Theorem cdleme9tN

Description: Part of proof of Lemma E in Crawley p. 113, 2nd paragraph on p. 114. X and F represent t_1 and f(t) respectively. In their notation, we prove f(t) \/ t_1 = q \/ t_1. (Contributed by NM, 8-Oct-2012) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cdleme9t.l	⊢ ≤ = ( le ‘ 𝐾 )
	cdleme9t.j	⊢ ∨ = ( join ‘ 𝐾 )
	cdleme9t.m	⊢ ∧ = ( meet ‘ 𝐾 )
	cdleme9t.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
	cdleme9t.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	cdleme9t.u	⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
	cdleme9t.g	⊢ 𝐹 = ( ( 𝑇 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑇 ) ∧ 𝑊 ) ) )
	cdleme9t.x	⊢ 𝑋 = ( ( 𝑃 ∨ 𝑇 ) ∧ 𝑊 )
Assertion	cdleme9tN	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ) → ( 𝐹 ∨ 𝑋 ) = ( 𝑄 ∨ 𝑋 ) )

Proof

Step	Hyp	Ref	Expression
1		cdleme9t.l	⊢ ≤ = ( le ‘ 𝐾 )
2		cdleme9t.j	⊢ ∨ = ( join ‘ 𝐾 )
3		cdleme9t.m	⊢ ∧ = ( meet ‘ 𝐾 )
4		cdleme9t.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5		cdleme9t.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
6		cdleme9t.u	⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
7		cdleme9t.g	⊢ 𝐹 = ( ( 𝑇 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑇 ) ∧ 𝑊 ) ) )
8		cdleme9t.x	⊢ 𝑋 = ( ( 𝑃 ∨ 𝑇 ) ∧ 𝑊 )
9	1 2 3 4 5 6 7 8	cdleme9	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ) → ( 𝐹 ∨ 𝑋 ) = ( 𝑄 ∨ 𝑋 ) )