Step |
Hyp |
Ref |
Expression |
1 |
|
dibval3.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
dibval3.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
dibval3.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
4 |
|
dibval3.t |
⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
5 |
|
dibval3.r |
⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) |
6 |
|
dibval3.o |
⊢ 0 = ( 𝑔 ∈ 𝑇 ↦ ( I ↾ 𝐵 ) ) |
7 |
|
dibval3.i |
⊢ 𝐼 = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) |
8 |
|
eqid |
⊢ ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) |
9 |
1 2 3 4 6 8 7
|
dibopelval2 |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( 〈 𝐹 , 𝑆 〉 ∈ ( 𝐼 ‘ 𝑋 ) ↔ ( 𝐹 ∈ ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑋 ) ∧ 𝑆 = 0 ) ) ) |
10 |
1 2 3 4 5 8
|
diaelval |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( 𝐹 ∈ ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑋 ) ↔ ( 𝐹 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝐹 ) ≤ 𝑋 ) ) ) |
11 |
10
|
anbi1d |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( ( 𝐹 ∈ ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑋 ) ∧ 𝑆 = 0 ) ↔ ( ( 𝐹 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝐹 ) ≤ 𝑋 ) ∧ 𝑆 = 0 ) ) ) |
12 |
9 11
|
bitrd |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( 〈 𝐹 , 𝑆 〉 ∈ ( 𝐼 ‘ 𝑋 ) ↔ ( ( 𝐹 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝐹 ) ≤ 𝑋 ) ∧ 𝑆 = 0 ) ) ) |