Step |
Hyp |
Ref |
Expression |
1 |
|
atlex.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
atlex.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
atlex.z |
⊢ 0 = ( 0. ‘ 𝐾 ) |
4 |
|
atlex.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
5 |
|
eqid |
⊢ ( glb ‘ 𝐾 ) = ( glb ‘ 𝐾 ) |
6 |
1 5 2 3 4
|
isatl |
⊢ ( 𝐾 ∈ AtLat ↔ ( 𝐾 ∈ Lat ∧ 𝐵 ∈ dom ( glb ‘ 𝐾 ) ∧ ∀ 𝑥 ∈ 𝐵 ( 𝑥 ≠ 0 → ∃ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ) ) ) |
7 |
6
|
simp3bi |
⊢ ( 𝐾 ∈ AtLat → ∀ 𝑥 ∈ 𝐵 ( 𝑥 ≠ 0 → ∃ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ) ) |
8 |
|
neeq1 |
⊢ ( 𝑥 = 𝑋 → ( 𝑥 ≠ 0 ↔ 𝑋 ≠ 0 ) ) |
9 |
|
breq2 |
⊢ ( 𝑥 = 𝑋 → ( 𝑦 ≤ 𝑥 ↔ 𝑦 ≤ 𝑋 ) ) |
10 |
9
|
rexbidv |
⊢ ( 𝑥 = 𝑋 → ( ∃ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ↔ ∃ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑋 ) ) |
11 |
8 10
|
imbi12d |
⊢ ( 𝑥 = 𝑋 → ( ( 𝑥 ≠ 0 → ∃ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ) ↔ ( 𝑋 ≠ 0 → ∃ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑋 ) ) ) |
12 |
11
|
rspccv |
⊢ ( ∀ 𝑥 ∈ 𝐵 ( 𝑥 ≠ 0 → ∃ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ) → ( 𝑋 ∈ 𝐵 → ( 𝑋 ≠ 0 → ∃ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑋 ) ) ) |
13 |
7 12
|
syl |
⊢ ( 𝐾 ∈ AtLat → ( 𝑋 ∈ 𝐵 → ( 𝑋 ≠ 0 → ∃ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑋 ) ) ) |
14 |
13
|
3imp |
⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → ∃ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑋 ) |