Step |
Hyp |
Ref |
Expression |
1 |
|
ltrnel.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
2 |
|
ltrnel.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
3 |
|
ltrnel.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
4 |
|
ltrnel.t |
⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
5 |
|
simp3 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ 𝐴 ) → 𝑃 ∈ 𝐴 ) |
6 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
7 |
6 2
|
atbase |
⊢ ( 𝑃 ∈ 𝐴 → 𝑃 ∈ ( Base ‘ 𝐾 ) ) |
8 |
6 2 3 4
|
ltrncnvatb |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ ( Base ‘ 𝐾 ) ) → ( 𝑃 ∈ 𝐴 ↔ ( ◡ 𝐹 ‘ 𝑃 ) ∈ 𝐴 ) ) |
9 |
7 8
|
syl3an3 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ 𝐴 ) → ( 𝑃 ∈ 𝐴 ↔ ( ◡ 𝐹 ‘ 𝑃 ) ∈ 𝐴 ) ) |
10 |
5 9
|
mpbid |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ 𝐴 ) → ( ◡ 𝐹 ‘ 𝑃 ) ∈ 𝐴 ) |