Theorem polpmapN

Description: The polarity of a projective map. (Contributed by NM, 24-Jan-2012) (New usage is discouraged.)

	Ref	Expression
Hypotheses	polpmap.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	polpmap.o	⊢ ⊥ = ( oc ‘ 𝐾 )
	polpmap.m	⊢ 𝑀 = ( pmap ‘ 𝐾 )
	polpmap.p	⊢ 𝑃 = ( ⊥𝑃 ‘ 𝐾 )
Assertion	polpmapN	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) → ( 𝑃 ‘ ( 𝑀 ‘ 𝑋 ) ) = ( 𝑀 ‘ ( ⊥ ‘ 𝑋 ) ) )

Proof

Step	Hyp	Ref	Expression
1		polpmap.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		polpmap.o	⊢ ⊥ = ( oc ‘ 𝐾 )
3		polpmap.m	⊢ 𝑀 = ( pmap ‘ 𝐾 )
4		polpmap.p	⊢ 𝑃 = ( ⊥𝑃 ‘ 𝐾 )
5		eqid	⊢ ( Atoms ‘ 𝐾 ) = ( Atoms ‘ 𝐾 )
6	1 5 3	pmapssat	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) → ( 𝑀 ‘ 𝑋 ) ⊆ ( Atoms ‘ 𝐾 ) )
7		eqid	⊢ ( lub ‘ 𝐾 ) = ( lub ‘ 𝐾 )
8	7 2 5 3 4	polval2N	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑀 ‘ 𝑋 ) ⊆ ( Atoms ‘ 𝐾 ) ) → ( 𝑃 ‘ ( 𝑀 ‘ 𝑋 ) ) = ( 𝑀 ‘ ( ⊥ ‘ ( ( lub ‘ 𝐾 ) ‘ ( 𝑀 ‘ 𝑋 ) ) ) ) )
9	6 8	syldan	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) → ( 𝑃 ‘ ( 𝑀 ‘ 𝑋 ) ) = ( 𝑀 ‘ ( ⊥ ‘ ( ( lub ‘ 𝐾 ) ‘ ( 𝑀 ‘ 𝑋 ) ) ) ) )
10		eqid	⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
11	1 10 5 3	pmapval	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) → ( 𝑀 ‘ 𝑋 ) = { 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∣ 𝑝 ( le ‘ 𝐾 ) 𝑋 } )
12	11	fveq2d	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) → ( ( lub ‘ 𝐾 ) ‘ ( 𝑀 ‘ 𝑋 ) ) = ( ( lub ‘ 𝐾 ) ‘ { 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∣ 𝑝 ( le ‘ 𝐾 ) 𝑋 } ) )
13		hlomcmat	⊢ ( 𝐾 ∈ HL → ( 𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat ) )
14	1 10 7 5	atlatmstc	⊢ ( ( ( 𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat ) ∧ 𝑋 ∈ 𝐵 ) → ( ( lub ‘ 𝐾 ) ‘ { 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∣ 𝑝 ( le ‘ 𝐾 ) 𝑋 } ) = 𝑋 )
15	13 14	sylan	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) → ( ( lub ‘ 𝐾 ) ‘ { 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∣ 𝑝 ( le ‘ 𝐾 ) 𝑋 } ) = 𝑋 )
16	12 15	eqtrd	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) → ( ( lub ‘ 𝐾 ) ‘ ( 𝑀 ‘ 𝑋 ) ) = 𝑋 )
17	16	fveq2d	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) → ( ⊥ ‘ ( ( lub ‘ 𝐾 ) ‘ ( 𝑀 ‘ 𝑋 ) ) ) = ( ⊥ ‘ 𝑋 ) )
18	17	fveq2d	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) → ( 𝑀 ‘ ( ⊥ ‘ ( ( lub ‘ 𝐾 ) ‘ ( 𝑀 ‘ 𝑋 ) ) ) ) = ( 𝑀 ‘ ( ⊥ ‘ 𝑋 ) ) )
19	9 18	eqtrd	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) → ( 𝑃 ‘ ( 𝑀 ‘ 𝑋 ) ) = ( 𝑀 ‘ ( ⊥ ‘ 𝑋 ) ) )