Step |
Hyp |
Ref |
Expression |
1 |
|
pjoml2.1 |
⊢ 𝐴 ∈ Cℋ |
2 |
|
pjoml2.2 |
⊢ 𝐵 ∈ Cℋ |
3 |
1
|
choccli |
⊢ ( ⊥ ‘ 𝐴 ) ∈ Cℋ |
4 |
2
|
choccli |
⊢ ( ⊥ ‘ 𝐵 ) ∈ Cℋ |
5 |
3 4
|
pjoml2i |
⊢ ( ( ⊥ ‘ 𝐴 ) ⊆ ( ⊥ ‘ 𝐵 ) → ( ( ⊥ ‘ 𝐴 ) ∨ℋ ( ( ⊥ ‘ ( ⊥ ‘ 𝐴 ) ) ∩ ( ⊥ ‘ 𝐵 ) ) ) = ( ⊥ ‘ 𝐵 ) ) |
6 |
2 1
|
chsscon3i |
⊢ ( 𝐵 ⊆ 𝐴 ↔ ( ⊥ ‘ 𝐴 ) ⊆ ( ⊥ ‘ 𝐵 ) ) |
7 |
|
eqcom |
⊢ ( ( ⊥ ‘ ( ( ⊥ ‘ 𝐴 ) ∨ℋ ( ( ⊥ ‘ ( ⊥ ‘ 𝐴 ) ) ∩ ( ⊥ ‘ 𝐵 ) ) ) ) = 𝐵 ↔ 𝐵 = ( ⊥ ‘ ( ( ⊥ ‘ 𝐴 ) ∨ℋ ( ( ⊥ ‘ ( ⊥ ‘ 𝐴 ) ) ∩ ( ⊥ ‘ 𝐵 ) ) ) ) ) |
8 |
3
|
choccli |
⊢ ( ⊥ ‘ ( ⊥ ‘ 𝐴 ) ) ∈ Cℋ |
9 |
8 4
|
chincli |
⊢ ( ( ⊥ ‘ ( ⊥ ‘ 𝐴 ) ) ∩ ( ⊥ ‘ 𝐵 ) ) ∈ Cℋ |
10 |
1 9
|
chdmj2i |
⊢ ( ⊥ ‘ ( ( ⊥ ‘ 𝐴 ) ∨ℋ ( ( ⊥ ‘ ( ⊥ ‘ 𝐴 ) ) ∩ ( ⊥ ‘ 𝐵 ) ) ) ) = ( 𝐴 ∩ ( ⊥ ‘ ( ( ⊥ ‘ ( ⊥ ‘ 𝐴 ) ) ∩ ( ⊥ ‘ 𝐵 ) ) ) ) |
11 |
3 2
|
chdmm4i |
⊢ ( ⊥ ‘ ( ( ⊥ ‘ ( ⊥ ‘ 𝐴 ) ) ∩ ( ⊥ ‘ 𝐵 ) ) ) = ( ( ⊥ ‘ 𝐴 ) ∨ℋ 𝐵 ) |
12 |
11
|
ineq2i |
⊢ ( 𝐴 ∩ ( ⊥ ‘ ( ( ⊥ ‘ ( ⊥ ‘ 𝐴 ) ) ∩ ( ⊥ ‘ 𝐵 ) ) ) ) = ( 𝐴 ∩ ( ( ⊥ ‘ 𝐴 ) ∨ℋ 𝐵 ) ) |
13 |
10 12
|
eqtri |
⊢ ( ⊥ ‘ ( ( ⊥ ‘ 𝐴 ) ∨ℋ ( ( ⊥ ‘ ( ⊥ ‘ 𝐴 ) ) ∩ ( ⊥ ‘ 𝐵 ) ) ) ) = ( 𝐴 ∩ ( ( ⊥ ‘ 𝐴 ) ∨ℋ 𝐵 ) ) |
14 |
13
|
eqeq1i |
⊢ ( ( ⊥ ‘ ( ( ⊥ ‘ 𝐴 ) ∨ℋ ( ( ⊥ ‘ ( ⊥ ‘ 𝐴 ) ) ∩ ( ⊥ ‘ 𝐵 ) ) ) ) = 𝐵 ↔ ( 𝐴 ∩ ( ( ⊥ ‘ 𝐴 ) ∨ℋ 𝐵 ) ) = 𝐵 ) |
15 |
3 9
|
chjcli |
⊢ ( ( ⊥ ‘ 𝐴 ) ∨ℋ ( ( ⊥ ‘ ( ⊥ ‘ 𝐴 ) ) ∩ ( ⊥ ‘ 𝐵 ) ) ) ∈ Cℋ |
16 |
2 15
|
chcon2i |
⊢ ( 𝐵 = ( ⊥ ‘ ( ( ⊥ ‘ 𝐴 ) ∨ℋ ( ( ⊥ ‘ ( ⊥ ‘ 𝐴 ) ) ∩ ( ⊥ ‘ 𝐵 ) ) ) ) ↔ ( ( ⊥ ‘ 𝐴 ) ∨ℋ ( ( ⊥ ‘ ( ⊥ ‘ 𝐴 ) ) ∩ ( ⊥ ‘ 𝐵 ) ) ) = ( ⊥ ‘ 𝐵 ) ) |
17 |
7 14 16
|
3bitr3i |
⊢ ( ( 𝐴 ∩ ( ( ⊥ ‘ 𝐴 ) ∨ℋ 𝐵 ) ) = 𝐵 ↔ ( ( ⊥ ‘ 𝐴 ) ∨ℋ ( ( ⊥ ‘ ( ⊥ ‘ 𝐴 ) ) ∩ ( ⊥ ‘ 𝐵 ) ) ) = ( ⊥ ‘ 𝐵 ) ) |
18 |
5 6 17
|
3imtr4i |
⊢ ( 𝐵 ⊆ 𝐴 → ( 𝐴 ∩ ( ( ⊥ ‘ 𝐴 ) ∨ℋ 𝐵 ) ) = 𝐵 ) |