Metamath Proof Explorer


Theorem dochsnkr2

Description: Kernel of the explicit functional G determined by a nonzero vector X . Compare the more general lshpkr . (Contributed by NM, 27-Oct-2014)

Ref Expression
Hypotheses dochsnkr2.h 𝐻 = ( LHyp ‘ 𝐾 )
dochsnkr2.o = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
dochsnkr2.u 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
dochsnkr2.v 𝑉 = ( Base ‘ 𝑈 )
dochsnkr2.z 0 = ( 0g𝑈 )
dochsnkr2.a + = ( +g𝑈 )
dochsnkr2.t · = ( ·𝑠𝑈 )
dochsnkr2.l 𝐿 = ( LKer ‘ 𝑈 )
dochsnkr2.d 𝐷 = ( Scalar ‘ 𝑈 )
dochsnkr2.r 𝑅 = ( Base ‘ 𝐷 )
dochsnkr2.g 𝐺 = ( 𝑣𝑉 ↦ ( 𝑘𝑅𝑤 ∈ ( ‘ { 𝑋 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑋 ) ) ) )
dochsnkr2.k ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
dochsnkr2.x ( 𝜑𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
Assertion dochsnkr2 ( 𝜑 → ( 𝐿𝐺 ) = ( ‘ { 𝑋 } ) )

Proof

Step Hyp Ref Expression
1 dochsnkr2.h 𝐻 = ( LHyp ‘ 𝐾 )
2 dochsnkr2.o = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
3 dochsnkr2.u 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
4 dochsnkr2.v 𝑉 = ( Base ‘ 𝑈 )
5 dochsnkr2.z 0 = ( 0g𝑈 )
6 dochsnkr2.a + = ( +g𝑈 )
7 dochsnkr2.t · = ( ·𝑠𝑈 )
8 dochsnkr2.l 𝐿 = ( LKer ‘ 𝑈 )
9 dochsnkr2.d 𝐷 = ( Scalar ‘ 𝑈 )
10 dochsnkr2.r 𝑅 = ( Base ‘ 𝐷 )
11 dochsnkr2.g 𝐺 = ( 𝑣𝑉 ↦ ( 𝑘𝑅𝑤 ∈ ( ‘ { 𝑋 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑋 ) ) ) )
12 dochsnkr2.k ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
13 dochsnkr2.x ( 𝜑𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
14 eqid ( LSpan ‘ 𝑈 ) = ( LSpan ‘ 𝑈 )
15 eqid ( LSSum ‘ 𝑈 ) = ( LSSum ‘ 𝑈 )
16 eqid ( LSHyp ‘ 𝑈 ) = ( LSHyp ‘ 𝑈 )
17 1 3 12 dvhlvec ( 𝜑𝑈 ∈ LVec )
18 1 2 3 4 5 16 12 13 dochsnshp ( 𝜑 → ( ‘ { 𝑋 } ) ∈ ( LSHyp ‘ 𝑈 ) )
19 13 eldifad ( 𝜑𝑋𝑉 )
20 1 2 3 4 5 14 15 12 13 dochexmidat ( 𝜑 → ( ( ‘ { 𝑋 } ) ( LSSum ‘ 𝑈 ) ( ( LSpan ‘ 𝑈 ) ‘ { 𝑋 } ) ) = 𝑉 )
21 4 6 14 15 16 17 18 19 20 9 10 7 11 8 lshpkr ( 𝜑 → ( 𝐿𝐺 ) = ( ‘ { 𝑋 } ) )