Theorem trlval5

Description: The value of the trace of a lattice translation in terms of itself. (Contributed by NM, 19-Jul-2013)

	Ref	Expression
Hypotheses	trlval3.l	⊢ ≤ = ( le ‘ 𝐾 )
	trlval3.j	⊢ ∨ = ( join ‘ 𝐾 )
	trlval3.m	⊢ ∧ = ( meet ‘ 𝐾 )
	trlval3.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
	trlval3.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	trlval3.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
	trlval3.r	⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
Assertion	trlval5	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝑅 ‘ 𝐹 ) = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) ∧ 𝑊 ) )

Proof

Step	Hyp	Ref	Expression
1		trlval3.l	⊢ ≤ = ( le ‘ 𝐾 )
2		trlval3.j	⊢ ∨ = ( join ‘ 𝐾 )
3		trlval3.m	⊢ ∧ = ( meet ‘ 𝐾 )
4		trlval3.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5		trlval3.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
6		trlval3.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
7		trlval3.r	⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
8	1 2 3 4 5 6 7	trlval2	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝑅 ‘ 𝐹 ) = ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ 𝑊 ) )
9	1 2 4 5 6 7	trljat1	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) = ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) )
10	9	oveq1d	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) ∧ 𝑊 ) = ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ 𝑊 ) )
11	8 10	eqtr4d	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝑅 ‘ 𝐹 ) = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) ∧ 𝑊 ) )