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Metamath Proof Explorer

Theorem 2polpmapN

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

Ref Expression
Hypotheses 2polpmap.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2polpmap.m ⊢ 𝑀 = ( pmap ‘ 𝐾 )
2polpmap.p ⊢ ⊥ = ( ⊥𝑃 ‘ 𝐾 )
Assertion 2polpmapN ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) → ( ⊥ ‘ ( ⊥ ‘ ( 𝑀 ‘ 𝑋 ) ) ) = ( 𝑀 ‘ 𝑋 ) )

Proof

Step Hyp Ref Expression
1 2polpmap.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 2polpmap.m ⊢ 𝑀 = ( pmap ‘ 𝐾 )
3 2polpmap.p ⊢ ⊥ = ( ⊥𝑃 ‘ 𝐾 )
4 eqid ⊢ ( oc ‘ 𝐾 ) = ( oc ‘ 𝐾 )
5 1 4 2 3 polpmapN ⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) → ( ⊥ ‘ ( 𝑀 ‘ 𝑋 ) ) = ( 𝑀 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ) )
6 5 fveq2d ⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) → ( ⊥ ‘ ( ⊥ ‘ ( 𝑀 ‘ 𝑋 ) ) ) = ( ⊥ ‘ ( 𝑀 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ) ) )
7 hlop ⊢ ( 𝐾 ∈ HL → 𝐾 ∈ OP )
8 1 4 opoccl ⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ) → ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∈ 𝐵 )
9 7 8 sylan ⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) → ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∈ 𝐵 )
10 1 4 2 3 polpmapN ⊢ ( ( 𝐾 ∈ HL ∧ ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∈ 𝐵 ) → ( ⊥ ‘ ( 𝑀 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ) ) = ( 𝑀 ‘ ( ( oc ‘ 𝐾 ) ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ) ) )
11 9 10 syldan ⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) → ( ⊥ ‘ ( 𝑀 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ) ) = ( 𝑀 ‘ ( ( oc ‘ 𝐾 ) ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ) ) )
12 1 4 opococ ⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ) → ( ( oc ‘ 𝐾 ) ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ) = 𝑋 )
13 7 12 sylan ⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) → ( ( oc ‘ 𝐾 ) ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ) = 𝑋 )
14 13 fveq2d ⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) → ( 𝑀 ‘ ( ( oc ‘ 𝐾 ) ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ) ) = ( 𝑀 ‘ 𝑋 ) )
15 6 11 14 3eqtrd ⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) → ( ⊥ ‘ ( ⊥ ‘ ( 𝑀 ‘ 𝑋 ) ) ) = ( 𝑀 ‘ 𝑋 ) )