Step |
Hyp |
Ref |
Expression |
1 |
|
joinle.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
joinle.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
joinle.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
4 |
|
joinle.k |
⊢ ( 𝜑 → 𝐾 ∈ Poset ) |
5 |
|
joinle.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝐵 ) |
6 |
|
joinle.y |
⊢ ( 𝜑 → 𝑌 ∈ 𝐵 ) |
7 |
|
joinle.z |
⊢ ( 𝜑 → 𝑍 ∈ 𝐵 ) |
8 |
|
joinle.e |
⊢ ( 𝜑 → 〈 𝑋 , 𝑌 〉 ∈ dom ∨ ) |
9 |
|
breq2 |
⊢ ( 𝑧 = 𝑍 → ( 𝑋 ≤ 𝑧 ↔ 𝑋 ≤ 𝑍 ) ) |
10 |
|
breq2 |
⊢ ( 𝑧 = 𝑍 → ( 𝑌 ≤ 𝑧 ↔ 𝑌 ≤ 𝑍 ) ) |
11 |
9 10
|
anbi12d |
⊢ ( 𝑧 = 𝑍 → ( ( 𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧 ) ↔ ( 𝑋 ≤ 𝑍 ∧ 𝑌 ≤ 𝑍 ) ) ) |
12 |
|
breq2 |
⊢ ( 𝑧 = 𝑍 → ( ( 𝑋 ∨ 𝑌 ) ≤ 𝑧 ↔ ( 𝑋 ∨ 𝑌 ) ≤ 𝑍 ) ) |
13 |
11 12
|
imbi12d |
⊢ ( 𝑧 = 𝑍 → ( ( ( 𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧 ) → ( 𝑋 ∨ 𝑌 ) ≤ 𝑧 ) ↔ ( ( 𝑋 ≤ 𝑍 ∧ 𝑌 ≤ 𝑍 ) → ( 𝑋 ∨ 𝑌 ) ≤ 𝑍 ) ) ) |
14 |
1 2 3 4 5 6 8
|
joinlem |
⊢ ( 𝜑 → ( ( 𝑋 ≤ ( 𝑋 ∨ 𝑌 ) ∧ 𝑌 ≤ ( 𝑋 ∨ 𝑌 ) ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧 ) → ( 𝑋 ∨ 𝑌 ) ≤ 𝑧 ) ) ) |
15 |
14
|
simprd |
⊢ ( 𝜑 → ∀ 𝑧 ∈ 𝐵 ( ( 𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧 ) → ( 𝑋 ∨ 𝑌 ) ≤ 𝑧 ) ) |
16 |
13 15 7
|
rspcdva |
⊢ ( 𝜑 → ( ( 𝑋 ≤ 𝑍 ∧ 𝑌 ≤ 𝑍 ) → ( 𝑋 ∨ 𝑌 ) ≤ 𝑍 ) ) |
17 |
1 2 3 4 5 6 8
|
lejoin1 |
⊢ ( 𝜑 → 𝑋 ≤ ( 𝑋 ∨ 𝑌 ) ) |
18 |
1 3 4 5 6 8
|
joincl |
⊢ ( 𝜑 → ( 𝑋 ∨ 𝑌 ) ∈ 𝐵 ) |
19 |
1 2
|
postr |
⊢ ( ( 𝐾 ∈ Poset ∧ ( 𝑋 ∈ 𝐵 ∧ ( 𝑋 ∨ 𝑌 ) ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 ≤ ( 𝑋 ∨ 𝑌 ) ∧ ( 𝑋 ∨ 𝑌 ) ≤ 𝑍 ) → 𝑋 ≤ 𝑍 ) ) |
20 |
4 5 18 7 19
|
syl13anc |
⊢ ( 𝜑 → ( ( 𝑋 ≤ ( 𝑋 ∨ 𝑌 ) ∧ ( 𝑋 ∨ 𝑌 ) ≤ 𝑍 ) → 𝑋 ≤ 𝑍 ) ) |
21 |
17 20
|
mpand |
⊢ ( 𝜑 → ( ( 𝑋 ∨ 𝑌 ) ≤ 𝑍 → 𝑋 ≤ 𝑍 ) ) |
22 |
1 2 3 4 5 6 8
|
lejoin2 |
⊢ ( 𝜑 → 𝑌 ≤ ( 𝑋 ∨ 𝑌 ) ) |
23 |
1 2
|
postr |
⊢ ( ( 𝐾 ∈ Poset ∧ ( 𝑌 ∈ 𝐵 ∧ ( 𝑋 ∨ 𝑌 ) ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑌 ≤ ( 𝑋 ∨ 𝑌 ) ∧ ( 𝑋 ∨ 𝑌 ) ≤ 𝑍 ) → 𝑌 ≤ 𝑍 ) ) |
24 |
4 6 18 7 23
|
syl13anc |
⊢ ( 𝜑 → ( ( 𝑌 ≤ ( 𝑋 ∨ 𝑌 ) ∧ ( 𝑋 ∨ 𝑌 ) ≤ 𝑍 ) → 𝑌 ≤ 𝑍 ) ) |
25 |
22 24
|
mpand |
⊢ ( 𝜑 → ( ( 𝑋 ∨ 𝑌 ) ≤ 𝑍 → 𝑌 ≤ 𝑍 ) ) |
26 |
21 25
|
jcad |
⊢ ( 𝜑 → ( ( 𝑋 ∨ 𝑌 ) ≤ 𝑍 → ( 𝑋 ≤ 𝑍 ∧ 𝑌 ≤ 𝑍 ) ) ) |
27 |
16 26
|
impbid |
⊢ ( 𝜑 → ( ( 𝑋 ≤ 𝑍 ∧ 𝑌 ≤ 𝑍 ) ↔ ( 𝑋 ∨ 𝑌 ) ≤ 𝑍 ) ) |