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Metamath Proof Explorer

Theorem dochoc1

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

Ref Expression
Hypotheses dochoc1.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
dochoc1.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
dochoc1.o ⊢ ⊥ = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
dochoc1.v ⊢ 𝑉 = ( Base ‘ 𝑈 )
dochoc1.k ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
Assertion dochoc1 ( 𝜑 → ( ⊥ ‘ ( ⊥ ‘ 𝑉 ) ) = 𝑉 )

Proof

Step Hyp Ref Expression
1 dochoc1.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2 dochoc1.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
3 dochoc1.o ⊢ ⊥ = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
4 dochoc1.v ⊢ 𝑉 = ( Base ‘ 𝑈 )
5 dochoc1.k ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
6 eqid ⊢ ( 0g ‘ 𝑈 ) = ( 0g ‘ 𝑈 )
7 1 2 3 4 6 doch1 ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( ⊥ ‘ 𝑉 ) = { ( 0g ‘ 𝑈 ) } )
8 7 fveq2d ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( ⊥ ‘ ( ⊥ ‘ 𝑉 ) ) = ( ⊥ ‘ { ( 0g ‘ 𝑈 ) } ) )
9 1 2 3 4 6 doch0 ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( ⊥ ‘ { ( 0g ‘ 𝑈 ) } ) = 𝑉 )
10 8 9 eqtrd ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( ⊥ ‘ ( ⊥ ‘ 𝑉 ) ) = 𝑉 )
11 5 10 syl ⊢ ( 𝜑 → ( ⊥ ‘ ( ⊥ ‘ 𝑉 ) ) = 𝑉 )