This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem dochoc0

Description: The zero subspace is closed. (Contributed by NM, 16-Feb-2015)

Ref Expression
Hypotheses dochoc0.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
dochoc0.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
dochoc0.o ⊢ ⊥ = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
dochoc0.z ⊢ 0 = ( 0g ‘ 𝑈 )
dochoc0.k ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
Assertion dochoc0 ( 𝜑 → ( ⊥ ‘ ( ⊥ ‘ { 0 } ) ) = { 0 } )

Proof

Step Hyp Ref Expression
1 dochoc0.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2 dochoc0.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
3 dochoc0.o ⊢ ⊥ = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
4 dochoc0.z ⊢ 0 = ( 0g ‘ 𝑈 )
5 dochoc0.k ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
6 eqid ⊢ ( Base ‘ 𝑈 ) = ( Base ‘ 𝑈 )
7 1 2 3 6 4 doch0 ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( ⊥ ‘ { 0 } ) = ( Base ‘ 𝑈 ) )
8 7 fveq2d ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( ⊥ ‘ ( ⊥ ‘ { 0 } ) ) = ( ⊥ ‘ ( Base ‘ 𝑈 ) ) )
9 1 2 3 6 4 doch1 ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( ⊥ ‘ ( Base ‘ 𝑈 ) ) = { 0 } )
10 8 9 eqtrd ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( ⊥ ‘ ( ⊥ ‘ { 0 } ) ) = { 0 } )
11 5 10 syl ⊢ ( 𝜑 → ( ⊥ ‘ ( ⊥ ‘ { 0 } ) ) = { 0 } )