Step |
Hyp |
Ref |
Expression |
1 |
|
dicval.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
2 |
|
dicval.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
3 |
|
dicval.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
4 |
|
dicval.p |
⊢ 𝑃 = ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) |
5 |
|
dicval.t |
⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
6 |
|
dicval.e |
⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) |
7 |
|
dicval.i |
⊢ 𝐼 = ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) |
8 |
|
dicelval.f |
⊢ 𝐹 ∈ V |
9 |
|
dicelval.s |
⊢ 𝑆 ∈ V |
10 |
1 2 3 4 5 6 7
|
dicval |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑄 ) = { 〈 𝑓 , 𝑠 〉 ∣ ( 𝑓 = ( 𝑠 ‘ ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑄 ) ) ∧ 𝑠 ∈ 𝐸 ) } ) |
11 |
10
|
eleq2d |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( 〈 𝐹 , 𝑆 〉 ∈ ( 𝐼 ‘ 𝑄 ) ↔ 〈 𝐹 , 𝑆 〉 ∈ { 〈 𝑓 , 𝑠 〉 ∣ ( 𝑓 = ( 𝑠 ‘ ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑄 ) ) ∧ 𝑠 ∈ 𝐸 ) } ) ) |
12 |
|
eqeq1 |
⊢ ( 𝑓 = 𝐹 → ( 𝑓 = ( 𝑠 ‘ ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑄 ) ) ↔ 𝐹 = ( 𝑠 ‘ ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑄 ) ) ) ) |
13 |
12
|
anbi1d |
⊢ ( 𝑓 = 𝐹 → ( ( 𝑓 = ( 𝑠 ‘ ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑄 ) ) ∧ 𝑠 ∈ 𝐸 ) ↔ ( 𝐹 = ( 𝑠 ‘ ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑄 ) ) ∧ 𝑠 ∈ 𝐸 ) ) ) |
14 |
|
fveq1 |
⊢ ( 𝑠 = 𝑆 → ( 𝑠 ‘ ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑄 ) ) = ( 𝑆 ‘ ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑄 ) ) ) |
15 |
14
|
eqeq2d |
⊢ ( 𝑠 = 𝑆 → ( 𝐹 = ( 𝑠 ‘ ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑄 ) ) ↔ 𝐹 = ( 𝑆 ‘ ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑄 ) ) ) ) |
16 |
|
eleq1 |
⊢ ( 𝑠 = 𝑆 → ( 𝑠 ∈ 𝐸 ↔ 𝑆 ∈ 𝐸 ) ) |
17 |
15 16
|
anbi12d |
⊢ ( 𝑠 = 𝑆 → ( ( 𝐹 = ( 𝑠 ‘ ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑄 ) ) ∧ 𝑠 ∈ 𝐸 ) ↔ ( 𝐹 = ( 𝑆 ‘ ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑄 ) ) ∧ 𝑆 ∈ 𝐸 ) ) ) |
18 |
8 9 13 17
|
opelopab |
⊢ ( 〈 𝐹 , 𝑆 〉 ∈ { 〈 𝑓 , 𝑠 〉 ∣ ( 𝑓 = ( 𝑠 ‘ ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑄 ) ) ∧ 𝑠 ∈ 𝐸 ) } ↔ ( 𝐹 = ( 𝑆 ‘ ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑄 ) ) ∧ 𝑆 ∈ 𝐸 ) ) |
19 |
11 18
|
bitrdi |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( 〈 𝐹 , 𝑆 〉 ∈ ( 𝐼 ‘ 𝑄 ) ↔ ( 𝐹 = ( 𝑆 ‘ ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑄 ) ) ∧ 𝑆 ∈ 𝐸 ) ) ) |