Metamath Proof Explorer


Theorem lhpmcvr2

Description: Alternate way to express that the meet of a lattice hyperplane with an element not under it is covered by the element. (Contributed by NM, 9-Apr-2013)

Ref Expression
Hypotheses lhpmcvr2.b 𝐵 = ( Base ‘ 𝐾 )
lhpmcvr2.l = ( le ‘ 𝐾 )
lhpmcvr2.j = ( join ‘ 𝐾 )
lhpmcvr2.m = ( meet ‘ 𝐾 )
lhpmcvr2.a 𝐴 = ( Atoms ‘ 𝐾 )
lhpmcvr2.h 𝐻 = ( LHyp ‘ 𝐾 )
Assertion lhpmcvr2 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ) → ∃ 𝑝𝐴 ( ¬ 𝑝 𝑊 ∧ ( 𝑝 ( 𝑋 𝑊 ) ) = 𝑋 ) )

Proof

Step Hyp Ref Expression
1 lhpmcvr2.b 𝐵 = ( Base ‘ 𝐾 )
2 lhpmcvr2.l = ( le ‘ 𝐾 )
3 lhpmcvr2.j = ( join ‘ 𝐾 )
4 lhpmcvr2.m = ( meet ‘ 𝐾 )
5 lhpmcvr2.a 𝐴 = ( Atoms ‘ 𝐾 )
6 lhpmcvr2.h 𝐻 = ( LHyp ‘ 𝐾 )
7 eqid ( ⋖ ‘ 𝐾 ) = ( ⋖ ‘ 𝐾 )
8 1 2 4 7 6 lhpmcvr ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ) → ( 𝑋 𝑊 ) ( ⋖ ‘ 𝐾 ) 𝑋 )
9 simpll ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ) → 𝐾 ∈ HL )
10 simprl ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ) → 𝑋𝐵 )
11 1 6 lhpbase ( 𝑊𝐻𝑊𝐵 )
12 11 ad2antlr ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ) → 𝑊𝐵 )
13 1 2 3 4 7 5 cvrval5 ( ( 𝐾 ∈ HL ∧ 𝑋𝐵𝑊𝐵 ) → ( ( 𝑋 𝑊 ) ( ⋖ ‘ 𝐾 ) 𝑋 ↔ ∃ 𝑝𝐴 ( ¬ 𝑝 𝑊 ∧ ( 𝑝 ( 𝑋 𝑊 ) ) = 𝑋 ) ) )
14 9 10 12 13 syl3anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ) → ( ( 𝑋 𝑊 ) ( ⋖ ‘ 𝐾 ) 𝑋 ↔ ∃ 𝑝𝐴 ( ¬ 𝑝 𝑊 ∧ ( 𝑝 ( 𝑋 𝑊 ) ) = 𝑋 ) ) )
15 8 14 mpbid ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ) → ∃ 𝑝𝐴 ( ¬ 𝑝 𝑊 ∧ ( 𝑝 ( 𝑋 𝑊 ) ) = 𝑋 ) )