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 ) ) )