Metamath Proof Explorer

Theorem dochsncom

Description: Swap vectors in an orthocomplement of a singleton. (Contributed by NM, 17-Jun-2015)

Ref Expression
Hypotheses dochsncom.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
dochsncom.o ⊢ ⊥ = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
dochsncom.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
dochsncom.v ⊢ 𝑉 = ( Base ‘ 𝑈 )
dochsncom.k ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
dochsncom.x ⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
dochsncom.y ⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
Assertion dochsncom ( 𝜑 → ( 𝑋 ∈ ( ⊥ ‘ { 𝑌 } ) ↔ 𝑌 ∈ ( ⊥ ‘ { 𝑋 } ) ) )

Proof

Step Hyp Ref Expression
1 dochsncom.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2 dochsncom.o ⊢ ⊥ = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
3 dochsncom.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
4 dochsncom.v ⊢ 𝑉 = ( Base ‘ 𝑈 )
5 dochsncom.k ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
6 dochsncom.x ⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
7 dochsncom.y ⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
8 eqid ⊢ ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
9 eqid ⊢ ( LSpan ‘ 𝑈 ) = ( LSpan ‘ 𝑈 )
10 1 3 4 9 8 dihlsprn ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝑉 ) → ( ( LSpan ‘ 𝑈 ) ‘ { 𝑋 } ) ∈ ran ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) )
11 5 6 10 syl2anc ⊢ ( 𝜑 → ( ( LSpan ‘ 𝑈 ) ‘ { 𝑋 } ) ∈ ran ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) )
12 1 3 4 9 8 dihlsprn ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ∈ 𝑉 ) → ( ( LSpan ‘ 𝑈 ) ‘ { 𝑌 } ) ∈ ran ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) )
13 5 7 12 syl2anc ⊢ ( 𝜑 → ( ( LSpan ‘ 𝑈 ) ‘ { 𝑌 } ) ∈ ran ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) )
14 1 8 2 5 11 13 dochord3 ⊢ ( 𝜑 → ( ( ( LSpan ‘ 𝑈 ) ‘ { 𝑋 } ) ⊆ ( ⊥ ‘ ( ( LSpan ‘ 𝑈 ) ‘ { 𝑌 } ) ) ↔ ( ( LSpan ‘ 𝑈 ) ‘ { 𝑌 } ) ⊆ ( ⊥ ‘ ( ( LSpan ‘ 𝑈 ) ‘ { 𝑋 } ) ) ) )
15 7 snssd ⊢ ( 𝜑 → { 𝑌 } ⊆ 𝑉 )
16 1 3 2 4 9 5 15 dochocsp ⊢ ( 𝜑 → ( ⊥ ‘ ( ( LSpan ‘ 𝑈 ) ‘ { 𝑌 } ) ) = ( ⊥ ‘ { 𝑌 } ) )
17 16 sseq2d ⊢ ( 𝜑 → ( ( ( LSpan ‘ 𝑈 ) ‘ { 𝑋 } ) ⊆ ( ⊥ ‘ ( ( LSpan ‘ 𝑈 ) ‘ { 𝑌 } ) ) ↔ ( ( LSpan ‘ 𝑈 ) ‘ { 𝑋 } ) ⊆ ( ⊥ ‘ { 𝑌 } ) ) )
18 6 snssd ⊢ ( 𝜑 → { 𝑋 } ⊆ 𝑉 )
19 1 3 2 4 9 5 18 dochocsp ⊢ ( 𝜑 → ( ⊥ ‘ ( ( LSpan ‘ 𝑈 ) ‘ { 𝑋 } ) ) = ( ⊥ ‘ { 𝑋 } ) )
20 19 sseq2d ⊢ ( 𝜑 → ( ( ( LSpan ‘ 𝑈 ) ‘ { 𝑌 } ) ⊆ ( ⊥ ‘ ( ( LSpan ‘ 𝑈 ) ‘ { 𝑋 } ) ) ↔ ( ( LSpan ‘ 𝑈 ) ‘ { 𝑌 } ) ⊆ ( ⊥ ‘ { 𝑋 } ) ) )
21 14 17 20 3bitr3d ⊢ ( 𝜑 → ( ( ( LSpan ‘ 𝑈 ) ‘ { 𝑋 } ) ⊆ ( ⊥ ‘ { 𝑌 } ) ↔ ( ( LSpan ‘ 𝑈 ) ‘ { 𝑌 } ) ⊆ ( ⊥ ‘ { 𝑋 } ) ) )
22 eqid ⊢ ( LSubSp ‘ 𝑈 ) = ( LSubSp ‘ 𝑈 )
23 1 3 5 dvhlmod ⊢ ( 𝜑 → 𝑈 ∈ LMod )
24 1 3 4 22 2 dochlss ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ { 𝑌 } ⊆ 𝑉 ) → ( ⊥ ‘ { 𝑌 } ) ∈ ( LSubSp ‘ 𝑈 ) )
25 5 15 24 syl2anc ⊢ ( 𝜑 → ( ⊥ ‘ { 𝑌 } ) ∈ ( LSubSp ‘ 𝑈 ) )
26 4 22 9 23 25 6 ellspsn5b ⊢ ( 𝜑 → ( 𝑋 ∈ ( ⊥ ‘ { 𝑌 } ) ↔ ( ( LSpan ‘ 𝑈 ) ‘ { 𝑋 } ) ⊆ ( ⊥ ‘ { 𝑌 } ) ) )
27 1 3 4 22 2 dochlss ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ { 𝑋 } ⊆ 𝑉 ) → ( ⊥ ‘ { 𝑋 } ) ∈ ( LSubSp ‘ 𝑈 ) )
28 5 18 27 syl2anc ⊢ ( 𝜑 → ( ⊥ ‘ { 𝑋 } ) ∈ ( LSubSp ‘ 𝑈 ) )
29 4 22 9 23 28 7 ellspsn5b ⊢ ( 𝜑 → ( 𝑌 ∈ ( ⊥ ‘ { 𝑋 } ) ↔ ( ( LSpan ‘ 𝑈 ) ‘ { 𝑌 } ) ⊆ ( ⊥ ‘ { 𝑋 } ) ) )
30 21 26 29 3bitr4d ⊢ ( 𝜑 → ( 𝑋 ∈ ( ⊥ ‘ { 𝑌 } ) ↔ 𝑌 ∈ ( ⊥ ‘ { 𝑋 } ) ) )