Theorem cdleme17a

Description: Part of proof of Lemma E in Crawley p. 114, first part of 4th paragraph. F , G , and C represent f(s), f_s(p), and s_1 respectively. We show, in their notation, f_s(p)=(p \/ q) /\ (q \/ s_1). (Contributed by NM, 11-Oct-2012)

	Ref	Expression
Hypotheses	cdleme17.l	⊢ ≤ = ( le ‘ 𝐾 )
	cdleme17.j	⊢ ∨ = ( join ‘ 𝐾 )
	cdleme17.m	⊢ ∧ = ( meet ‘ 𝐾 )
	cdleme17.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
	cdleme17.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	cdleme17.u	⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
	cdleme17.f	⊢ 𝐹 = ( ( 𝑆 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ) )
	cdleme17.g	⊢ 𝐺 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐹 ∨ ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ) )
	cdleme17.c	⊢ 𝐶 = ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 )
Assertion	cdleme17a	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝐺 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑄 ∨ 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		cdleme17.l	⊢ ≤ = ( le ‘ 𝐾 )
2		cdleme17.j	⊢ ∨ = ( join ‘ 𝐾 )
3		cdleme17.m	⊢ ∧ = ( meet ‘ 𝐾 )
4		cdleme17.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5		cdleme17.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
6		cdleme17.u	⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
7		cdleme17.f	⊢ 𝐹 = ( ( 𝑆 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ) )
8		cdleme17.g	⊢ 𝐺 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐹 ∨ ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ) )
9		cdleme17.c	⊢ 𝐶 = ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 )
10	1 2 3 4 5 6 7 8 9	cdleme7a	⊢ 𝐺 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐹 ∨ 𝐶 ) )
11	1 2 3 4 5 6 7 9	cdleme9	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) → ( 𝐹 ∨ 𝐶 ) = ( 𝑄 ∨ 𝐶 ) )
12	11	oveq2d	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) → ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐹 ∨ 𝐶 ) ) = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑄 ∨ 𝐶 ) ) )
13	10 12	eqtrid	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝐺 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑄 ∨ 𝐶 ) ) )