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 ) )

Metamath Proof Explorer

Theorem cdlemkuu

Proof