Step |
Hyp |
Ref |
Expression |
1 |
|
mdsl.1 |
⊢ 𝐴 ∈ Cℋ |
2 |
|
mdsl.2 |
⊢ 𝐵 ∈ Cℋ |
3 |
|
anass |
⊢ ( ( ( ( 𝐴 ∩ 𝐵 ) ⊆ 𝑥 ∧ 𝑥 ⊆ ( 𝐴 ∨ℋ 𝐵 ) ) ∧ 𝑥 ⊆ 𝐵 ) ↔ ( ( 𝐴 ∩ 𝐵 ) ⊆ 𝑥 ∧ ( 𝑥 ⊆ ( 𝐴 ∨ℋ 𝐵 ) ∧ 𝑥 ⊆ 𝐵 ) ) ) |
4 |
2 1
|
chub2i |
⊢ 𝐵 ⊆ ( 𝐴 ∨ℋ 𝐵 ) |
5 |
|
sstr |
⊢ ( ( 𝑥 ⊆ 𝐵 ∧ 𝐵 ⊆ ( 𝐴 ∨ℋ 𝐵 ) ) → 𝑥 ⊆ ( 𝐴 ∨ℋ 𝐵 ) ) |
6 |
4 5
|
mpan2 |
⊢ ( 𝑥 ⊆ 𝐵 → 𝑥 ⊆ ( 𝐴 ∨ℋ 𝐵 ) ) |
7 |
6
|
pm4.71ri |
⊢ ( 𝑥 ⊆ 𝐵 ↔ ( 𝑥 ⊆ ( 𝐴 ∨ℋ 𝐵 ) ∧ 𝑥 ⊆ 𝐵 ) ) |
8 |
7
|
anbi2i |
⊢ ( ( ( 𝐴 ∩ 𝐵 ) ⊆ 𝑥 ∧ 𝑥 ⊆ 𝐵 ) ↔ ( ( 𝐴 ∩ 𝐵 ) ⊆ 𝑥 ∧ ( 𝑥 ⊆ ( 𝐴 ∨ℋ 𝐵 ) ∧ 𝑥 ⊆ 𝐵 ) ) ) |
9 |
3 8
|
bitr4i |
⊢ ( ( ( ( 𝐴 ∩ 𝐵 ) ⊆ 𝑥 ∧ 𝑥 ⊆ ( 𝐴 ∨ℋ 𝐵 ) ) ∧ 𝑥 ⊆ 𝐵 ) ↔ ( ( 𝐴 ∩ 𝐵 ) ⊆ 𝑥 ∧ 𝑥 ⊆ 𝐵 ) ) |
10 |
1 2
|
chincli |
⊢ ( 𝐴 ∩ 𝐵 ) ∈ Cℋ |
11 |
|
cvnbtwn4 |
⊢ ( ( ( 𝐴 ∩ 𝐵 ) ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝑥 ∈ Cℋ ) → ( ( 𝐴 ∩ 𝐵 ) ⋖ℋ 𝐵 → ( ( ( 𝐴 ∩ 𝐵 ) ⊆ 𝑥 ∧ 𝑥 ⊆ 𝐵 ) → ( 𝑥 = ( 𝐴 ∩ 𝐵 ) ∨ 𝑥 = 𝐵 ) ) ) ) |
12 |
10 2 11
|
mp3an12 |
⊢ ( 𝑥 ∈ Cℋ → ( ( 𝐴 ∩ 𝐵 ) ⋖ℋ 𝐵 → ( ( ( 𝐴 ∩ 𝐵 ) ⊆ 𝑥 ∧ 𝑥 ⊆ 𝐵 ) → ( 𝑥 = ( 𝐴 ∩ 𝐵 ) ∨ 𝑥 = 𝐵 ) ) ) ) |
13 |
12
|
impcom |
⊢ ( ( ( 𝐴 ∩ 𝐵 ) ⋖ℋ 𝐵 ∧ 𝑥 ∈ Cℋ ) → ( ( ( 𝐴 ∩ 𝐵 ) ⊆ 𝑥 ∧ 𝑥 ⊆ 𝐵 ) → ( 𝑥 = ( 𝐴 ∩ 𝐵 ) ∨ 𝑥 = 𝐵 ) ) ) |
14 |
10 1
|
chjcomi |
⊢ ( ( 𝐴 ∩ 𝐵 ) ∨ℋ 𝐴 ) = ( 𝐴 ∨ℋ ( 𝐴 ∩ 𝐵 ) ) |
15 |
1 2
|
chabs1i |
⊢ ( 𝐴 ∨ℋ ( 𝐴 ∩ 𝐵 ) ) = 𝐴 |
16 |
14 15
|
eqtri |
⊢ ( ( 𝐴 ∩ 𝐵 ) ∨ℋ 𝐴 ) = 𝐴 |
17 |
16
|
ineq1i |
⊢ ( ( ( 𝐴 ∩ 𝐵 ) ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( 𝐴 ∩ 𝐵 ) |
18 |
10
|
chjidmi |
⊢ ( ( 𝐴 ∩ 𝐵 ) ∨ℋ ( 𝐴 ∩ 𝐵 ) ) = ( 𝐴 ∩ 𝐵 ) |
19 |
17 18
|
eqtr4i |
⊢ ( ( ( 𝐴 ∩ 𝐵 ) ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( ( 𝐴 ∩ 𝐵 ) ∨ℋ ( 𝐴 ∩ 𝐵 ) ) |
20 |
|
oveq1 |
⊢ ( 𝑥 = ( 𝐴 ∩ 𝐵 ) → ( 𝑥 ∨ℋ 𝐴 ) = ( ( 𝐴 ∩ 𝐵 ) ∨ℋ 𝐴 ) ) |
21 |
20
|
ineq1d |
⊢ ( 𝑥 = ( 𝐴 ∩ 𝐵 ) → ( ( 𝑥 ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( ( ( 𝐴 ∩ 𝐵 ) ∨ℋ 𝐴 ) ∩ 𝐵 ) ) |
22 |
|
oveq1 |
⊢ ( 𝑥 = ( 𝐴 ∩ 𝐵 ) → ( 𝑥 ∨ℋ ( 𝐴 ∩ 𝐵 ) ) = ( ( 𝐴 ∩ 𝐵 ) ∨ℋ ( 𝐴 ∩ 𝐵 ) ) ) |
23 |
19 21 22
|
3eqtr4a |
⊢ ( 𝑥 = ( 𝐴 ∩ 𝐵 ) → ( ( 𝑥 ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( 𝑥 ∨ℋ ( 𝐴 ∩ 𝐵 ) ) ) |
24 |
|
incom |
⊢ ( ( 𝐵 ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( 𝐵 ∩ ( 𝐵 ∨ℋ 𝐴 ) ) |
25 |
2 1
|
chabs2i |
⊢ ( 𝐵 ∩ ( 𝐵 ∨ℋ 𝐴 ) ) = 𝐵 |
26 |
2 1
|
chabs1i |
⊢ ( 𝐵 ∨ℋ ( 𝐵 ∩ 𝐴 ) ) = 𝐵 |
27 |
|
incom |
⊢ ( 𝐵 ∩ 𝐴 ) = ( 𝐴 ∩ 𝐵 ) |
28 |
27
|
oveq2i |
⊢ ( 𝐵 ∨ℋ ( 𝐵 ∩ 𝐴 ) ) = ( 𝐵 ∨ℋ ( 𝐴 ∩ 𝐵 ) ) |
29 |
25 26 28
|
3eqtr2i |
⊢ ( 𝐵 ∩ ( 𝐵 ∨ℋ 𝐴 ) ) = ( 𝐵 ∨ℋ ( 𝐴 ∩ 𝐵 ) ) |
30 |
24 29
|
eqtri |
⊢ ( ( 𝐵 ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( 𝐵 ∨ℋ ( 𝐴 ∩ 𝐵 ) ) |
31 |
|
oveq1 |
⊢ ( 𝑥 = 𝐵 → ( 𝑥 ∨ℋ 𝐴 ) = ( 𝐵 ∨ℋ 𝐴 ) ) |
32 |
31
|
ineq1d |
⊢ ( 𝑥 = 𝐵 → ( ( 𝑥 ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( ( 𝐵 ∨ℋ 𝐴 ) ∩ 𝐵 ) ) |
33 |
|
oveq1 |
⊢ ( 𝑥 = 𝐵 → ( 𝑥 ∨ℋ ( 𝐴 ∩ 𝐵 ) ) = ( 𝐵 ∨ℋ ( 𝐴 ∩ 𝐵 ) ) ) |
34 |
30 32 33
|
3eqtr4a |
⊢ ( 𝑥 = 𝐵 → ( ( 𝑥 ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( 𝑥 ∨ℋ ( 𝐴 ∩ 𝐵 ) ) ) |
35 |
23 34
|
jaoi |
⊢ ( ( 𝑥 = ( 𝐴 ∩ 𝐵 ) ∨ 𝑥 = 𝐵 ) → ( ( 𝑥 ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( 𝑥 ∨ℋ ( 𝐴 ∩ 𝐵 ) ) ) |
36 |
13 35
|
syl6 |
⊢ ( ( ( 𝐴 ∩ 𝐵 ) ⋖ℋ 𝐵 ∧ 𝑥 ∈ Cℋ ) → ( ( ( 𝐴 ∩ 𝐵 ) ⊆ 𝑥 ∧ 𝑥 ⊆ 𝐵 ) → ( ( 𝑥 ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( 𝑥 ∨ℋ ( 𝐴 ∩ 𝐵 ) ) ) ) |
37 |
9 36
|
syl5bi |
⊢ ( ( ( 𝐴 ∩ 𝐵 ) ⋖ℋ 𝐵 ∧ 𝑥 ∈ Cℋ ) → ( ( ( ( 𝐴 ∩ 𝐵 ) ⊆ 𝑥 ∧ 𝑥 ⊆ ( 𝐴 ∨ℋ 𝐵 ) ) ∧ 𝑥 ⊆ 𝐵 ) → ( ( 𝑥 ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( 𝑥 ∨ℋ ( 𝐴 ∩ 𝐵 ) ) ) ) |
38 |
37
|
exp4b |
⊢ ( ( 𝐴 ∩ 𝐵 ) ⋖ℋ 𝐵 → ( 𝑥 ∈ Cℋ → ( ( ( 𝐴 ∩ 𝐵 ) ⊆ 𝑥 ∧ 𝑥 ⊆ ( 𝐴 ∨ℋ 𝐵 ) ) → ( 𝑥 ⊆ 𝐵 → ( ( 𝑥 ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( 𝑥 ∨ℋ ( 𝐴 ∩ 𝐵 ) ) ) ) ) ) |
39 |
38
|
ralrimiv |
⊢ ( ( 𝐴 ∩ 𝐵 ) ⋖ℋ 𝐵 → ∀ 𝑥 ∈ Cℋ ( ( ( 𝐴 ∩ 𝐵 ) ⊆ 𝑥 ∧ 𝑥 ⊆ ( 𝐴 ∨ℋ 𝐵 ) ) → ( 𝑥 ⊆ 𝐵 → ( ( 𝑥 ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( 𝑥 ∨ℋ ( 𝐴 ∩ 𝐵 ) ) ) ) ) |
40 |
1 2
|
mdsl1i |
⊢ ( ∀ 𝑥 ∈ Cℋ ( ( ( 𝐴 ∩ 𝐵 ) ⊆ 𝑥 ∧ 𝑥 ⊆ ( 𝐴 ∨ℋ 𝐵 ) ) → ( 𝑥 ⊆ 𝐵 → ( ( 𝑥 ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( 𝑥 ∨ℋ ( 𝐴 ∩ 𝐵 ) ) ) ) ↔ 𝐴 𝑀ℋ 𝐵 ) |
41 |
39 40
|
sylib |
⊢ ( ( 𝐴 ∩ 𝐵 ) ⋖ℋ 𝐵 → 𝐴 𝑀ℋ 𝐵 ) |