Step |
Hyp |
Ref |
Expression |
1 |
|
mdbr |
⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 𝑀ℋ 𝐵 ↔ ∀ 𝑦 ∈ Cℋ ( 𝑦 ⊆ 𝐵 → ( ( 𝑦 ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( 𝑦 ∨ℋ ( 𝐴 ∩ 𝐵 ) ) ) ) ) |
2 |
|
chincl |
⊢ ( ( 𝑥 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝑥 ∩ 𝐵 ) ∈ Cℋ ) |
3 |
|
inss2 |
⊢ ( 𝑥 ∩ 𝐵 ) ⊆ 𝐵 |
4 |
|
sseq1 |
⊢ ( 𝑦 = ( 𝑥 ∩ 𝐵 ) → ( 𝑦 ⊆ 𝐵 ↔ ( 𝑥 ∩ 𝐵 ) ⊆ 𝐵 ) ) |
5 |
|
oveq1 |
⊢ ( 𝑦 = ( 𝑥 ∩ 𝐵 ) → ( 𝑦 ∨ℋ 𝐴 ) = ( ( 𝑥 ∩ 𝐵 ) ∨ℋ 𝐴 ) ) |
6 |
5
|
ineq1d |
⊢ ( 𝑦 = ( 𝑥 ∩ 𝐵 ) → ( ( 𝑦 ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( ( ( 𝑥 ∩ 𝐵 ) ∨ℋ 𝐴 ) ∩ 𝐵 ) ) |
7 |
|
oveq1 |
⊢ ( 𝑦 = ( 𝑥 ∩ 𝐵 ) → ( 𝑦 ∨ℋ ( 𝐴 ∩ 𝐵 ) ) = ( ( 𝑥 ∩ 𝐵 ) ∨ℋ ( 𝐴 ∩ 𝐵 ) ) ) |
8 |
6 7
|
eqeq12d |
⊢ ( 𝑦 = ( 𝑥 ∩ 𝐵 ) → ( ( ( 𝑦 ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( 𝑦 ∨ℋ ( 𝐴 ∩ 𝐵 ) ) ↔ ( ( ( 𝑥 ∩ 𝐵 ) ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( ( 𝑥 ∩ 𝐵 ) ∨ℋ ( 𝐴 ∩ 𝐵 ) ) ) ) |
9 |
4 8
|
imbi12d |
⊢ ( 𝑦 = ( 𝑥 ∩ 𝐵 ) → ( ( 𝑦 ⊆ 𝐵 → ( ( 𝑦 ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( 𝑦 ∨ℋ ( 𝐴 ∩ 𝐵 ) ) ) ↔ ( ( 𝑥 ∩ 𝐵 ) ⊆ 𝐵 → ( ( ( 𝑥 ∩ 𝐵 ) ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( ( 𝑥 ∩ 𝐵 ) ∨ℋ ( 𝐴 ∩ 𝐵 ) ) ) ) ) |
10 |
9
|
rspcv |
⊢ ( ( 𝑥 ∩ 𝐵 ) ∈ Cℋ → ( ∀ 𝑦 ∈ Cℋ ( 𝑦 ⊆ 𝐵 → ( ( 𝑦 ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( 𝑦 ∨ℋ ( 𝐴 ∩ 𝐵 ) ) ) → ( ( 𝑥 ∩ 𝐵 ) ⊆ 𝐵 → ( ( ( 𝑥 ∩ 𝐵 ) ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( ( 𝑥 ∩ 𝐵 ) ∨ℋ ( 𝐴 ∩ 𝐵 ) ) ) ) ) |
11 |
3 10
|
mpii |
⊢ ( ( 𝑥 ∩ 𝐵 ) ∈ Cℋ → ( ∀ 𝑦 ∈ Cℋ ( 𝑦 ⊆ 𝐵 → ( ( 𝑦 ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( 𝑦 ∨ℋ ( 𝐴 ∩ 𝐵 ) ) ) → ( ( ( 𝑥 ∩ 𝐵 ) ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( ( 𝑥 ∩ 𝐵 ) ∨ℋ ( 𝐴 ∩ 𝐵 ) ) ) ) |
12 |
2 11
|
syl |
⊢ ( ( 𝑥 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( ∀ 𝑦 ∈ Cℋ ( 𝑦 ⊆ 𝐵 → ( ( 𝑦 ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( 𝑦 ∨ℋ ( 𝐴 ∩ 𝐵 ) ) ) → ( ( ( 𝑥 ∩ 𝐵 ) ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( ( 𝑥 ∩ 𝐵 ) ∨ℋ ( 𝐴 ∩ 𝐵 ) ) ) ) |
13 |
12
|
ex |
⊢ ( 𝑥 ∈ Cℋ → ( 𝐵 ∈ Cℋ → ( ∀ 𝑦 ∈ Cℋ ( 𝑦 ⊆ 𝐵 → ( ( 𝑦 ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( 𝑦 ∨ℋ ( 𝐴 ∩ 𝐵 ) ) ) → ( ( ( 𝑥 ∩ 𝐵 ) ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( ( 𝑥 ∩ 𝐵 ) ∨ℋ ( 𝐴 ∩ 𝐵 ) ) ) ) ) |
14 |
13
|
com3l |
⊢ ( 𝐵 ∈ Cℋ → ( ∀ 𝑦 ∈ Cℋ ( 𝑦 ⊆ 𝐵 → ( ( 𝑦 ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( 𝑦 ∨ℋ ( 𝐴 ∩ 𝐵 ) ) ) → ( 𝑥 ∈ Cℋ → ( ( ( 𝑥 ∩ 𝐵 ) ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( ( 𝑥 ∩ 𝐵 ) ∨ℋ ( 𝐴 ∩ 𝐵 ) ) ) ) ) |
15 |
14
|
ralrimdv |
⊢ ( 𝐵 ∈ Cℋ → ( ∀ 𝑦 ∈ Cℋ ( 𝑦 ⊆ 𝐵 → ( ( 𝑦 ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( 𝑦 ∨ℋ ( 𝐴 ∩ 𝐵 ) ) ) → ∀ 𝑥 ∈ Cℋ ( ( ( 𝑥 ∩ 𝐵 ) ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( ( 𝑥 ∩ 𝐵 ) ∨ℋ ( 𝐴 ∩ 𝐵 ) ) ) ) |
16 |
|
dfss |
⊢ ( 𝑥 ⊆ 𝐵 ↔ 𝑥 = ( 𝑥 ∩ 𝐵 ) ) |
17 |
16
|
biimpi |
⊢ ( 𝑥 ⊆ 𝐵 → 𝑥 = ( 𝑥 ∩ 𝐵 ) ) |
18 |
17
|
oveq1d |
⊢ ( 𝑥 ⊆ 𝐵 → ( 𝑥 ∨ℋ 𝐴 ) = ( ( 𝑥 ∩ 𝐵 ) ∨ℋ 𝐴 ) ) |
19 |
18
|
ineq1d |
⊢ ( 𝑥 ⊆ 𝐵 → ( ( 𝑥 ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( ( ( 𝑥 ∩ 𝐵 ) ∨ℋ 𝐴 ) ∩ 𝐵 ) ) |
20 |
17
|
oveq1d |
⊢ ( 𝑥 ⊆ 𝐵 → ( 𝑥 ∨ℋ ( 𝐴 ∩ 𝐵 ) ) = ( ( 𝑥 ∩ 𝐵 ) ∨ℋ ( 𝐴 ∩ 𝐵 ) ) ) |
21 |
19 20
|
eqeq12d |
⊢ ( 𝑥 ⊆ 𝐵 → ( ( ( 𝑥 ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( 𝑥 ∨ℋ ( 𝐴 ∩ 𝐵 ) ) ↔ ( ( ( 𝑥 ∩ 𝐵 ) ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( ( 𝑥 ∩ 𝐵 ) ∨ℋ ( 𝐴 ∩ 𝐵 ) ) ) ) |
22 |
21
|
biimprcd |
⊢ ( ( ( ( 𝑥 ∩ 𝐵 ) ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( ( 𝑥 ∩ 𝐵 ) ∨ℋ ( 𝐴 ∩ 𝐵 ) ) → ( 𝑥 ⊆ 𝐵 → ( ( 𝑥 ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( 𝑥 ∨ℋ ( 𝐴 ∩ 𝐵 ) ) ) ) |
23 |
22
|
ralimi |
⊢ ( ∀ 𝑥 ∈ Cℋ ( ( ( 𝑥 ∩ 𝐵 ) ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( ( 𝑥 ∩ 𝐵 ) ∨ℋ ( 𝐴 ∩ 𝐵 ) ) → ∀ 𝑥 ∈ Cℋ ( 𝑥 ⊆ 𝐵 → ( ( 𝑥 ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( 𝑥 ∨ℋ ( 𝐴 ∩ 𝐵 ) ) ) ) |
24 |
|
sseq1 |
⊢ ( 𝑥 = 𝑦 → ( 𝑥 ⊆ 𝐵 ↔ 𝑦 ⊆ 𝐵 ) ) |
25 |
|
oveq1 |
⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∨ℋ 𝐴 ) = ( 𝑦 ∨ℋ 𝐴 ) ) |
26 |
25
|
ineq1d |
⊢ ( 𝑥 = 𝑦 → ( ( 𝑥 ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( ( 𝑦 ∨ℋ 𝐴 ) ∩ 𝐵 ) ) |
27 |
|
oveq1 |
⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∨ℋ ( 𝐴 ∩ 𝐵 ) ) = ( 𝑦 ∨ℋ ( 𝐴 ∩ 𝐵 ) ) ) |
28 |
26 27
|
eqeq12d |
⊢ ( 𝑥 = 𝑦 → ( ( ( 𝑥 ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( 𝑥 ∨ℋ ( 𝐴 ∩ 𝐵 ) ) ↔ ( ( 𝑦 ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( 𝑦 ∨ℋ ( 𝐴 ∩ 𝐵 ) ) ) ) |
29 |
24 28
|
imbi12d |
⊢ ( 𝑥 = 𝑦 → ( ( 𝑥 ⊆ 𝐵 → ( ( 𝑥 ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( 𝑥 ∨ℋ ( 𝐴 ∩ 𝐵 ) ) ) ↔ ( 𝑦 ⊆ 𝐵 → ( ( 𝑦 ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( 𝑦 ∨ℋ ( 𝐴 ∩ 𝐵 ) ) ) ) ) |
30 |
29
|
cbvralvw |
⊢ ( ∀ 𝑥 ∈ Cℋ ( 𝑥 ⊆ 𝐵 → ( ( 𝑥 ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( 𝑥 ∨ℋ ( 𝐴 ∩ 𝐵 ) ) ) ↔ ∀ 𝑦 ∈ Cℋ ( 𝑦 ⊆ 𝐵 → ( ( 𝑦 ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( 𝑦 ∨ℋ ( 𝐴 ∩ 𝐵 ) ) ) ) |
31 |
23 30
|
sylib |
⊢ ( ∀ 𝑥 ∈ Cℋ ( ( ( 𝑥 ∩ 𝐵 ) ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( ( 𝑥 ∩ 𝐵 ) ∨ℋ ( 𝐴 ∩ 𝐵 ) ) → ∀ 𝑦 ∈ Cℋ ( 𝑦 ⊆ 𝐵 → ( ( 𝑦 ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( 𝑦 ∨ℋ ( 𝐴 ∩ 𝐵 ) ) ) ) |
32 |
15 31
|
impbid1 |
⊢ ( 𝐵 ∈ Cℋ → ( ∀ 𝑦 ∈ Cℋ ( 𝑦 ⊆ 𝐵 → ( ( 𝑦 ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( 𝑦 ∨ℋ ( 𝐴 ∩ 𝐵 ) ) ) ↔ ∀ 𝑥 ∈ Cℋ ( ( ( 𝑥 ∩ 𝐵 ) ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( ( 𝑥 ∩ 𝐵 ) ∨ℋ ( 𝐴 ∩ 𝐵 ) ) ) ) |
33 |
32
|
adantl |
⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( ∀ 𝑦 ∈ Cℋ ( 𝑦 ⊆ 𝐵 → ( ( 𝑦 ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( 𝑦 ∨ℋ ( 𝐴 ∩ 𝐵 ) ) ) ↔ ∀ 𝑥 ∈ Cℋ ( ( ( 𝑥 ∩ 𝐵 ) ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( ( 𝑥 ∩ 𝐵 ) ∨ℋ ( 𝐴 ∩ 𝐵 ) ) ) ) |
34 |
1 33
|
bitrd |
⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 𝑀ℋ 𝐵 ↔ ∀ 𝑥 ∈ Cℋ ( ( ( 𝑥 ∩ 𝐵 ) ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( ( 𝑥 ∩ 𝐵 ) ∨ℋ ( 𝐴 ∩ 𝐵 ) ) ) ) |