Metamath Proof Explorer

Theorem lclkrlem2p

Description: Lemma for lclkr . When B is zero, X and Y must colinear, so their orthocomplements must be comparable. (Contributed by NM, 17-Jan-2015)

Ref Expression
Hypotheses lclkrlem2m.v ⊢ 𝑉 = ( Base ‘ 𝑈 )
lclkrlem2m.t ⊢ · = ( ·𝑠 ‘ 𝑈 )
lclkrlem2m.s ⊢ 𝑆 = ( Scalar ‘ 𝑈 )
lclkrlem2m.q ⊢ × = ( .r ‘ 𝑆 )
lclkrlem2m.z ⊢ 0 = ( 0g ‘ 𝑆 )
lclkrlem2m.i ⊢ 𝐼 = ( invr ‘ 𝑆 )
lclkrlem2m.m ⊢ − = ( -g ‘ 𝑈 )
lclkrlem2m.f ⊢ 𝐹 = ( LFnl ‘ 𝑈 )
lclkrlem2m.d ⊢ 𝐷 = ( LDual ‘ 𝑈 )
lclkrlem2m.p ⊢ + = ( +g ‘ 𝐷 )
lclkrlem2m.x ⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
lclkrlem2m.y ⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
lclkrlem2m.e ⊢ ( 𝜑 → 𝐸 ∈ 𝐹 )
lclkrlem2m.g ⊢ ( 𝜑 → 𝐺 ∈ 𝐹 )
lclkrlem2n.n ⊢ 𝑁 = ( LSpan ‘ 𝑈 )
lclkrlem2n.l ⊢ 𝐿 = ( LKer ‘ 𝑈 )
lclkrlem2o.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
lclkrlem2o.o ⊢ ⊥ = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
lclkrlem2o.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
lclkrlem2o.a ⊢ ⊕ = ( LSSum ‘ 𝑈 )
lclkrlem2o.k ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
lclkrlem2o.b ⊢ 𝐵 = ( 𝑋 − ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) )
lclkrlem2o.n ⊢ ( 𝜑 → ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ≠ 0 )
lclkrlem2p.bn ⊢ ( 𝜑 → 𝐵 = ( 0g ‘ 𝑈 ) )
Assertion lclkrlem2p ( 𝜑 → ( ⊥ ‘ { 𝑌 } ) ⊆ ( ⊥ ‘ { 𝑋 } ) )

Proof

Step Hyp Ref Expression
1 lclkrlem2m.v ⊢ 𝑉 = ( Base ‘ 𝑈 )
2 lclkrlem2m.t ⊢ · = ( ·𝑠 ‘ 𝑈 )
3 lclkrlem2m.s ⊢ 𝑆 = ( Scalar ‘ 𝑈 )
4 lclkrlem2m.q ⊢ × = ( .r ‘ 𝑆 )
5 lclkrlem2m.z ⊢ 0 = ( 0g ‘ 𝑆 )
6 lclkrlem2m.i ⊢ 𝐼 = ( invr ‘ 𝑆 )
7 lclkrlem2m.m ⊢ − = ( -g ‘ 𝑈 )
8 lclkrlem2m.f ⊢ 𝐹 = ( LFnl ‘ 𝑈 )
9 lclkrlem2m.d ⊢ 𝐷 = ( LDual ‘ 𝑈 )
10 lclkrlem2m.p ⊢ + = ( +g ‘ 𝐷 )
11 lclkrlem2m.x ⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
12 lclkrlem2m.y ⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
13 lclkrlem2m.e ⊢ ( 𝜑 → 𝐸 ∈ 𝐹 )
14 lclkrlem2m.g ⊢ ( 𝜑 → 𝐺 ∈ 𝐹 )
15 lclkrlem2n.n ⊢ 𝑁 = ( LSpan ‘ 𝑈 )
16 lclkrlem2n.l ⊢ 𝐿 = ( LKer ‘ 𝑈 )
17 lclkrlem2o.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
18 lclkrlem2o.o ⊢ ⊥ = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
19 lclkrlem2o.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
20 lclkrlem2o.a ⊢ ⊕ = ( LSSum ‘ 𝑈 )
21 lclkrlem2o.k ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
22 lclkrlem2o.b ⊢ 𝐵 = ( 𝑋 − ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) )
23 lclkrlem2o.n ⊢ ( 𝜑 → ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ≠ 0 )
24 lclkrlem2p.bn ⊢ ( 𝜑 → 𝐵 = ( 0g ‘ 𝑈 ) )
25 17 19 21 dvhlmod ⊢ ( 𝜑 → 𝑈 ∈ LMod )
26 eqid ⊢ ( LSubSp ‘ 𝑈 ) = ( LSubSp ‘ 𝑈 )
27 1 26 15 lspsncl ⊢ ( ( 𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉 ) → ( 𝑁 ‘ { 𝑌 } ) ∈ ( LSubSp ‘ 𝑈 ) )
28 25 12 27 syl2anc ⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) ∈ ( LSubSp ‘ 𝑈 ) )
29 1 26 lssss ⊢ ( ( 𝑁 ‘ { 𝑌 } ) ∈ ( LSubSp ‘ 𝑈 ) → ( 𝑁 ‘ { 𝑌 } ) ⊆ 𝑉 )
30 28 29 syl ⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) ⊆ 𝑉 )
31 22 24 eqtr3id ⊢ ( 𝜑 → ( 𝑋 − ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) ) = ( 0g ‘ 𝑈 ) )
32 3 lmodring ⊢ ( 𝑈 ∈ LMod → 𝑆 ∈ Ring )
33 25 32 syl ⊢ ( 𝜑 → 𝑆 ∈ Ring )
34 8 9 10 25 13 14 ldualvaddcl ⊢ ( 𝜑 → ( 𝐸 + 𝐺 ) ∈ 𝐹 )
35 eqid ⊢ ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 )
36 3 35 1 8 lflcl ⊢ ( ( 𝑈 ∈ LMod ∧ ( 𝐸 + 𝐺 ) ∈ 𝐹 ∧ 𝑋 ∈ 𝑉 ) → ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) ∈ ( Base ‘ 𝑆 ) )
37 25 34 11 36 syl3anc ⊢ ( 𝜑 → ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) ∈ ( Base ‘ 𝑆 ) )
38 17 19 21 dvhlvec ⊢ ( 𝜑 → 𝑈 ∈ LVec )
39 3 lvecdrng ⊢ ( 𝑈 ∈ LVec → 𝑆 ∈ DivRing )
40 38 39 syl ⊢ ( 𝜑 → 𝑆 ∈ DivRing )
41 3 35 1 8 lflcl ⊢ ( ( 𝑈 ∈ LMod ∧ ( 𝐸 + 𝐺 ) ∈ 𝐹 ∧ 𝑌 ∈ 𝑉 ) → ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ∈ ( Base ‘ 𝑆 ) )
42 25 34 12 41 syl3anc ⊢ ( 𝜑 → ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ∈ ( Base ‘ 𝑆 ) )
43 35 5 6 drnginvrcl ⊢ ( ( 𝑆 ∈ DivRing ∧ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ∈ ( Base ‘ 𝑆 ) ∧ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ≠ 0 ) → ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ∈ ( Base ‘ 𝑆 ) )
44 40 42 23 43 syl3anc ⊢ ( 𝜑 → ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ∈ ( Base ‘ 𝑆 ) )
45 35 4 ringcl ⊢ ( ( 𝑆 ∈ Ring ∧ ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) ∈ ( Base ‘ 𝑆 ) ∧ ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ∈ ( Base ‘ 𝑆 ) ) → ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) ∈ ( Base ‘ 𝑆 ) )
46 33 37 44 45 syl3anc ⊢ ( 𝜑 → ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) ∈ ( Base ‘ 𝑆 ) )
47 1 3 2 35 lmodvscl ⊢ ( ( 𝑈 ∈ LMod ∧ ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) ∈ ( Base ‘ 𝑆 ) ∧ 𝑌 ∈ 𝑉 ) → ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) ∈ 𝑉 )
48 25 46 12 47 syl3anc ⊢ ( 𝜑 → ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) ∈ 𝑉 )
49 eqid ⊢ ( 0g ‘ 𝑈 ) = ( 0g ‘ 𝑈 )
50 1 49 7 lmodsubeq0 ⊢ ( ( 𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) ∈ 𝑉 ) → ( ( 𝑋 − ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) ) = ( 0g ‘ 𝑈 ) ↔ 𝑋 = ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) ) )
51 25 11 48 50 syl3anc ⊢ ( 𝜑 → ( ( 𝑋 − ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) ) = ( 0g ‘ 𝑈 ) ↔ 𝑋 = ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) ) )
52 31 51 mpbid ⊢ ( 𝜑 → 𝑋 = ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) )
53 52 sneqd ⊢ ( 𝜑 → { 𝑋 } = { ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) } )
54 53 fveq2d ⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) = ( 𝑁 ‘ { ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) } ) )
55 3 35 1 2 15 lspsnvsi ⊢ ( ( 𝑈 ∈ LMod ∧ ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) ∈ ( Base ‘ 𝑆 ) ∧ 𝑌 ∈ 𝑉 ) → ( 𝑁 ‘ { ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) } ) ⊆ ( 𝑁 ‘ { 𝑌 } ) )
56 25 46 12 55 syl3anc ⊢ ( 𝜑 → ( 𝑁 ‘ { ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) } ) ⊆ ( 𝑁 ‘ { 𝑌 } ) )
57 54 56 eqsstrd ⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ⊆ ( 𝑁 ‘ { 𝑌 } ) )
58 17 19 1 18 dochss ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑁 ‘ { 𝑌 } ) ⊆ 𝑉 ∧ ( 𝑁 ‘ { 𝑋 } ) ⊆ ( 𝑁 ‘ { 𝑌 } ) ) → ( ⊥ ‘ ( 𝑁 ‘ { 𝑌 } ) ) ⊆ ( ⊥ ‘ ( 𝑁 ‘ { 𝑋 } ) ) )
59 21 30 57 58 syl3anc ⊢ ( 𝜑 → ( ⊥ ‘ ( 𝑁 ‘ { 𝑌 } ) ) ⊆ ( ⊥ ‘ ( 𝑁 ‘ { 𝑋 } ) ) )
60 12 snssd ⊢ ( 𝜑 → { 𝑌 } ⊆ 𝑉 )
61 17 19 18 1 15 21 60 dochocsp ⊢ ( 𝜑 → ( ⊥ ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( ⊥ ‘ { 𝑌 } ) )
62 11 snssd ⊢ ( 𝜑 → { 𝑋 } ⊆ 𝑉 )
63 17 19 18 1 15 21 62 dochocsp ⊢ ( 𝜑 → ( ⊥ ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( ⊥ ‘ { 𝑋 } ) )
64 59 61 63 3sstr3d ⊢ ( 𝜑 → ( ⊥ ‘ { 𝑌 } ) ⊆ ( ⊥ ‘ { 𝑋 } ) )