Step |
Hyp |
Ref |
Expression |
1 |
|
cdlemk3.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
cdlemk3.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
cdlemk3.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
4 |
|
cdlemk3.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
5 |
|
cdlemk3.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
6 |
|
cdlemk3.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
7 |
|
cdlemk3.t |
⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
8 |
|
cdlemk3.r |
⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) |
9 |
|
cdlemk3.s |
⊢ 𝑆 = ( 𝑓 ∈ 𝑇 ↦ ( ℩ 𝑖 ∈ 𝑇 ( 𝑖 ‘ 𝑃 ) = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝑓 ) ) ∧ ( ( 𝑁 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝑓 ∘ ◡ 𝐹 ) ) ) ) ) ) |
10 |
|
cdlemk3.u1 |
⊢ 𝑌 = ( 𝑑 ∈ 𝑇 , 𝑒 ∈ 𝑇 ↦ ( ℩ 𝑗 ∈ 𝑇 ( 𝑗 ‘ 𝑃 ) = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝑒 ) ) ∧ ( ( ( 𝑆 ‘ 𝑑 ) ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝑒 ∘ ◡ 𝑑 ) ) ) ) ) ) |
11 |
|
cdlemk3.o2 |
⊢ 𝑄 = ( 𝑆 ‘ 𝐷 ) |
12 |
|
cdlemk3.u2 |
⊢ 𝑍 = ( 𝑒 ∈ 𝑇 ↦ ( ℩ 𝑗 ∈ 𝑇 ( 𝑗 ‘ 𝑃 ) = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝑒 ) ) ∧ ( ( 𝑄 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝑒 ∘ ◡ 𝐷 ) ) ) ) ) ) |
13 |
|
fveq2 |
⊢ ( 𝑑 = 𝐷 → ( 𝑆 ‘ 𝑑 ) = ( 𝑆 ‘ 𝐷 ) ) |
14 |
13 11
|
eqtr4di |
⊢ ( 𝑑 = 𝐷 → ( 𝑆 ‘ 𝑑 ) = 𝑄 ) |
15 |
14
|
fveq1d |
⊢ ( 𝑑 = 𝐷 → ( ( 𝑆 ‘ 𝑑 ) ‘ 𝑃 ) = ( 𝑄 ‘ 𝑃 ) ) |
16 |
|
cnveq |
⊢ ( 𝑑 = 𝐷 → ◡ 𝑑 = ◡ 𝐷 ) |
17 |
16
|
coeq2d |
⊢ ( 𝑑 = 𝐷 → ( 𝑒 ∘ ◡ 𝑑 ) = ( 𝑒 ∘ ◡ 𝐷 ) ) |
18 |
17
|
fveq2d |
⊢ ( 𝑑 = 𝐷 → ( 𝑅 ‘ ( 𝑒 ∘ ◡ 𝑑 ) ) = ( 𝑅 ‘ ( 𝑒 ∘ ◡ 𝐷 ) ) ) |
19 |
15 18
|
oveq12d |
⊢ ( 𝑑 = 𝐷 → ( ( ( 𝑆 ‘ 𝑑 ) ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝑒 ∘ ◡ 𝑑 ) ) ) = ( ( 𝑄 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝑒 ∘ ◡ 𝐷 ) ) ) ) |
20 |
19
|
oveq2d |
⊢ ( 𝑑 = 𝐷 → ( ( 𝑃 ∨ ( 𝑅 ‘ 𝑒 ) ) ∧ ( ( ( 𝑆 ‘ 𝑑 ) ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝑒 ∘ ◡ 𝑑 ) ) ) ) = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝑒 ) ) ∧ ( ( 𝑄 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝑒 ∘ ◡ 𝐷 ) ) ) ) ) |
21 |
20
|
eqeq2d |
⊢ ( 𝑑 = 𝐷 → ( ( 𝑗 ‘ 𝑃 ) = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝑒 ) ) ∧ ( ( ( 𝑆 ‘ 𝑑 ) ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝑒 ∘ ◡ 𝑑 ) ) ) ) ↔ ( 𝑗 ‘ 𝑃 ) = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝑒 ) ) ∧ ( ( 𝑄 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝑒 ∘ ◡ 𝐷 ) ) ) ) ) ) |
22 |
21
|
riotabidv |
⊢ ( 𝑑 = 𝐷 → ( ℩ 𝑗 ∈ 𝑇 ( 𝑗 ‘ 𝑃 ) = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝑒 ) ) ∧ ( ( ( 𝑆 ‘ 𝑑 ) ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝑒 ∘ ◡ 𝑑 ) ) ) ) ) = ( ℩ 𝑗 ∈ 𝑇 ( 𝑗 ‘ 𝑃 ) = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝑒 ) ) ∧ ( ( 𝑄 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝑒 ∘ ◡ 𝐷 ) ) ) ) ) ) |
23 |
|
fveq2 |
⊢ ( 𝑒 = 𝐺 → ( 𝑅 ‘ 𝑒 ) = ( 𝑅 ‘ 𝐺 ) ) |
24 |
23
|
oveq2d |
⊢ ( 𝑒 = 𝐺 → ( 𝑃 ∨ ( 𝑅 ‘ 𝑒 ) ) = ( 𝑃 ∨ ( 𝑅 ‘ 𝐺 ) ) ) |
25 |
|
coeq1 |
⊢ ( 𝑒 = 𝐺 → ( 𝑒 ∘ ◡ 𝐷 ) = ( 𝐺 ∘ ◡ 𝐷 ) ) |
26 |
25
|
fveq2d |
⊢ ( 𝑒 = 𝐺 → ( 𝑅 ‘ ( 𝑒 ∘ ◡ 𝐷 ) ) = ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐷 ) ) ) |
27 |
26
|
oveq2d |
⊢ ( 𝑒 = 𝐺 → ( ( 𝑄 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝑒 ∘ ◡ 𝐷 ) ) ) = ( ( 𝑄 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐷 ) ) ) ) |
28 |
24 27
|
oveq12d |
⊢ ( 𝑒 = 𝐺 → ( ( 𝑃 ∨ ( 𝑅 ‘ 𝑒 ) ) ∧ ( ( 𝑄 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝑒 ∘ ◡ 𝐷 ) ) ) ) = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝐺 ) ) ∧ ( ( 𝑄 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐷 ) ) ) ) ) |
29 |
28
|
eqeq2d |
⊢ ( 𝑒 = 𝐺 → ( ( 𝑗 ‘ 𝑃 ) = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝑒 ) ) ∧ ( ( 𝑄 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝑒 ∘ ◡ 𝐷 ) ) ) ) ↔ ( 𝑗 ‘ 𝑃 ) = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝐺 ) ) ∧ ( ( 𝑄 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐷 ) ) ) ) ) ) |
30 |
29
|
riotabidv |
⊢ ( 𝑒 = 𝐺 → ( ℩ 𝑗 ∈ 𝑇 ( 𝑗 ‘ 𝑃 ) = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝑒 ) ) ∧ ( ( 𝑄 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝑒 ∘ ◡ 𝐷 ) ) ) ) ) = ( ℩ 𝑗 ∈ 𝑇 ( 𝑗 ‘ 𝑃 ) = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝐺 ) ) ∧ ( ( 𝑄 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐷 ) ) ) ) ) ) |
31 |
|
riotaex |
⊢ ( ℩ 𝑗 ∈ 𝑇 ( 𝑗 ‘ 𝑃 ) = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝐺 ) ) ∧ ( ( 𝑄 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐷 ) ) ) ) ) ∈ V |
32 |
22 30 10 31
|
ovmpo |
⊢ ( ( 𝐷 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) → ( 𝐷 𝑌 𝐺 ) = ( ℩ 𝑗 ∈ 𝑇 ( 𝑗 ‘ 𝑃 ) = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝐺 ) ) ∧ ( ( 𝑄 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐷 ) ) ) ) ) ) |
33 |
1 2 3 5 6 7 8 4 12
|
cdlemksv |
⊢ ( 𝐺 ∈ 𝑇 → ( 𝑍 ‘ 𝐺 ) = ( ℩ 𝑗 ∈ 𝑇 ( 𝑗 ‘ 𝑃 ) = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝐺 ) ) ∧ ( ( 𝑄 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐷 ) ) ) ) ) ) |
34 |
33
|
adantl |
⊢ ( ( 𝐷 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) → ( 𝑍 ‘ 𝐺 ) = ( ℩ 𝑗 ∈ 𝑇 ( 𝑗 ‘ 𝑃 ) = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝐺 ) ) ∧ ( ( 𝑄 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐷 ) ) ) ) ) ) |
35 |
32 34
|
eqtr4d |
⊢ ( ( 𝐷 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) → ( 𝐷 𝑌 𝐺 ) = ( 𝑍 ‘ 𝐺 ) ) |