Theorem cdlemg17j

Description: TODO: fix comment. (Contributed by NM, 11-May-2013)

	Ref	Expression
Hypotheses	cdlemg12.l	⊢ ≤ = ( le ‘ 𝐾 )
	cdlemg12.j	⊢ ∨ = ( join ‘ 𝐾 )
	cdlemg12.m	⊢ ∧ = ( meet ‘ 𝐾 )
	cdlemg12.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
	cdlemg12.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	cdlemg12.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
	cdlemg12b.r	⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
Assertion	cdlemg17j	⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) ) ) → ( 𝐺 ‘ ( 𝐹 ‘ 𝑃 ) ) = ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) )

Proof

Step	Hyp	Ref	Expression
1		cdlemg12.l	⊢ ≤ = ( le ‘ 𝐾 )
2		cdlemg12.j	⊢ ∨ = ( join ‘ 𝐾 )
3		cdlemg12.m	⊢ ∧ = ( meet ‘ 𝐾 )
4		cdlemg12.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5		cdlemg12.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
6		cdlemg12.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
7		cdlemg12b.r	⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
8	1 2 3 4 5 6 7	cdlemg17i	⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) ) ) → ( 𝐺 ‘ ( 𝐹 ‘ 𝑃 ) ) = ( 𝐹 ‘ 𝑄 ) )
9	1 2 3 4 5 6 7	cdlemg17ir	⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) ) ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = ( 𝐹 ‘ 𝑄 ) )
10	8 9	eqtr4d	⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) ) ) → ( 𝐺 ‘ ( 𝐹 ‘ 𝑃 ) ) = ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) )