Metamath Proof Explorer

Theorem cdlemkuu

Description: Convert between function and operation forms of Y . TODO: Use operation form everywhere. (Contributed by NM, 6-Jul-2013)

Ref Expression
Hypotheses cdlemk3.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
cdlemk3.l ⊢ ≤ = ( le ‘ 𝐾 )
cdlemk3.j ⊢ ∨ = ( join ‘ 𝐾 )
cdlemk3.m ⊢ ∧ = ( meet ‘ 𝐾 )
cdlemk3.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdlemk3.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdlemk3.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
cdlemk3.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
cdlemk3.s ⊢ 𝑆 = ( 𝑓 ∈ 𝑇 ↦ ( ℩ 𝑖 ∈ 𝑇 ( 𝑖 ‘ 𝑃 ) = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝑓 ) ) ∧ ( ( 𝑁 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝑓 ∘ ◡ 𝐹 ) ) ) ) ) )
cdlemk3.u1 ⊢ 𝑌 = ( 𝑑 ∈ 𝑇 , 𝑒 ∈ 𝑇 ↦ ( ℩ 𝑗 ∈ 𝑇 ( 𝑗 ‘ 𝑃 ) = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝑒 ) ) ∧ ( ( ( 𝑆 ‘ 𝑑 ) ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝑒 ∘ ◡ 𝑑 ) ) ) ) ) )
cdlemk3.o2 ⊢ 𝑄 = ( 𝑆 ‘ 𝐷 )
cdlemk3.u2 ⊢ 𝑍 = ( 𝑒 ∈ 𝑇 ↦ ( ℩ 𝑗 ∈ 𝑇 ( 𝑗 ‘ 𝑃 ) = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝑒 ) ) ∧ ( ( 𝑄 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝑒 ∘ ◡ 𝐷 ) ) ) ) ) )
Assertion cdlemkuu ( ( 𝐷 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) → ( 𝐷 𝑌 𝐺 ) = ( 𝑍 ‘ 𝐺 ) )

Proof

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 ) )