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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 ⊢ · = ( ·𝑠 ‘ 𝐷 )
lkrss.w ⊢ ( 𝜑 → 𝑊 ∈ LVec )
lkrss.g ⊢ ( 𝜑 → 𝐺 ∈ 𝐹 )
lkrss.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐾 )
Assertion lkrss ( 𝜑 → ( 𝐿 ‘ 𝐺 ) ⊆ ( 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 ⊢ · = ( ·𝑠 ‘ 𝐷 )
7 lkrss.w ⊢ ( 𝜑 → 𝑊 ∈ LVec )
8 lkrss.g ⊢ ( 𝜑 → 𝐺 ∈ 𝐹 )
9 lkrss.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐾 )
10 eqid ⊢ ( Base ‘ 𝑊 ) = ( Base ‘ 𝑊 )
11 eqid ⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 )
12 10 1 2 11 3 4 7 8 9 lkrscss ⊢ ( 𝜑 → ( 𝐿 ‘ 𝐺 ) ⊆ ( 𝐿 ‘ ( 𝐺 ∘f ( .r ‘ 𝑅 ) ( ( Base ‘ 𝑊 ) × { 𝑋 } ) ) ) )
13 3 10 1 2 11 5 6 7 9 8 ldualvs ⊢ ( 𝜑 → ( 𝑋 · 𝐺 ) = ( 𝐺 ∘f ( .r ‘ 𝑅 ) ( ( Base ‘ 𝑊 ) × { 𝑋 } ) ) )
14 13 fveq2d ⊢ ( 𝜑 → ( 𝐿 ‘ ( 𝑋 · 𝐺 ) ) = ( 𝐿 ‘ ( 𝐺 ∘f ( .r ‘ 𝑅 ) ( ( Base ‘ 𝑊 ) × { 𝑋 } ) ) ) )
15 12 14 sseqtrrd ⊢ ( 𝜑 → ( 𝐿 ‘ 𝐺 ) ⊆ ( 𝐿 ‘ ( 𝑋 · 𝐺 ) ) )