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Metamath Proof Explorer

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 ) ) )