Step |
Hyp |
Ref |
Expression |
1 |
|
joinval2.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
joinval2.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
joinval2.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
4 |
|
joinval2.k |
⊢ ( 𝜑 → 𝐾 ∈ 𝑉 ) |
5 |
|
joinval2.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝐵 ) |
6 |
|
joinval2.y |
⊢ ( 𝜑 → 𝑌 ∈ 𝐵 ) |
7 |
|
breq1 |
⊢ ( 𝑦 = 𝑋 → ( 𝑦 ≤ 𝑥 ↔ 𝑋 ≤ 𝑥 ) ) |
8 |
|
breq1 |
⊢ ( 𝑦 = 𝑌 → ( 𝑦 ≤ 𝑥 ↔ 𝑌 ≤ 𝑥 ) ) |
9 |
7 8
|
ralprg |
⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ∀ 𝑦 ∈ { 𝑋 , 𝑌 } 𝑦 ≤ 𝑥 ↔ ( 𝑋 ≤ 𝑥 ∧ 𝑌 ≤ 𝑥 ) ) ) |
10 |
|
breq1 |
⊢ ( 𝑦 = 𝑋 → ( 𝑦 ≤ 𝑧 ↔ 𝑋 ≤ 𝑧 ) ) |
11 |
|
breq1 |
⊢ ( 𝑦 = 𝑌 → ( 𝑦 ≤ 𝑧 ↔ 𝑌 ≤ 𝑧 ) ) |
12 |
10 11
|
ralprg |
⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ∀ 𝑦 ∈ { 𝑋 , 𝑌 } 𝑦 ≤ 𝑧 ↔ ( 𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧 ) ) ) |
13 |
12
|
imbi1d |
⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( ∀ 𝑦 ∈ { 𝑋 , 𝑌 } 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧 ) ↔ ( ( 𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧 ) → 𝑥 ≤ 𝑧 ) ) ) |
14 |
13
|
ralbidv |
⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ∀ 𝑧 ∈ 𝐵 ( ∀ 𝑦 ∈ { 𝑋 , 𝑌 } 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧 ) ↔ ∀ 𝑧 ∈ 𝐵 ( ( 𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧 ) → 𝑥 ≤ 𝑧 ) ) ) |
15 |
9 14
|
anbi12d |
⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( ∀ 𝑦 ∈ { 𝑋 , 𝑌 } 𝑦 ≤ 𝑥 ∧ ∀ 𝑧 ∈ 𝐵 ( ∀ 𝑦 ∈ { 𝑋 , 𝑌 } 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧 ) ) ↔ ( ( 𝑋 ≤ 𝑥 ∧ 𝑌 ≤ 𝑥 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧 ) → 𝑥 ≤ 𝑧 ) ) ) ) |