Metamath Proof Explorer

Theorem lclkrlem2g

Description: Lemma for lclkr . Comparable hyperplanes are equal, so the kernel of the sum is closed. (Contributed by NM, 16-Jan-2015)

Ref Expression
Hypotheses lclkrlem2f.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
lclkrlem2f.o ⊢ ⊥ = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
lclkrlem2f.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
lclkrlem2f.v ⊢ 𝑉 = ( Base ‘ 𝑈 )
lclkrlem2f.s ⊢ 𝑆 = ( Scalar ‘ 𝑈 )
lclkrlem2f.q ⊢ 𝑄 = ( 0g ‘ 𝑆 )
lclkrlem2f.z ⊢ 0 = ( 0g ‘ 𝑈 )
lclkrlem2f.a ⊢ ⊕ = ( LSSum ‘ 𝑈 )
lclkrlem2f.n ⊢ 𝑁 = ( LSpan ‘ 𝑈 )
lclkrlem2f.f ⊢ 𝐹 = ( LFnl ‘ 𝑈 )
lclkrlem2f.j ⊢ 𝐽 = ( LSHyp ‘ 𝑈 )
lclkrlem2f.l ⊢ 𝐿 = ( LKer ‘ 𝑈 )
lclkrlem2f.d ⊢ 𝐷 = ( LDual ‘ 𝑈 )
lclkrlem2f.p ⊢ + = ( +g ‘ 𝐷 )
lclkrlem2f.k ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
lclkrlem2f.b ⊢ ( 𝜑 → 𝐵 ∈ ( 𝑉 ∖ { 0 } ) )
lclkrlem2f.e ⊢ ( 𝜑 → 𝐸 ∈ 𝐹 )
lclkrlem2f.g ⊢ ( 𝜑 → 𝐺 ∈ 𝐹 )
lclkrlem2f.le ⊢ ( 𝜑 → ( 𝐿 ‘ 𝐸 ) = ( ⊥ ‘ { 𝑋 } ) )
lclkrlem2f.lg ⊢ ( 𝜑 → ( 𝐿 ‘ 𝐺 ) = ( ⊥ ‘ { 𝑌 } ) )
lclkrlem2f.kb ⊢ ( 𝜑 → ( ( 𝐸 + 𝐺 ) ‘ 𝐵 ) = 𝑄 )
lclkrlem2f.nx ⊢ ( 𝜑 → ( ¬ 𝑋 ∈ ( ⊥ ‘ { 𝐵 } ) ∨ ¬ 𝑌 ∈ ( ⊥ ‘ { 𝐵 } ) ) )
lclkrlem2f.x ⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
lclkrlem2f.y ⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
lclkrlem2f.ne ⊢ ( 𝜑 → ( 𝐿 ‘ 𝐸 ) ≠ ( 𝐿 ‘ 𝐺 ) )
lclkrlem2f.lp ⊢ ( 𝜑 → ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) ∈ 𝐽 )
Assertion lclkrlem2g ( 𝜑 → ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) ) ) = ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) )

Proof

Step Hyp Ref Expression
1 lclkrlem2f.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2 lclkrlem2f.o ⊢ ⊥ = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
3 lclkrlem2f.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
4 lclkrlem2f.v ⊢ 𝑉 = ( Base ‘ 𝑈 )
5 lclkrlem2f.s ⊢ 𝑆 = ( Scalar ‘ 𝑈 )
6 lclkrlem2f.q ⊢ 𝑄 = ( 0g ‘ 𝑆 )
7 lclkrlem2f.z ⊢ 0 = ( 0g ‘ 𝑈 )
8 lclkrlem2f.a ⊢ ⊕ = ( LSSum ‘ 𝑈 )
9 lclkrlem2f.n ⊢ 𝑁 = ( LSpan ‘ 𝑈 )
10 lclkrlem2f.f ⊢ 𝐹 = ( LFnl ‘ 𝑈 )
11 lclkrlem2f.j ⊢ 𝐽 = ( LSHyp ‘ 𝑈 )
12 lclkrlem2f.l ⊢ 𝐿 = ( LKer ‘ 𝑈 )
13 lclkrlem2f.d ⊢ 𝐷 = ( LDual ‘ 𝑈 )
14 lclkrlem2f.p ⊢ + = ( +g ‘ 𝐷 )
15 lclkrlem2f.k ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
16 lclkrlem2f.b ⊢ ( 𝜑 → 𝐵 ∈ ( 𝑉 ∖ { 0 } ) )
17 lclkrlem2f.e ⊢ ( 𝜑 → 𝐸 ∈ 𝐹 )
18 lclkrlem2f.g ⊢ ( 𝜑 → 𝐺 ∈ 𝐹 )
19 lclkrlem2f.le ⊢ ( 𝜑 → ( 𝐿 ‘ 𝐸 ) = ( ⊥ ‘ { 𝑋 } ) )
20 lclkrlem2f.lg ⊢ ( 𝜑 → ( 𝐿 ‘ 𝐺 ) = ( ⊥ ‘ { 𝑌 } ) )
21 lclkrlem2f.kb ⊢ ( 𝜑 → ( ( 𝐸 + 𝐺 ) ‘ 𝐵 ) = 𝑄 )
22 lclkrlem2f.nx ⊢ ( 𝜑 → ( ¬ 𝑋 ∈ ( ⊥ ‘ { 𝐵 } ) ∨ ¬ 𝑌 ∈ ( ⊥ ‘ { 𝐵 } ) ) )
23 lclkrlem2f.x ⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
24 lclkrlem2f.y ⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
25 lclkrlem2f.ne ⊢ ( 𝜑 → ( 𝐿 ‘ 𝐸 ) ≠ ( 𝐿 ‘ 𝐺 ) )
26 lclkrlem2f.lp ⊢ ( 𝜑 → ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) ∈ 𝐽 )
27 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 lclkrlem2f ⊢ ( 𝜑 → ( ( ( 𝐿 ‘ 𝐸 ) ∩ ( 𝐿 ‘ 𝐺 ) ) ⊕ ( 𝑁 ‘ { 𝐵 } ) ) ⊆ ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) )
28 1 3 15 dvhlvec ⊢ ( 𝜑 → 𝑈 ∈ LVec )
29 19 20 ineq12d ⊢ ( 𝜑 → ( ( 𝐿 ‘ 𝐸 ) ∩ ( 𝐿 ‘ 𝐺 ) ) = ( ( ⊥ ‘ { 𝑋 } ) ∩ ( ⊥ ‘ { 𝑌 } ) ) )
30 29 oveq1d ⊢ ( 𝜑 → ( ( ( 𝐿 ‘ 𝐸 ) ∩ ( 𝐿 ‘ 𝐺 ) ) ⊕ ( 𝑁 ‘ { 𝐵 } ) ) = ( ( ( ⊥ ‘ { 𝑋 } ) ∩ ( ⊥ ‘ { 𝑌 } ) ) ⊕ ( 𝑁 ‘ { 𝐵 } ) ) )
31 eqid ⊢ ( LSAtoms ‘ 𝑈 ) = ( LSAtoms ‘ 𝑈 )
32 25 19 20 3netr3d ⊢ ( 𝜑 → ( ⊥ ‘ { 𝑋 } ) ≠ ( ⊥ ‘ { 𝑌 } ) )
33 1 2 3 4 7 8 9 31 15 16 23 24 32 22 11 lclkrlem2c ⊢ ( 𝜑 → ( ( ( ⊥ ‘ { 𝑋 } ) ∩ ( ⊥ ‘ { 𝑌 } ) ) ⊕ ( 𝑁 ‘ { 𝐵 } ) ) ∈ 𝐽 )
34 30 33 eqeltrd ⊢ ( 𝜑 → ( ( ( 𝐿 ‘ 𝐸 ) ∩ ( 𝐿 ‘ 𝐺 ) ) ⊕ ( 𝑁 ‘ { 𝐵 } ) ) ∈ 𝐽 )
35 11 28 34 26 lshpcmp ⊢ ( 𝜑 → ( ( ( ( 𝐿 ‘ 𝐸 ) ∩ ( 𝐿 ‘ 𝐺 ) ) ⊕ ( 𝑁 ‘ { 𝐵 } ) ) ⊆ ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) ↔ ( ( ( 𝐿 ‘ 𝐸 ) ∩ ( 𝐿 ‘ 𝐺 ) ) ⊕ ( 𝑁 ‘ { 𝐵 } ) ) = ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) ) )
36 27 35 mpbid ⊢ ( 𝜑 → ( ( ( 𝐿 ‘ 𝐸 ) ∩ ( 𝐿 ‘ 𝐺 ) ) ⊕ ( 𝑁 ‘ { 𝐵 } ) ) = ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) )
37 eqid ⊢ ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
38 1 2 3 4 7 8 9 31 15 16 23 24 32 22 37 lclkrlem2d ⊢ ( 𝜑 → ( ( ( ⊥ ‘ { 𝑋 } ) ∩ ( ⊥ ‘ { 𝑌 } ) ) ⊕ ( 𝑁 ‘ { 𝐵 } ) ) ∈ ran ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) )
39 30 38 eqeltrd ⊢ ( 𝜑 → ( ( ( 𝐿 ‘ 𝐸 ) ∩ ( 𝐿 ‘ 𝐺 ) ) ⊕ ( 𝑁 ‘ { 𝐵 } ) ) ∈ ran ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) )
40 36 39 eqeltrrd ⊢ ( 𝜑 → ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) ∈ ran ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) )
41 1 3 37 4 dihrnss ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) ∈ ran ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ) → ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) ⊆ 𝑉 )
42 15 40 41 syl2anc ⊢ ( 𝜑 → ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) ⊆ 𝑉 )
43 1 37 3 4 2 15 42 dochoccl ⊢ ( 𝜑 → ( ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) ∈ ran ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ↔ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) ) ) = ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) ) )
44 40 43 mpbid ⊢ ( 𝜑 → ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) ) ) = ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) )