Metamath Proof Explorer

Theorem lclkrlem2f

Description: Lemma for lclkr . Construct a closed hyperplane under the kernel of the sum. (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 lclkrlem2f ( 𝜑 → ( ( ( 𝐿 ‘ 𝐸 ) ∩ ( 𝐿 ‘ 𝐺 ) ) ⊕ ( 𝑁 ‘ { 𝐵 } ) ) ⊆ ( 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 3 15 dvhlmod ⊢ ( 𝜑 → 𝑈 ∈ LMod )
28 10 12 13 14 27 17 18 lkrin ⊢ ( 𝜑 → ( ( 𝐿 ‘ 𝐸 ) ∩ ( 𝐿 ‘ 𝐺 ) ) ⊆ ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) )
29 eqid ⊢ ( LSubSp ‘ 𝑈 ) = ( LSubSp ‘ 𝑈 )
30 29 11 27 26 lshplss ⊢ ( 𝜑 → ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) ∈ ( LSubSp ‘ 𝑈 ) )
31 10 13 14 27 17 18 ldualvaddcl ⊢ ( 𝜑 → ( 𝐸 + 𝐺 ) ∈ 𝐹 )
32 16 eldifad ⊢ ( 𝜑 → 𝐵 ∈ 𝑉 )
33 4 5 6 10 12 27 31 32 ellkr2 ⊢ ( 𝜑 → ( 𝐵 ∈ ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) ↔ ( ( 𝐸 + 𝐺 ) ‘ 𝐵 ) = 𝑄 ) )
34 21 33 mpbird ⊢ ( 𝜑 → 𝐵 ∈ ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) )
35 29 9 27 30 34 lspsnel5a ⊢ ( 𝜑 → ( 𝑁 ‘ { 𝐵 } ) ⊆ ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) )
36 29 lsssssubg ⊢ ( 𝑈 ∈ LMod → ( LSubSp ‘ 𝑈 ) ⊆ ( SubGrp ‘ 𝑈 ) )
37 27 36 syl ⊢ ( 𝜑 → ( LSubSp ‘ 𝑈 ) ⊆ ( SubGrp ‘ 𝑈 ) )
38 10 12 29 lkrlss ⊢ ( ( 𝑈 ∈ LMod ∧ 𝐸 ∈ 𝐹 ) → ( 𝐿 ‘ 𝐸 ) ∈ ( LSubSp ‘ 𝑈 ) )
39 27 17 38 syl2anc ⊢ ( 𝜑 → ( 𝐿 ‘ 𝐸 ) ∈ ( LSubSp ‘ 𝑈 ) )
40 10 12 29 lkrlss ⊢ ( ( 𝑈 ∈ LMod ∧ 𝐺 ∈ 𝐹 ) → ( 𝐿 ‘ 𝐺 ) ∈ ( LSubSp ‘ 𝑈 ) )
41 27 18 40 syl2anc ⊢ ( 𝜑 → ( 𝐿 ‘ 𝐺 ) ∈ ( LSubSp ‘ 𝑈 ) )
42 29 lssincl ⊢ ( ( 𝑈 ∈ LMod ∧ ( 𝐿 ‘ 𝐸 ) ∈ ( LSubSp ‘ 𝑈 ) ∧ ( 𝐿 ‘ 𝐺 ) ∈ ( LSubSp ‘ 𝑈 ) ) → ( ( 𝐿 ‘ 𝐸 ) ∩ ( 𝐿 ‘ 𝐺 ) ) ∈ ( LSubSp ‘ 𝑈 ) )
43 27 39 41 42 syl3anc ⊢ ( 𝜑 → ( ( 𝐿 ‘ 𝐸 ) ∩ ( 𝐿 ‘ 𝐺 ) ) ∈ ( LSubSp ‘ 𝑈 ) )
44 37 43 sseldd ⊢ ( 𝜑 → ( ( 𝐿 ‘ 𝐸 ) ∩ ( 𝐿 ‘ 𝐺 ) ) ∈ ( SubGrp ‘ 𝑈 ) )
45 4 29 9 lspsncl ⊢ ( ( 𝑈 ∈ LMod ∧ 𝐵 ∈ 𝑉 ) → ( 𝑁 ‘ { 𝐵 } ) ∈ ( LSubSp ‘ 𝑈 ) )
46 27 32 45 syl2anc ⊢ ( 𝜑 → ( 𝑁 ‘ { 𝐵 } ) ∈ ( LSubSp ‘ 𝑈 ) )
47 37 46 sseldd ⊢ ( 𝜑 → ( 𝑁 ‘ { 𝐵 } ) ∈ ( SubGrp ‘ 𝑈 ) )
48 37 30 sseldd ⊢ ( 𝜑 → ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) ∈ ( SubGrp ‘ 𝑈 ) )
49 8 lsmlub ⊢ ( ( ( ( 𝐿 ‘ 𝐸 ) ∩ ( 𝐿 ‘ 𝐺 ) ) ∈ ( SubGrp ‘ 𝑈 ) ∧ ( 𝑁 ‘ { 𝐵 } ) ∈ ( SubGrp ‘ 𝑈 ) ∧ ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) ∈ ( SubGrp ‘ 𝑈 ) ) → ( ( ( ( 𝐿 ‘ 𝐸 ) ∩ ( 𝐿 ‘ 𝐺 ) ) ⊆ ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) ∧ ( 𝑁 ‘ { 𝐵 } ) ⊆ ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) ) ↔ ( ( ( 𝐿 ‘ 𝐸 ) ∩ ( 𝐿 ‘ 𝐺 ) ) ⊕ ( 𝑁 ‘ { 𝐵 } ) ) ⊆ ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) ) )
50 44 47 48 49 syl3anc ⊢ ( 𝜑 → ( ( ( ( 𝐿 ‘ 𝐸 ) ∩ ( 𝐿 ‘ 𝐺 ) ) ⊆ ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) ∧ ( 𝑁 ‘ { 𝐵 } ) ⊆ ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) ) ↔ ( ( ( 𝐿 ‘ 𝐸 ) ∩ ( 𝐿 ‘ 𝐺 ) ) ⊕ ( 𝑁 ‘ { 𝐵 } ) ) ⊆ ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) ) )
51 28 35 50 mpbi2and ⊢ ( 𝜑 → ( ( ( 𝐿 ‘ 𝐸 ) ∩ ( 𝐿 ‘ 𝐺 ) ) ⊕ ( 𝑁 ‘ { 𝐵 } ) ) ⊆ ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) )