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 = ( 𝑓𝑇 ↦ ( 𝑖𝑇 ( 𝑖𝑃 ) = ( ( 𝑃 ( 𝑅𝑓 ) ) ( ( 𝑁𝑃 ) ( 𝑅 ‘ ( 𝑓 𝐹 ) ) ) ) ) )
cdlemk3.u1 𝑌 = ( 𝑑𝑇 , 𝑒𝑇 ↦ ( 𝑗𝑇 ( 𝑗𝑃 ) = ( ( 𝑃 ( 𝑅𝑒 ) ) ( ( ( 𝑆𝑑 ) ‘ 𝑃 ) ( 𝑅 ‘ ( 𝑒 𝑑 ) ) ) ) ) )
cdlemk3.o2 𝑄 = ( 𝑆𝐷 )
cdlemk3.u2 𝑍 = ( 𝑒𝑇 ↦ ( 𝑗𝑇 ( 𝑗𝑃 ) = ( ( 𝑃 ( 𝑅𝑒 ) ) ( ( 𝑄𝑃 ) ( 𝑅 ‘ ( 𝑒 𝐷 ) ) ) ) ) )
Assertion cdlemkuu ( ( 𝐷𝑇𝐺𝑇 ) → ( 𝐷 𝑌 𝐺 ) = ( 𝑍𝐺 ) )

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 = ( 𝑓𝑇 ↦ ( 𝑖𝑇 ( 𝑖𝑃 ) = ( ( 𝑃 ( 𝑅𝑓 ) ) ( ( 𝑁𝑃 ) ( 𝑅 ‘ ( 𝑓 𝐹 ) ) ) ) ) )
10 cdlemk3.u1 𝑌 = ( 𝑑𝑇 , 𝑒𝑇 ↦ ( 𝑗𝑇 ( 𝑗𝑃 ) = ( ( 𝑃 ( 𝑅𝑒 ) ) ( ( ( 𝑆𝑑 ) ‘ 𝑃 ) ( 𝑅 ‘ ( 𝑒 𝑑 ) ) ) ) ) )
11 cdlemk3.o2 𝑄 = ( 𝑆𝐷 )
12 cdlemk3.u2 𝑍 = ( 𝑒𝑇 ↦ ( 𝑗𝑇 ( 𝑗𝑃 ) = ( ( 𝑃 ( 𝑅𝑒 ) ) ( ( 𝑄𝑃 ) ( 𝑅 ‘ ( 𝑒 𝐷 ) ) ) ) ) )
13 fveq2 ( 𝑑 = 𝐷 → ( 𝑆𝑑 ) = ( 𝑆𝐷 ) )
14 13 11 eqtr4di ( 𝑑 = 𝐷 → ( 𝑆𝑑 ) = 𝑄 )
15 14 fveq1d ( 𝑑 = 𝐷 → ( ( 𝑆𝑑 ) ‘ 𝑃 ) = ( 𝑄𝑃 ) )
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 → ( ( ( 𝑆𝑑 ) ‘ 𝑃 ) ( 𝑅 ‘ ( 𝑒 𝑑 ) ) ) = ( ( 𝑄𝑃 ) ( 𝑅 ‘ ( 𝑒 𝐷 ) ) ) )
20 19 oveq2d ( 𝑑 = 𝐷 → ( ( 𝑃 ( 𝑅𝑒 ) ) ( ( ( 𝑆𝑑 ) ‘ 𝑃 ) ( 𝑅 ‘ ( 𝑒 𝑑 ) ) ) ) = ( ( 𝑃 ( 𝑅𝑒 ) ) ( ( 𝑄𝑃 ) ( 𝑅 ‘ ( 𝑒 𝐷 ) ) ) ) )
21 20 eqeq2d ( 𝑑 = 𝐷 → ( ( 𝑗𝑃 ) = ( ( 𝑃 ( 𝑅𝑒 ) ) ( ( ( 𝑆𝑑 ) ‘ 𝑃 ) ( 𝑅 ‘ ( 𝑒 𝑑 ) ) ) ) ↔ ( 𝑗𝑃 ) = ( ( 𝑃 ( 𝑅𝑒 ) ) ( ( 𝑄𝑃 ) ( 𝑅 ‘ ( 𝑒 𝐷 ) ) ) ) ) )
22 21 riotabidv ( 𝑑 = 𝐷 → ( 𝑗𝑇 ( 𝑗𝑃 ) = ( ( 𝑃 ( 𝑅𝑒 ) ) ( ( ( 𝑆𝑑 ) ‘ 𝑃 ) ( 𝑅 ‘ ( 𝑒 𝑑 ) ) ) ) ) = ( 𝑗𝑇 ( 𝑗𝑃 ) = ( ( 𝑃 ( 𝑅𝑒 ) ) ( ( 𝑄𝑃 ) ( 𝑅 ‘ ( 𝑒 𝐷 ) ) ) ) ) )
23 fveq2 ( 𝑒 = 𝐺 → ( 𝑅𝑒 ) = ( 𝑅𝐺 ) )
24 23 oveq2d ( 𝑒 = 𝐺 → ( 𝑃 ( 𝑅𝑒 ) ) = ( 𝑃 ( 𝑅𝐺 ) ) )
25 coeq1 ( 𝑒 = 𝐺 → ( 𝑒 𝐷 ) = ( 𝐺 𝐷 ) )
26 25 fveq2d ( 𝑒 = 𝐺 → ( 𝑅 ‘ ( 𝑒 𝐷 ) ) = ( 𝑅 ‘ ( 𝐺 𝐷 ) ) )
27 26 oveq2d ( 𝑒 = 𝐺 → ( ( 𝑄𝑃 ) ( 𝑅 ‘ ( 𝑒 𝐷 ) ) ) = ( ( 𝑄𝑃 ) ( 𝑅 ‘ ( 𝐺 𝐷 ) ) ) )
28 24 27 oveq12d ( 𝑒 = 𝐺 → ( ( 𝑃 ( 𝑅𝑒 ) ) ( ( 𝑄𝑃 ) ( 𝑅 ‘ ( 𝑒 𝐷 ) ) ) ) = ( ( 𝑃 ( 𝑅𝐺 ) ) ( ( 𝑄𝑃 ) ( 𝑅 ‘ ( 𝐺 𝐷 ) ) ) ) )
29 28 eqeq2d ( 𝑒 = 𝐺 → ( ( 𝑗𝑃 ) = ( ( 𝑃 ( 𝑅𝑒 ) ) ( ( 𝑄𝑃 ) ( 𝑅 ‘ ( 𝑒 𝐷 ) ) ) ) ↔ ( 𝑗𝑃 ) = ( ( 𝑃 ( 𝑅𝐺 ) ) ( ( 𝑄𝑃 ) ( 𝑅 ‘ ( 𝐺 𝐷 ) ) ) ) ) )
30 29 riotabidv ( 𝑒 = 𝐺 → ( 𝑗𝑇 ( 𝑗𝑃 ) = ( ( 𝑃 ( 𝑅𝑒 ) ) ( ( 𝑄𝑃 ) ( 𝑅 ‘ ( 𝑒 𝐷 ) ) ) ) ) = ( 𝑗𝑇 ( 𝑗𝑃 ) = ( ( 𝑃 ( 𝑅𝐺 ) ) ( ( 𝑄𝑃 ) ( 𝑅 ‘ ( 𝐺 𝐷 ) ) ) ) ) )
31 riotaex ( 𝑗𝑇 ( 𝑗𝑃 ) = ( ( 𝑃 ( 𝑅𝐺 ) ) ( ( 𝑄𝑃 ) ( 𝑅 ‘ ( 𝐺 𝐷 ) ) ) ) ) ∈ V
32 22 30 10 31 ovmpo ( ( 𝐷𝑇𝐺𝑇 ) → ( 𝐷 𝑌 𝐺 ) = ( 𝑗𝑇 ( 𝑗𝑃 ) = ( ( 𝑃 ( 𝑅𝐺 ) ) ( ( 𝑄𝑃 ) ( 𝑅 ‘ ( 𝐺 𝐷 ) ) ) ) ) )
33 1 2 3 5 6 7 8 4 12 cdlemksv ( 𝐺𝑇 → ( 𝑍𝐺 ) = ( 𝑗𝑇 ( 𝑗𝑃 ) = ( ( 𝑃 ( 𝑅𝐺 ) ) ( ( 𝑄𝑃 ) ( 𝑅 ‘ ( 𝐺 𝐷 ) ) ) ) ) )
34 33 adantl ( ( 𝐷𝑇𝐺𝑇 ) → ( 𝑍𝐺 ) = ( 𝑗𝑇 ( 𝑗𝑃 ) = ( ( 𝑃 ( 𝑅𝐺 ) ) ( ( 𝑄𝑃 ) ( 𝑅 ‘ ( 𝐺 𝐷 ) ) ) ) ) )
35 32 34 eqtr4d ( ( 𝐷𝑇𝐺𝑇 ) → ( 𝐷 𝑌 𝐺 ) = ( 𝑍𝐺 ) )