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Metamath Proof Explorer

Theorem pmapsubclN

Description: A projective map value is a closed projective subspace. (Contributed by NM, 24-Jan-2012) (New usage is discouraged.)

Ref Expression
Hypotheses pmapsubcl.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
pmapsubcl.m ⊢ 𝑀 = ( pmap ‘ 𝐾 )
pmapsubcl.c ⊢ 𝐶 = ( PSubCl ‘ 𝐾 )
Assertion pmapsubclN ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) → ( 𝑀 ‘ 𝑋 ) ∈ 𝐶 )

Proof

Step Hyp Ref Expression
1 pmapsubcl.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 pmapsubcl.m ⊢ 𝑀 = ( pmap ‘ 𝐾 )
3 pmapsubcl.c ⊢ 𝐶 = ( PSubCl ‘ 𝐾 )
4 eqid ⊢ ( Atoms ‘ 𝐾 ) = ( Atoms ‘ 𝐾 )
5 1 4 2 pmapssat ⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) → ( 𝑀 ‘ 𝑋 ) ⊆ ( Atoms ‘ 𝐾 ) )
6 eqid ⊢ ( ⊥𝑃 ‘ 𝐾 ) = ( ⊥𝑃 ‘ 𝐾 )
7 1 2 6 2polpmapN ⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) → ( ( ⊥𝑃 ‘ 𝐾 ) ‘ ( ( ⊥𝑃 ‘ 𝐾 ) ‘ ( 𝑀 ‘ 𝑋 ) ) ) = ( 𝑀 ‘ 𝑋 ) )
8 4 6 3 ispsubclN ⊢ ( 𝐾 ∈ HL → ( ( 𝑀 ‘ 𝑋 ) ∈ 𝐶 ↔ ( ( 𝑀 ‘ 𝑋 ) ⊆ ( Atoms ‘ 𝐾 ) ∧ ( ( ⊥𝑃 ‘ 𝐾 ) ‘ ( ( ⊥𝑃 ‘ 𝐾 ) ‘ ( 𝑀 ‘ 𝑋 ) ) ) = ( 𝑀 ‘ 𝑋 ) ) ) )
9 8 adantr ⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) → ( ( 𝑀 ‘ 𝑋 ) ∈ 𝐶 ↔ ( ( 𝑀 ‘ 𝑋 ) ⊆ ( Atoms ‘ 𝐾 ) ∧ ( ( ⊥𝑃 ‘ 𝐾 ) ‘ ( ( ⊥𝑃 ‘ 𝐾 ) ‘ ( 𝑀 ‘ 𝑋 ) ) ) = ( 𝑀 ‘ 𝑋 ) ) ) )
10 5 7 9 mpbir2and ⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) → ( 𝑀 ‘ 𝑋 ) ∈ 𝐶 )