Metamath Proof Explorer

Theorem 2polcon4bN

Description: Contraposition law for polarity. (Contributed by NM, 6-Mar-2012) (New usage is discouraged.)

Ref Expression
Hypotheses 2polss.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
2polss.p ⊢ ⊥ = ( ⊥𝑃 ‘ 𝐾 )
Assertion 2polcon4bN ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) → ( ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) ⊆ ( ⊥ ‘ ( ⊥ ‘ 𝑌 ) ) ↔ ( ⊥ ‘ 𝑌 ) ⊆ ( ⊥ ‘ 𝑋 ) ) )

Proof

Step Hyp Ref Expression
1 2polss.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
2 2polss.p ⊢ ⊥ = ( ⊥𝑃 ‘ 𝐾 )
3 simpl1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) ⊆ ( ⊥ ‘ ( ⊥ ‘ 𝑌 ) ) ) → 𝐾 ∈ HL )
4 simp1 ⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) → 𝐾 ∈ HL )
5 1 2 polssatN ⊢ ( ( 𝐾 ∈ HL ∧ 𝑌 ⊆ 𝐴 ) → ( ⊥ ‘ 𝑌 ) ⊆ 𝐴 )
6 5 3adant2 ⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) → ( ⊥ ‘ 𝑌 ) ⊆ 𝐴 )
7 1 2 polssatN ⊢ ( ( 𝐾 ∈ HL ∧ ( ⊥ ‘ 𝑌 ) ⊆ 𝐴 ) → ( ⊥ ‘ ( ⊥ ‘ 𝑌 ) ) ⊆ 𝐴 )
8 4 6 7 syl2anc ⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) → ( ⊥ ‘ ( ⊥ ‘ 𝑌 ) ) ⊆ 𝐴 )
9 8 adantr ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) ⊆ ( ⊥ ‘ ( ⊥ ‘ 𝑌 ) ) ) → ( ⊥ ‘ ( ⊥ ‘ 𝑌 ) ) ⊆ 𝐴 )
10 simpr ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) ⊆ ( ⊥ ‘ ( ⊥ ‘ 𝑌 ) ) ) → ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) ⊆ ( ⊥ ‘ ( ⊥ ‘ 𝑌 ) ) )
11 1 2 polcon3N ⊢ ( ( 𝐾 ∈ HL ∧ ( ⊥ ‘ ( ⊥ ‘ 𝑌 ) ) ⊆ 𝐴 ∧ ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) ⊆ ( ⊥ ‘ ( ⊥ ‘ 𝑌 ) ) ) → ( ⊥ ‘ ( ⊥ ‘ ( ⊥ ‘ 𝑌 ) ) ) ⊆ ( ⊥ ‘ ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) ) )
12 3 9 10 11 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) ⊆ ( ⊥ ‘ ( ⊥ ‘ 𝑌 ) ) ) → ( ⊥ ‘ ( ⊥ ‘ ( ⊥ ‘ 𝑌 ) ) ) ⊆ ( ⊥ ‘ ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) ) )
13 12 ex ⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) → ( ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) ⊆ ( ⊥ ‘ ( ⊥ ‘ 𝑌 ) ) → ( ⊥ ‘ ( ⊥ ‘ ( ⊥ ‘ 𝑌 ) ) ) ⊆ ( ⊥ ‘ ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) ) ) )
14 1 2 3polN ⊢ ( ( 𝐾 ∈ HL ∧ 𝑌 ⊆ 𝐴 ) → ( ⊥ ‘ ( ⊥ ‘ ( ⊥ ‘ 𝑌 ) ) ) = ( ⊥ ‘ 𝑌 ) )
15 14 3adant2 ⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) → ( ⊥ ‘ ( ⊥ ‘ ( ⊥ ‘ 𝑌 ) ) ) = ( ⊥ ‘ 𝑌 ) )
16 1 2 3polN ⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ) → ( ⊥ ‘ ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) ) = ( ⊥ ‘ 𝑋 ) )
17 16 3adant3 ⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) → ( ⊥ ‘ ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) ) = ( ⊥ ‘ 𝑋 ) )
18 15 17 sseq12d ⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) → ( ( ⊥ ‘ ( ⊥ ‘ ( ⊥ ‘ 𝑌 ) ) ) ⊆ ( ⊥ ‘ ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) ) ↔ ( ⊥ ‘ 𝑌 ) ⊆ ( ⊥ ‘ 𝑋 ) ) )
19 13 18 sylibd ⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) → ( ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) ⊆ ( ⊥ ‘ ( ⊥ ‘ 𝑌 ) ) → ( ⊥ ‘ 𝑌 ) ⊆ ( ⊥ ‘ 𝑋 ) ) )
20 simpl1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ ( ⊥ ‘ 𝑌 ) ⊆ ( ⊥ ‘ 𝑋 ) ) → 𝐾 ∈ HL )
21 1 2 polssatN ⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ) → ( ⊥ ‘ 𝑋 ) ⊆ 𝐴 )
22 21 3adant3 ⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) → ( ⊥ ‘ 𝑋 ) ⊆ 𝐴 )
23 22 adantr ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ ( ⊥ ‘ 𝑌 ) ⊆ ( ⊥ ‘ 𝑋 ) ) → ( ⊥ ‘ 𝑋 ) ⊆ 𝐴 )
24 simpr ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ ( ⊥ ‘ 𝑌 ) ⊆ ( ⊥ ‘ 𝑋 ) ) → ( ⊥ ‘ 𝑌 ) ⊆ ( ⊥ ‘ 𝑋 ) )
25 1 2 polcon3N ⊢ ( ( 𝐾 ∈ HL ∧ ( ⊥ ‘ 𝑋 ) ⊆ 𝐴 ∧ ( ⊥ ‘ 𝑌 ) ⊆ ( ⊥ ‘ 𝑋 ) ) → ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) ⊆ ( ⊥ ‘ ( ⊥ ‘ 𝑌 ) ) )
26 20 23 24 25 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ ( ⊥ ‘ 𝑌 ) ⊆ ( ⊥ ‘ 𝑋 ) ) → ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) ⊆ ( ⊥ ‘ ( ⊥ ‘ 𝑌 ) ) )
27 26 ex ⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) → ( ( ⊥ ‘ 𝑌 ) ⊆ ( ⊥ ‘ 𝑋 ) → ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) ⊆ ( ⊥ ‘ ( ⊥ ‘ 𝑌 ) ) ) )
28 19 27 impbid ⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) → ( ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) ⊆ ( ⊥ ‘ ( ⊥ ‘ 𝑌 ) ) ↔ ( ⊥ ‘ 𝑌 ) ⊆ ( ⊥ ‘ 𝑋 ) ) )