Theorem polcon2N

Description: Contraposition law for polarity. (Contributed by NM, 23-Mar-2012) (New usage is discouraged.)

	Ref	Expression
Hypotheses	2polss.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
	2polss.p	⊢ ⊥ = ( ⊥𝑃 ‘ 𝐾 )
Assertion	polcon2N	⊢ ( ( 𝐾 ∈ HL ∧ 𝑌 ⊆ 𝐴 ∧ 𝑋 ⊆ ( ⊥ ‘ 𝑌 ) ) → 𝑌 ⊆ ( ⊥ ‘ 𝑋 ) )

Proof

Step	Hyp	Ref	Expression
1		2polss.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
2		2polss.p	⊢ ⊥ = ( ⊥𝑃 ‘ 𝐾 )
3	1 2	2polssN	⊢ ( ( 𝐾 ∈ HL ∧ 𝑌 ⊆ 𝐴 ) → 𝑌 ⊆ ( ⊥ ‘ ( ⊥ ‘ 𝑌 ) ) )
4	3	3adant3	⊢ ( ( 𝐾 ∈ HL ∧ 𝑌 ⊆ 𝐴 ∧ 𝑋 ⊆ ( ⊥ ‘ 𝑌 ) ) → 𝑌 ⊆ ( ⊥ ‘ ( ⊥ ‘ 𝑌 ) ) )
5	1 2	polssatN	⊢ ( ( 𝐾 ∈ HL ∧ 𝑌 ⊆ 𝐴 ) → ( ⊥ ‘ 𝑌 ) ⊆ 𝐴 )
6	5	3adant3	⊢ ( ( 𝐾 ∈ HL ∧ 𝑌 ⊆ 𝐴 ∧ 𝑋 ⊆ ( ⊥ ‘ 𝑌 ) ) → ( ⊥ ‘ 𝑌 ) ⊆ 𝐴 )
7	1 2	polcon3N	⊢ ( ( 𝐾 ∈ HL ∧ ( ⊥ ‘ 𝑌 ) ⊆ 𝐴 ∧ 𝑋 ⊆ ( ⊥ ‘ 𝑌 ) ) → ( ⊥ ‘ ( ⊥ ‘ 𝑌 ) ) ⊆ ( ⊥ ‘ 𝑋 ) )
8	6 7	syld3an2	⊢ ( ( 𝐾 ∈ HL ∧ 𝑌 ⊆ 𝐴 ∧ 𝑋 ⊆ ( ⊥ ‘ 𝑌 ) ) → ( ⊥ ‘ ( ⊥ ‘ 𝑌 ) ) ⊆ ( ⊥ ‘ 𝑋 ) )
9	4 8	sstrd	⊢ ( ( 𝐾 ∈ HL ∧ 𝑌 ⊆ 𝐴 ∧ 𝑋 ⊆ ( ⊥ ‘ 𝑌 ) ) → 𝑌 ⊆ ( ⊥ ‘ 𝑋 ) )