Theorem cdleme10tN

Description: Part of proof of Lemma E in Crawley p. 113, 2nd paragraph on p. 114. Y represents t_2. In their notation, we prove t \/ t_2 = t \/ r. (Contributed by NM, 8-Oct-2012) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cdleme10t.l	⊢ ≤ = ( le ‘ 𝐾 )
	cdleme10t.j	⊢ ∨ = ( join ‘ 𝐾 )
	cdleme10t.m	⊢ ∧ = ( meet ‘ 𝐾 )
	cdleme10t.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
	cdleme10t.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	cdleme10t.y	⊢ 𝑌 = ( ( 𝑅 ∨ 𝑇 ) ∧ 𝑊 )
Assertion	cdleme10tN	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑅 ∈ 𝐴 ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ) → ( 𝑇 ∨ 𝑌 ) = ( 𝑇 ∨ 𝑅 ) )

Proof

Step	Hyp	Ref	Expression
1		cdleme10t.l	⊢ ≤ = ( le ‘ 𝐾 )
2		cdleme10t.j	⊢ ∨ = ( join ‘ 𝐾 )
3		cdleme10t.m	⊢ ∧ = ( meet ‘ 𝐾 )
4		cdleme10t.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5		cdleme10t.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
6		cdleme10t.y	⊢ 𝑌 = ( ( 𝑅 ∨ 𝑇 ) ∧ 𝑊 )
7	1 2 3 4 5 6	cdleme10	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑅 ∈ 𝐴 ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ) → ( 𝑇 ∨ 𝑌 ) = ( 𝑇 ∨ 𝑅 ) )