Step |
Hyp |
Ref |
Expression |
1 |
|
trlne.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
2 |
|
trlne.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
3 |
|
trlne.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
4 |
|
trlne.t |
⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
5 |
|
trlne.r |
⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) |
6 |
|
simp3r |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ¬ 𝑃 ≤ 𝑊 ) |
7 |
1 3 4 5
|
trlle |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝑅 ‘ 𝐹 ) ≤ 𝑊 ) |
8 |
7
|
3adant3 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝑅 ‘ 𝐹 ) ≤ 𝑊 ) |
9 |
|
breq1 |
⊢ ( 𝑃 = ( 𝑅 ‘ 𝐹 ) → ( 𝑃 ≤ 𝑊 ↔ ( 𝑅 ‘ 𝐹 ) ≤ 𝑊 ) ) |
10 |
8 9
|
syl5ibrcom |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝑃 = ( 𝑅 ‘ 𝐹 ) → 𝑃 ≤ 𝑊 ) ) |
11 |
10
|
necon3bd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ¬ 𝑃 ≤ 𝑊 → 𝑃 ≠ ( 𝑅 ‘ 𝐹 ) ) ) |
12 |
6 11
|
mpd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝑃 ≠ ( 𝑅 ‘ 𝐹 ) ) |