Theorem trljat3

Description: The value of a translation of an atom P not under the fiducial co-atom W , joined with trace. Equation above Lemma C in Crawley p. 112. (Contributed by NM, 22-May-2012)

	Ref	Expression
Hypotheses	trljat.l	⊢ ≤ = ( le ‘ 𝐾 )
	trljat.j	⊢ ∨ = ( join ‘ 𝐾 )
	trljat.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
	trljat.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	trljat.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
	trljat.r	⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
Assertion	trljat3	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) = ( ( 𝐹 ‘ 𝑃 ) ∨ ( 𝑅 ‘ 𝐹 ) ) )

Proof

Step	Hyp	Ref	Expression
1		trljat.l	⊢ ≤ = ( le ‘ 𝐾 )
2		trljat.j	⊢ ∨ = ( join ‘ 𝐾 )
3		trljat.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
4		trljat.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
5		trljat.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
6		trljat.r	⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
7	1 2 3 4 5 6	trljat1	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) = ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) )
8	1 2 3 4 5 6	trljat2	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( 𝐹 ‘ 𝑃 ) ∨ ( 𝑅 ‘ 𝐹 ) ) = ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) )
9	7 8	eqtr4d	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) = ( ( 𝐹 ‘ 𝑃 ) ∨ ( 𝑅 ‘ 𝐹 ) ) )