Theorem cdlemg6e

Description: TODO: FIX COMMENT. (Contributed by NM, 27-Apr-2013)

	Ref	Expression
Hypotheses	cdlemg4.l	⊢ ≤ = ( le ‘ 𝐾 )
	cdlemg4.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
	cdlemg4.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	cdlemg4.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
	cdlemg4.r	⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
	cdlemg4.j	⊢ ∨ = ( join ‘ 𝐾 )
	cdlemg4b.v	⊢ 𝑉 = ( 𝑅 ‘ 𝐺 )
Assertion	cdlemg6e	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) = 𝑄 )

Proof

Step	Hyp	Ref	Expression
1		cdlemg4.l	⊢ ≤ = ( le ‘ 𝐾 )
2		cdlemg4.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
3		cdlemg4.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4		cdlemg4.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
5		cdlemg4.r	⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
6		cdlemg4.j	⊢ ∨ = ( join ‘ 𝐾 )
7		cdlemg4b.v	⊢ 𝑉 = ( 𝑅 ‘ 𝐺 )
8		simp1	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
9		simp21	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
10		simp31	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → 𝐺 ∈ 𝑇 )
11	1 2 3 4	ltrnel	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐺 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( 𝐺 ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐺 ‘ 𝑃 ) ≤ 𝑊 ) )
12	8 10 9 11	syl3anc	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → ( ( 𝐺 ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐺 ‘ 𝑃 ) ≤ 𝑊 ) )
13	1 6 2 3	cdlemb3	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( ( 𝐺 ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐺 ‘ 𝑃 ) ≤ 𝑊 ) ) → ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ¬ 𝑟 ≤ ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ) )
14	8 9 12 13	syl3anc	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ¬ 𝑟 ≤ ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ) )
15	1 2 3 4 5 6 7	cdlemg6d	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → ( ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ¬ 𝑟 ≤ ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) = 𝑄 ) )
16	15	exp4c	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → ( 𝑟 ∈ 𝐴 → ( ¬ 𝑟 ≤ 𝑊 → ( ¬ 𝑟 ≤ ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) = 𝑄 ) ) ) )
17	16	imp4a	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → ( 𝑟 ∈ 𝐴 → ( ( ¬ 𝑟 ≤ 𝑊 ∧ ¬ 𝑟 ≤ ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) = 𝑄 ) ) )
18	17	rexlimdv	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → ( ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ¬ 𝑟 ≤ ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) = 𝑄 ) )
19	14 18	mpd	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) = 𝑄 )