Theorem cdlemefs29pre00N

Description: FIX COMMENT. TODO: see if this is the optimal utility theorem using lhpmat . (Contributed by NM, 27-Mar-2013) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cdlemefs29.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	cdlemefs29.l	⊢ ≤ = ( le ‘ 𝐾 )
	cdlemefs29.j	⊢ ∨ = ( join ‘ 𝐾 )
	cdlemefs29.m	⊢ ∧ = ( meet ‘ 𝐾 )
	cdlemefs29.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
	cdlemefs29.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
Assertion	cdlemefs29pre00N	⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑠 ∈ 𝐴 ) → ( ( ( ¬ 𝑠 ≤ 𝑊 ∧ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑠 ∨ ( 𝑅 ∧ 𝑊 ) ) = 𝑅 ) ↔ ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ∨ ( 𝑅 ∧ 𝑊 ) ) = 𝑅 ) ) )

Proof

Step	Hyp	Ref	Expression
1		cdlemefs29.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		cdlemefs29.l	⊢ ≤ = ( le ‘ 𝐾 )
3		cdlemefs29.j	⊢ ∨ = ( join ‘ 𝐾 )
4		cdlemefs29.m	⊢ ∧ = ( meet ‘ 𝐾 )
5		cdlemefs29.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
6		cdlemefs29.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
7		breq1	⊢ ( 𝑠 = 𝑅 → ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ↔ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) )
8	1 2 3 4 5 6 7	cdlemefrs29pre00	⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑠 ∈ 𝐴 ) → ( ( ( ¬ 𝑠 ≤ 𝑊 ∧ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑠 ∨ ( 𝑅 ∧ 𝑊 ) ) = 𝑅 ) ↔ ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ∨ ( 𝑅 ∧ 𝑊 ) ) = 𝑅 ) ) )