 𝐷 |