Theorem cvr2N

Description: Less-than and covers equivalence in a Hilbert lattice. ( chcv2 analog.) (Contributed by NM, 7-Feb-2012) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cvr2.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	cvr2.s	⊢ < = ( lt ‘ 𝐾 )
	cvr2.j	⊢ ∨ = ( join ‘ 𝐾 )
	cvr2.c	⊢ 𝐶 = ( ⋖ ‘ 𝐾 )
	cvr2.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
Assertion	cvr2N	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) → ( 𝑋 < ( 𝑋 ∨ 𝑃 ) ↔ 𝑋 𝐶 ( 𝑋 ∨ 𝑃 ) ) )

Proof

Step	Hyp	Ref	Expression
1		cvr2.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		cvr2.s	⊢ < = ( lt ‘ 𝐾 )
3		cvr2.j	⊢ ∨ = ( join ‘ 𝐾 )
4		cvr2.c	⊢ 𝐶 = ( ⋖ ‘ 𝐾 )
5		cvr2.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
6		hllat	⊢ ( 𝐾 ∈ HL → 𝐾 ∈ Lat )
7	6	3ad2ant1	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) → 𝐾 ∈ Lat )
8		simp2	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) → 𝑋 ∈ 𝐵 )
9	1 5	atbase	⊢ ( 𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵 )
10	9	3ad2ant3	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) → 𝑃 ∈ 𝐵 )
11		eqid	⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
12	1 11 2 3	latnle	⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐵 ) → ( ¬ 𝑃 ( le ‘ 𝐾 ) 𝑋 ↔ 𝑋 < ( 𝑋 ∨ 𝑃 ) ) )
13	7 8 10 12	syl3anc	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) → ( ¬ 𝑃 ( le ‘ 𝐾 ) 𝑋 ↔ 𝑋 < ( 𝑋 ∨ 𝑃 ) ) )
14	1 11 3 4 5	cvr1	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) → ( ¬ 𝑃 ( le ‘ 𝐾 ) 𝑋 ↔ 𝑋 𝐶 ( 𝑋 ∨ 𝑃 ) ) )
15	13 14	bitr3d	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) → ( 𝑋 < ( 𝑋 ∨ 𝑃 ) ↔ 𝑋 𝐶 ( 𝑋 ∨ 𝑃 ) ) )