Metamath Proof Explorer


Theorem lkrss

Description: The kernel of a scalar product of a functional includes the kernel of the functional. (Contributed by NM, 27-Jan-2015)

Ref Expression
Hypotheses lkrss.r 𝑅 = ( Scalar ‘ 𝑊 )
lkrss.k 𝐾 = ( Base ‘ 𝑅 )
lkrss.f 𝐹 = ( LFnl ‘ 𝑊 )
lkrss.l 𝐿 = ( LKer ‘ 𝑊 )
lkrss.d 𝐷 = ( LDual ‘ 𝑊 )
lkrss.s · = ( ·𝑠𝐷 )
lkrss.w ( 𝜑𝑊 ∈ LVec )
lkrss.g ( 𝜑𝐺𝐹 )
lkrss.x ( 𝜑𝑋𝐾 )
Assertion lkrss ( 𝜑 → ( 𝐿𝐺 ) ⊆ ( 𝐿 ‘ ( 𝑋 · 𝐺 ) ) )

Proof

Step Hyp Ref Expression
1 lkrss.r 𝑅 = ( Scalar ‘ 𝑊 )
2 lkrss.k 𝐾 = ( Base ‘ 𝑅 )
3 lkrss.f 𝐹 = ( LFnl ‘ 𝑊 )
4 lkrss.l 𝐿 = ( LKer ‘ 𝑊 )
5 lkrss.d 𝐷 = ( LDual ‘ 𝑊 )
6 lkrss.s · = ( ·𝑠𝐷 )
7 lkrss.w ( 𝜑𝑊 ∈ LVec )
8 lkrss.g ( 𝜑𝐺𝐹 )
9 lkrss.x ( 𝜑𝑋𝐾 )
10 eqid ( Base ‘ 𝑊 ) = ( Base ‘ 𝑊 )
11 eqid ( .r𝑅 ) = ( .r𝑅 )
12 10 1 2 11 3 4 7 8 9 lkrscss ( 𝜑 → ( 𝐿𝐺 ) ⊆ ( 𝐿 ‘ ( 𝐺f ( .r𝑅 ) ( ( Base ‘ 𝑊 ) × { 𝑋 } ) ) ) )
13 3 10 1 2 11 5 6 7 9 8 ldualvs ( 𝜑 → ( 𝑋 · 𝐺 ) = ( 𝐺f ( .r𝑅 ) ( ( Base ‘ 𝑊 ) × { 𝑋 } ) ) )
14 13 fveq2d ( 𝜑 → ( 𝐿 ‘ ( 𝑋 · 𝐺 ) ) = ( 𝐿 ‘ ( 𝐺f ( .r𝑅 ) ( ( Base ‘ 𝑊 ) × { 𝑋 } ) ) ) )
15 12 14 sseqtrrd ( 𝜑 → ( 𝐿𝐺 ) ⊆ ( 𝐿 ‘ ( 𝑋 · 𝐺 ) ) )