Step |
Hyp |
Ref |
Expression |
1 |
|
dibelval1.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
dibelval1.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
dibelval1.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
4 |
|
dibelval1.j |
⊢ 𝐽 = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) |
5 |
|
dibelval1.i |
⊢ 𝐼 = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) |
6 |
|
eqid |
⊢ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
7 |
|
eqid |
⊢ ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ 𝐵 ) ) = ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ 𝐵 ) ) |
8 |
1 2 3 6 7 4 5
|
dibval2 |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑋 ) = ( ( 𝐽 ‘ 𝑋 ) × { ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ 𝐵 ) ) } ) ) |
9 |
8
|
eleq2d |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( 𝑌 ∈ ( 𝐼 ‘ 𝑋 ) ↔ 𝑌 ∈ ( ( 𝐽 ‘ 𝑋 ) × { ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ 𝐵 ) ) } ) ) ) |
10 |
9
|
biimp3a |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ 𝑌 ∈ ( 𝐼 ‘ 𝑋 ) ) → 𝑌 ∈ ( ( 𝐽 ‘ 𝑋 ) × { ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ 𝐵 ) ) } ) ) |
11 |
|
xp1st |
⊢ ( 𝑌 ∈ ( ( 𝐽 ‘ 𝑋 ) × { ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ 𝐵 ) ) } ) → ( 1st ‘ 𝑌 ) ∈ ( 𝐽 ‘ 𝑋 ) ) |
12 |
10 11
|
syl |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ 𝑌 ∈ ( 𝐼 ‘ 𝑋 ) ) → ( 1st ‘ 𝑌 ) ∈ ( 𝐽 ‘ 𝑋 ) ) |