This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 ) ) )