Step |
Hyp |
Ref |
Expression |
1 |
|
dihelval2.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
2 |
|
dihelval2.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
3 |
|
dihelval2.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
4 |
|
dihelval2.p |
⊢ 𝑃 = ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) |
5 |
|
dihelval2.t |
⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
6 |
|
dihelval2.e |
⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) |
7 |
|
dihelval2.i |
⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) |
8 |
|
dihelval2.g |
⊢ 𝐺 = ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑄 ) |
9 |
|
dihelval2.f |
⊢ 𝐹 ∈ V |
10 |
|
dihelval2.s |
⊢ 𝑆 ∈ V |
11 |
|
eqid |
⊢ ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) = ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) |
12 |
1 2 3 11 7
|
dihvalcqat |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑄 ) = ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑄 ) ) |
13 |
12
|
eleq2d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( 〈 𝐹 , 𝑆 〉 ∈ ( 𝐼 ‘ 𝑄 ) ↔ 〈 𝐹 , 𝑆 〉 ∈ ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑄 ) ) ) |
14 |
1 2 3 4 5 6 11 8 9 10
|
dicopelval2 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( 〈 𝐹 , 𝑆 〉 ∈ ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑄 ) ↔ ( 𝐹 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐸 ) ) ) |
15 |
13 14
|
bitrd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( 〈 𝐹 , 𝑆 〉 ∈ ( 𝐼 ‘ 𝑄 ) ↔ ( 𝐹 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐸 ) ) ) |