Step |
Hyp |
Ref |
Expression |
1 |
|
ltrnval1.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
ltrnval1.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
ltrnval1.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
4 |
|
ltrnval1.t |
⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
5 |
|
eqid |
⊢ ( ( LDil ‘ 𝐾 ) ‘ 𝑊 ) = ( ( LDil ‘ 𝐾 ) ‘ 𝑊 ) |
6 |
3 5 4
|
ltrnldil |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → 𝐹 ∈ ( ( LDil ‘ 𝐾 ) ‘ 𝑊 ) ) |
7 |
6
|
3adant3 |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → 𝐹 ∈ ( ( LDil ‘ 𝐾 ) ‘ 𝑊 ) ) |
8 |
1 2 3 5
|
ldilval |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ ( ( LDil ‘ 𝐾 ) ‘ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( 𝐹 ‘ 𝑋 ) = 𝑋 ) |
9 |
7 8
|
syld3an2 |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( 𝐹 ‘ 𝑋 ) = 𝑋 ) |