Step |
Hyp |
Ref |
Expression |
1 |
|
ltrn2eq.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
2 |
|
ltrn2eq.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
3 |
|
ltrn2eq.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
4 |
|
ltrn2eq.t |
⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
5 |
1 2 3 4
|
ltrn2ateq |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ) → ( ( 𝐹 ‘ 𝑃 ) = 𝑃 ↔ ( 𝐹 ‘ 𝑄 ) = 𝑄 ) ) |
6 |
5
|
necon3bid |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ) → ( ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ↔ ( 𝐹 ‘ 𝑄 ) ≠ 𝑄 ) ) |
7 |
6
|
biimp3a |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) → ( 𝐹 ‘ 𝑄 ) ≠ 𝑄 ) |