Metamath Proof Explorer

Theorem lclkrlem2j

Description: Lemma for lclkr . Kernel closure when Y is zero. (Contributed by NM, 18-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 } ) ) )
lclkrlem2j.x ⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
lclkrlem2j.y ⊢ ( 𝜑 → 𝑌 = 0 )
Assertion lclkrlem2j ( 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 lclkrlem2j.x ⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
24 lclkrlem2j.y ⊢ ( 𝜑 → 𝑌 = 0 )
25 23 snssd ⊢ ( 𝜑 → { 𝑋 } ⊆ 𝑉 )
26 eqid ⊢ ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
27 1 26 3 4 2 dochcl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ { 𝑋 } ⊆ 𝑉 ) → ( ⊥ ‘ { 𝑋 } ) ∈ ran ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) )
28 15 25 27 syl2anc ⊢ ( 𝜑 → ( ⊥ ‘ { 𝑋 } ) ∈ ran ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) )
29 1 26 2 dochoc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ⊥ ‘ { 𝑋 } ) ∈ ran ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ) → ( ⊥ ‘ ( ⊥ ‘ ( ⊥ ‘ { 𝑋 } ) ) ) = ( ⊥ ‘ { 𝑋 } ) )
30 15 28 29 syl2anc ⊢ ( 𝜑 → ( ⊥ ‘ ( ⊥ ‘ ( ⊥ ‘ { 𝑋 } ) ) ) = ( ⊥ ‘ { 𝑋 } ) )
31 24 sneqd ⊢ ( 𝜑 → { 𝑌 } = { 0 } )
32 31 fveq2d ⊢ ( 𝜑 → ( ⊥ ‘ { 𝑌 } ) = ( ⊥ ‘ { 0 } ) )
33 eqid ⊢ ( Base ‘ 𝑈 ) = ( Base ‘ 𝑈 )
34 1 3 2 33 7 doch0 ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( ⊥ ‘ { 0 } ) = ( Base ‘ 𝑈 ) )
35 15 34 syl ⊢ ( 𝜑 → ( ⊥ ‘ { 0 } ) = ( Base ‘ 𝑈 ) )
36 20 32 35 3eqtrd ⊢ ( 𝜑 → ( 𝐿 ‘ 𝐺 ) = ( Base ‘ 𝑈 ) )
37 1 3 15 dvhlmod ⊢ ( 𝜑 → 𝑈 ∈ LMod )
38 5 6 33 10 12 lkr0f ⊢ ( ( 𝑈 ∈ LMod ∧ 𝐺 ∈ 𝐹 ) → ( ( 𝐿 ‘ 𝐺 ) = ( Base ‘ 𝑈 ) ↔ 𝐺 = ( ( Base ‘ 𝑈 ) × { 𝑄 } ) ) )
39 37 18 38 syl2anc ⊢ ( 𝜑 → ( ( 𝐿 ‘ 𝐺 ) = ( Base ‘ 𝑈 ) ↔ 𝐺 = ( ( Base ‘ 𝑈 ) × { 𝑄 } ) ) )
40 36 39 mpbid ⊢ ( 𝜑 → 𝐺 = ( ( Base ‘ 𝑈 ) × { 𝑄 } ) )
41 eqid ⊢ ( 0g ‘ 𝐷 ) = ( 0g ‘ 𝐷 )
42 33 5 6 13 41 37 ldual0v ⊢ ( 𝜑 → ( 0g ‘ 𝐷 ) = ( ( Base ‘ 𝑈 ) × { 𝑄 } ) )
43 40 42 eqtr4d ⊢ ( 𝜑 → 𝐺 = ( 0g ‘ 𝐷 ) )
44 43 oveq2d ⊢ ( 𝜑 → ( 𝐸 + 𝐺 ) = ( 𝐸 + ( 0g ‘ 𝐷 ) ) )
45 13 37 lduallmod ⊢ ( 𝜑 → 𝐷 ∈ LMod )
46 eqid ⊢ ( Base ‘ 𝐷 ) = ( Base ‘ 𝐷 )
47 10 13 46 37 17 ldualelvbase ⊢ ( 𝜑 → 𝐸 ∈ ( Base ‘ 𝐷 ) )
48 46 14 41 lmod0vrid ⊢ ( ( 𝐷 ∈ LMod ∧ 𝐸 ∈ ( Base ‘ 𝐷 ) ) → ( 𝐸 + ( 0g ‘ 𝐷 ) ) = 𝐸 )
49 45 47 48 syl2anc ⊢ ( 𝜑 → ( 𝐸 + ( 0g ‘ 𝐷 ) ) = 𝐸 )
50 44 49 eqtrd ⊢ ( 𝜑 → ( 𝐸 + 𝐺 ) = 𝐸 )
51 50 fveq2d ⊢ ( 𝜑 → ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) = ( 𝐿 ‘ 𝐸 ) )
52 51 19 eqtr2d ⊢ ( 𝜑 → ( ⊥ ‘ { 𝑋 } ) = ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) )
53 52 fveq2d ⊢ ( 𝜑 → ( ⊥ ‘ ( ⊥ ‘ { 𝑋 } ) ) = ( ⊥ ‘ ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) ) )
54 53 fveq2d ⊢ ( 𝜑 → ( ⊥ ‘ ( ⊥ ‘ ( ⊥ ‘ { 𝑋 } ) ) ) = ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) ) ) )
55 30 54 52 3eqtr3d ⊢ ( 𝜑 → ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) ) ) = ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) )