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Metamath Proof Explorer

Theorem dochnel2

Description: A nonzero member of a subspace doesn't belong to the orthocomplement of the subspace. (Contributed by NM, 28-Feb-2015)

Ref Expression
Hypotheses dochnoncon.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
dochnoncon.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
dochnoncon.s ⊢ 𝑆 = ( LSubSp ‘ 𝑈 )
dochnoncon.z ⊢ 0 = ( 0g ‘ 𝑈 )
dochnoncon.o ⊢ ⊥ = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
dochnel2.k ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
dochnel2.t ⊢ ( 𝜑 → 𝑇 ∈ 𝑆 )
dochnel2.x ⊢ ( 𝜑 → 𝑋 ∈ ( 𝑇 ∖ { 0 } ) )
Assertion dochnel2 ( 𝜑 → ¬ 𝑋 ∈ ( ⊥ ‘ 𝑇 ) )

Proof

Step Hyp Ref Expression
1 dochnoncon.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2 dochnoncon.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
3 dochnoncon.s ⊢ 𝑆 = ( LSubSp ‘ 𝑈 )
4 dochnoncon.z ⊢ 0 = ( 0g ‘ 𝑈 )
5 dochnoncon.o ⊢ ⊥ = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
6 dochnel2.k ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
7 dochnel2.t ⊢ ( 𝜑 → 𝑇 ∈ 𝑆 )
8 dochnel2.x ⊢ ( 𝜑 → 𝑋 ∈ ( 𝑇 ∖ { 0 } ) )
9 8 eldifbd ⊢ ( 𝜑 → ¬ 𝑋 ∈ { 0 } )
10 8 eldifad ⊢ ( 𝜑 → 𝑋 ∈ 𝑇 )
11 elin ⊢ ( 𝑋 ∈ ( 𝑇 ∩ ( ⊥ ‘ 𝑇 ) ) ↔ ( 𝑋 ∈ 𝑇 ∧ 𝑋 ∈ ( ⊥ ‘ 𝑇 ) ) )
12 1 2 3 4 5 dochnoncon ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑇 ∈ 𝑆 ) → ( 𝑇 ∩ ( ⊥ ‘ 𝑇 ) ) = { 0 } )
13 6 7 12 syl2anc ⊢ ( 𝜑 → ( 𝑇 ∩ ( ⊥ ‘ 𝑇 ) ) = { 0 } )
14 13 eleq2d ⊢ ( 𝜑 → ( 𝑋 ∈ ( 𝑇 ∩ ( ⊥ ‘ 𝑇 ) ) ↔ 𝑋 ∈ { 0 } ) )
15 11 14 bitr3id ⊢ ( 𝜑 → ( ( 𝑋 ∈ 𝑇 ∧ 𝑋 ∈ ( ⊥ ‘ 𝑇 ) ) ↔ 𝑋 ∈ { 0 } ) )
16 15 biimpd ⊢ ( 𝜑 → ( ( 𝑋 ∈ 𝑇 ∧ 𝑋 ∈ ( ⊥ ‘ 𝑇 ) ) → 𝑋 ∈ { 0 } ) )
17 10 16 mpand ⊢ ( 𝜑 → ( 𝑋 ∈ ( ⊥ ‘ 𝑇 ) → 𝑋 ∈ { 0 } ) )
18 9 17 mtod ⊢ ( 𝜑 → ¬ 𝑋 ∈ ( ⊥ ‘ 𝑇 ) )