Metamath Proof Explorer


Theorem dochnel

Description: A nonzero vector doesn't belong to the orthocomplement of its singleton. (Contributed by NM, 27-Oct-2014)

Ref Expression
Hypotheses dochnel.h 𝐻 = ( LHyp ‘ 𝐾 )
dochnel.o = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
dochnel.u 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
dochnel.v 𝑉 = ( Base ‘ 𝑈 )
dochnel.z 0 = ( 0g𝑈 )
dochnel.k ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
dochnel.x ( 𝜑𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
Assertion dochnel ( 𝜑 → ¬ 𝑋 ∈ ( ‘ { 𝑋 } ) )

Proof

Step Hyp Ref Expression
1 dochnel.h 𝐻 = ( LHyp ‘ 𝐾 )
2 dochnel.o = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
3 dochnel.u 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
4 dochnel.v 𝑉 = ( Base ‘ 𝑈 )
5 dochnel.z 0 = ( 0g𝑈 )
6 dochnel.k ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
7 dochnel.x ( 𝜑𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
8 eqid ( LSubSp ‘ 𝑈 ) = ( LSubSp ‘ 𝑈 )
9 1 3 6 dvhlmod ( 𝜑𝑈 ∈ LMod )
10 7 eldifad ( 𝜑𝑋𝑉 )
11 eqid ( LSpan ‘ 𝑈 ) = ( LSpan ‘ 𝑈 )
12 4 8 11 lspsncl ( ( 𝑈 ∈ LMod ∧ 𝑋𝑉 ) → ( ( LSpan ‘ 𝑈 ) ‘ { 𝑋 } ) ∈ ( LSubSp ‘ 𝑈 ) )
13 9 10 12 syl2anc ( 𝜑 → ( ( LSpan ‘ 𝑈 ) ‘ { 𝑋 } ) ∈ ( LSubSp ‘ 𝑈 ) )
14 4 11 lspsnid ( ( 𝑈 ∈ LMod ∧ 𝑋𝑉 ) → 𝑋 ∈ ( ( LSpan ‘ 𝑈 ) ‘ { 𝑋 } ) )
15 9 10 14 syl2anc ( 𝜑𝑋 ∈ ( ( LSpan ‘ 𝑈 ) ‘ { 𝑋 } ) )
16 eldifsni ( 𝑋 ∈ ( 𝑉 ∖ { 0 } ) → 𝑋0 )
17 7 16 syl ( 𝜑𝑋0 )
18 eldifsn ( 𝑋 ∈ ( ( ( LSpan ‘ 𝑈 ) ‘ { 𝑋 } ) ∖ { 0 } ) ↔ ( 𝑋 ∈ ( ( LSpan ‘ 𝑈 ) ‘ { 𝑋 } ) ∧ 𝑋0 ) )
19 15 17 18 sylanbrc ( 𝜑𝑋 ∈ ( ( ( LSpan ‘ 𝑈 ) ‘ { 𝑋 } ) ∖ { 0 } ) )
20 1 3 8 5 2 6 13 19 dochnel2 ( 𝜑 → ¬ 𝑋 ∈ ( ‘ ( ( LSpan ‘ 𝑈 ) ‘ { 𝑋 } ) ) )
21 10 snssd ( 𝜑 → { 𝑋 } ⊆ 𝑉 )
22 1 3 2 4 11 6 21 dochocsp ( 𝜑 → ( ‘ ( ( LSpan ‘ 𝑈 ) ‘ { 𝑋 } ) ) = ( ‘ { 𝑋 } ) )
23 20 22 neleqtrd ( 𝜑 → ¬ 𝑋 ∈ ( ‘ { 𝑋 } ) )