Theorem cdlemg2cex

Description: Any translation is one of our F s. TODO: fix comment, move to its own block maybe? Would this help for cdlemf ? (Contributed by NM, 22-Apr-2013)

	Ref	Expression
Hypotheses	cdlemg2.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	cdlemg2.l	⊢ ≤ = ( le ‘ 𝐾 )
	cdlemg2.j	⊢ ∨ = ( join ‘ 𝐾 )
	cdlemg2.m	⊢ ∧ = ( meet ‘ 𝐾 )
	cdlemg2.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
	cdlemg2.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	cdlemg2.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
	cdlemg2ex.u	⊢ 𝑈 = ( ( 𝑝 ∨ 𝑞 ) ∧ 𝑊 )
	cdlemg2ex.d	⊢ 𝐷 = ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑞 ∨ ( ( 𝑝 ∨ 𝑡 ) ∧ 𝑊 ) ) )
	cdlemg2ex.e	⊢ 𝐸 = ( ( 𝑝 ∨ 𝑞 ) ∧ ( 𝐷 ∨ ( ( 𝑠 ∨ 𝑡 ) ∧ 𝑊 ) ) )
	cdlemg2ex.g	⊢ 𝐺 = ( 𝑥 ∈ 𝐵 ↦ if ( ( 𝑝 ≠ 𝑞 ∧ ¬ 𝑥 ≤ 𝑊 ) , ( ℩ 𝑧 ∈ 𝐵 ∀ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ∨ ( 𝑥 ∧ 𝑊 ) ) = 𝑥 ) → 𝑧 = ( if ( 𝑠 ≤ ( 𝑝 ∨ 𝑞 ) , ( ℩ 𝑦 ∈ 𝐵 ∀ 𝑡 ∈ 𝐴 ( ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑝 ∨ 𝑞 ) ) → 𝑦 = 𝐸 ) ) , ⦋ 𝑠 / 𝑡 ⦌ 𝐷 ) ∨ ( 𝑥 ∧ 𝑊 ) ) ) ) , 𝑥 ) )
Assertion	cdlemg2cex	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( 𝐹 ∈ 𝑇 ↔ ∃ 𝑝 ∈ 𝐴 ∃ 𝑞 ∈ 𝐴 ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = 𝐺 ) ) )

Proof

Step	Hyp	Ref	Expression
1		cdlemg2.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		cdlemg2.l	⊢ ≤ = ( le ‘ 𝐾 )
3		cdlemg2.j	⊢ ∨ = ( join ‘ 𝐾 )
4		cdlemg2.m	⊢ ∧ = ( meet ‘ 𝐾 )
5		cdlemg2.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
6		cdlemg2.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
7		cdlemg2.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
8		cdlemg2ex.u	⊢ 𝑈 = ( ( 𝑝 ∨ 𝑞 ) ∧ 𝑊 )
9		cdlemg2ex.d	⊢ 𝐷 = ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑞 ∨ ( ( 𝑝 ∨ 𝑡 ) ∧ 𝑊 ) ) )
10		cdlemg2ex.e	⊢ 𝐸 = ( ( 𝑝 ∨ 𝑞 ) ∧ ( 𝐷 ∨ ( ( 𝑠 ∨ 𝑡 ) ∧ 𝑊 ) ) )
11		cdlemg2ex.g	⊢ 𝐺 = ( 𝑥 ∈ 𝐵 ↦ if ( ( 𝑝 ≠ 𝑞 ∧ ¬ 𝑥 ≤ 𝑊 ) , ( ℩ 𝑧 ∈ 𝐵 ∀ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ∨ ( 𝑥 ∧ 𝑊 ) ) = 𝑥 ) → 𝑧 = ( if ( 𝑠 ≤ ( 𝑝 ∨ 𝑞 ) , ( ℩ 𝑦 ∈ 𝐵 ∀ 𝑡 ∈ 𝐴 ( ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑝 ∨ 𝑞 ) ) → 𝑦 = 𝐸 ) ) , ⦋ 𝑠 / 𝑡 ⦌ 𝐷 ) ∨ ( 𝑥 ∧ 𝑊 ) ) ) ) , 𝑥 ) )
12	2 5 6 7	cdlemg1cex	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( 𝐹 ∈ 𝑇 ↔ ∃ 𝑝 ∈ 𝐴 ∃ 𝑞 ∈ 𝐴 ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑝 ) = 𝑞 ) ) ) )
13		simplll	⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴 ) ) ∧ ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ) ) → 𝐾 ∈ HL )
14		simpllr	⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴 ) ) ∧ ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ) ) → 𝑊 ∈ 𝐻 )
15		simplrl	⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴 ) ) ∧ ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ) ) → 𝑝 ∈ 𝐴 )
16		simprl	⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴 ) ) ∧ ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ) ) → ¬ 𝑝 ≤ 𝑊 )
17		simplrr	⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴 ) ) ∧ ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ) ) → 𝑞 ∈ 𝐴 )
18		simprr	⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴 ) ) ∧ ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ) ) → ¬ 𝑞 ≤ 𝑊 )
19		eqid	⊢ ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑝 ) = 𝑞 ) = ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑝 ) = 𝑞 )
20	1 2 3 4 5 6 8 9 10 11 7 19	cdlemg1b2	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ ( 𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊 ) ) → ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑝 ) = 𝑞 ) = 𝐺 )
21	13 14 15 16 17 18 20	syl222anc	⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴 ) ) ∧ ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ) ) → ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑝 ) = 𝑞 ) = 𝐺 )
22	21	eqeq2d	⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴 ) ) ∧ ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ) ) → ( 𝐹 = ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑝 ) = 𝑞 ) ↔ 𝐹 = 𝐺 ) )
23	22	pm5.32da	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴 ) ) → ( ( ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ) ∧ 𝐹 = ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑝 ) = 𝑞 ) ) ↔ ( ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ) ∧ 𝐹 = 𝐺 ) ) )
24		df-3an	⊢ ( ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑝 ) = 𝑞 ) ) ↔ ( ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ) ∧ 𝐹 = ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑝 ) = 𝑞 ) ) )
25		df-3an	⊢ ( ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = 𝐺 ) ↔ ( ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ) ∧ 𝐹 = 𝐺 ) )
26	23 24 25	3bitr4g	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴 ) ) → ( ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑝 ) = 𝑞 ) ) ↔ ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = 𝐺 ) ) )
27	26	2rexbidva	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( ∃ 𝑝 ∈ 𝐴 ∃ 𝑞 ∈ 𝐴 ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑝 ) = 𝑞 ) ) ↔ ∃ 𝑝 ∈ 𝐴 ∃ 𝑞 ∈ 𝐴 ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = 𝐺 ) ) )
28	12 27	bitrd	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( 𝐹 ∈ 𝑇 ↔ ∃ 𝑝 ∈ 𝐴 ∃ 𝑞 ∈ 𝐴 ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = 𝐺 ) ) )