Theorem trlcoabs

Description: Absorption into a composition by joining with trace. (Contributed by NM, 22-Jul-2013)

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

Proof

Step	Hyp	Ref	Expression
1		trlcoabs.l	⊢ ≤ = ( le ‘ 𝐾 )
2		trlcoabs.j	⊢ ∨ = ( join ‘ 𝐾 )
3		trlcoabs.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
4		trlcoabs.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
5		trlcoabs.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
6		trlcoabs.r	⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
7	1 3 4 5	ltrncoval	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑃 ∈ 𝐴 ) → ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑃 ) = ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) )
8	7	3adant3r	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑃 ) = ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) )
9	8	oveq1d	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑃 ) ∨ ( 𝑅 ‘ 𝐹 ) ) = ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝑅 ‘ 𝐹 ) ) )
10		simp1	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
11		simp2l	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝐹 ∈ 𝑇 )
12	1 3 4 5	ltrnel	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐺 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( 𝐺 ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐺 ‘ 𝑃 ) ≤ 𝑊 ) )
13	12	3adant2l	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( 𝐺 ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐺 ‘ 𝑃 ) ≤ 𝑊 ) )
14	1 2 3 4 5 6	trljat3	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝐺 ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐺 ‘ 𝑃 ) ≤ 𝑊 ) ) → ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝑅 ‘ 𝐹 ) ) = ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝑅 ‘ 𝐹 ) ) )
15	10 11 13 14	syl3anc	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝑅 ‘ 𝐹 ) ) = ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝑅 ‘ 𝐹 ) ) )
16	9 15	eqtr4d	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑃 ) ∨ ( 𝑅 ‘ 𝐹 ) ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝑅 ‘ 𝐹 ) ) )