Step |
Hyp |
Ref |
Expression |
1 |
|
dibval2.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
dibval2.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
dibval2.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
4 |
|
dibval2.t |
⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
5 |
|
dibval2.o |
⊢ 0 = ( 𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵 ) ) |
6 |
|
dibval2.j |
⊢ 𝐽 = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) |
7 |
|
dibval2.i |
⊢ 𝐼 = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) |
8 |
1 2 3 4 5 6 7
|
dibval2 |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑋 ) = ( ( 𝐽 ‘ 𝑋 ) × { 0 } ) ) |
9 |
8
|
eleq2d |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( 〈 𝐹 , 𝑆 〉 ∈ ( 𝐼 ‘ 𝑋 ) ↔ 〈 𝐹 , 𝑆 〉 ∈ ( ( 𝐽 ‘ 𝑋 ) × { 0 } ) ) ) |
10 |
|
opelxp |
⊢ ( 〈 𝐹 , 𝑆 〉 ∈ ( ( 𝐽 ‘ 𝑋 ) × { 0 } ) ↔ ( 𝐹 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑆 ∈ { 0 } ) ) |
11 |
4
|
fvexi |
⊢ 𝑇 ∈ V |
12 |
11
|
mptex |
⊢ ( 𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵 ) ) ∈ V |
13 |
5 12
|
eqeltri |
⊢ 0 ∈ V |
14 |
13
|
elsn2 |
⊢ ( 𝑆 ∈ { 0 } ↔ 𝑆 = 0 ) |
15 |
14
|
anbi2i |
⊢ ( ( 𝐹 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑆 ∈ { 0 } ) ↔ ( 𝐹 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑆 = 0 ) ) |
16 |
10 15
|
bitri |
⊢ ( 〈 𝐹 , 𝑆 〉 ∈ ( ( 𝐽 ‘ 𝑋 ) × { 0 } ) ↔ ( 𝐹 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑆 = 0 ) ) |
17 |
9 16
|
bitrdi |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( 〈 𝐹 , 𝑆 〉 ∈ ( 𝐼 ‘ 𝑋 ) ↔ ( 𝐹 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑆 = 0 ) ) ) |