Metamath Proof Explorer


Theorem cvlatcvr1

Description: An atom is covered by its join with a different atom. (Contributed by NM, 5-Nov-2012)

Ref Expression
Hypotheses cvlatcvr1.j = ( join ‘ 𝐾 )
cvlatcvr1.c 𝐶 = ( ⋖ ‘ 𝐾 )
cvlatcvr1.a 𝐴 = ( Atoms ‘ 𝐾 )
Assertion cvlatcvr1 ( ( ( 𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat ) ∧ 𝑃𝐴𝑄𝐴 ) → ( 𝑃𝑄𝑃 𝐶 ( 𝑃 𝑄 ) ) )

Proof

Step Hyp Ref Expression
1 cvlatcvr1.j = ( join ‘ 𝐾 )
2 cvlatcvr1.c 𝐶 = ( ⋖ ‘ 𝐾 )
3 cvlatcvr1.a 𝐴 = ( Atoms ‘ 𝐾 )
4 simp13 ( ( ( 𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat ) ∧ 𝑃𝐴𝑄𝐴 ) → 𝐾 ∈ CvLat )
5 cvlatl ( 𝐾 ∈ CvLat → 𝐾 ∈ AtLat )
6 4 5 syl ( ( ( 𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat ) ∧ 𝑃𝐴𝑄𝐴 ) → 𝐾 ∈ AtLat )
7 eqid ( meet ‘ 𝐾 ) = ( meet ‘ 𝐾 )
8 eqid ( 0. ‘ 𝐾 ) = ( 0. ‘ 𝐾 )
9 7 8 3 atnem0 ( ( 𝐾 ∈ AtLat ∧ 𝑃𝐴𝑄𝐴 ) → ( 𝑃𝑄 ↔ ( 𝑃 ( meet ‘ 𝐾 ) 𝑄 ) = ( 0. ‘ 𝐾 ) ) )
10 6 9 syld3an1 ( ( ( 𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat ) ∧ 𝑃𝐴𝑄𝐴 ) → ( 𝑃𝑄 ↔ ( 𝑃 ( meet ‘ 𝐾 ) 𝑄 ) = ( 0. ‘ 𝐾 ) ) )
11 eqid ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
12 11 3 atbase ( 𝑃𝐴𝑃 ∈ ( Base ‘ 𝐾 ) )
13 11 1 7 8 2 3 cvlcvrp ( ( ( 𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat ) ∧ 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ 𝑄𝐴 ) → ( ( 𝑃 ( meet ‘ 𝐾 ) 𝑄 ) = ( 0. ‘ 𝐾 ) ↔ 𝑃 𝐶 ( 𝑃 𝑄 ) ) )
14 12 13 syl3an2 ( ( ( 𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat ) ∧ 𝑃𝐴𝑄𝐴 ) → ( ( 𝑃 ( meet ‘ 𝐾 ) 𝑄 ) = ( 0. ‘ 𝐾 ) ↔ 𝑃 𝐶 ( 𝑃 𝑄 ) ) )
15 10 14 bitrd ( ( ( 𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat ) ∧ 𝑃𝐴𝑄𝐴 ) → ( 𝑃𝑄𝑃 𝐶 ( 𝑃 𝑄 ) ) )