Theorem ldualkrsc

Description: The kernel of a nonzero scalar product of a functional equals the kernel of the functional. (Contributed by NM, 28-Dec-2014)

	Ref	Expression
Hypotheses	ldualkrsc.r	⊢ 𝑅 = ( Scalar ‘ 𝑊 )
	ldualkrsc.k	⊢ 𝐾 = ( Base ‘ 𝑅 )
	ldualkrsc.o	⊢ 0 = ( 0g ‘ 𝑅 )
	ldualkrsc.f	⊢ 𝐹 = ( LFnl ‘ 𝑊 )
	ldualkrsc.l	⊢ 𝐿 = ( LKer ‘ 𝑊 )
	ldualkrsc.d	⊢ 𝐷 = ( LDual ‘ 𝑊 )
	ldualkrsc.s	⊢ · = ( ·𝑠 ‘ 𝐷 )
	ldualkrsc.w	⊢ ( 𝜑 → 𝑊 ∈ LVec )
	ldualkrsc.g	⊢ ( 𝜑 → 𝐺 ∈ 𝐹 )
	ldualkrsc.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐾 )
	ldualkrsc.e	⊢ ( 𝜑 → 𝑋 ≠ 0 )
Assertion	ldualkrsc	⊢ ( 𝜑 → ( 𝐿 ‘ ( 𝑋 · 𝐺 ) ) = ( 𝐿 ‘ 𝐺 ) )

Proof

Step	Hyp	Ref	Expression
1		ldualkrsc.r	⊢ 𝑅 = ( Scalar ‘ 𝑊 )
2		ldualkrsc.k	⊢ 𝐾 = ( Base ‘ 𝑅 )
3		ldualkrsc.o	⊢ 0 = ( 0g ‘ 𝑅 )
4		ldualkrsc.f	⊢ 𝐹 = ( LFnl ‘ 𝑊 )
5		ldualkrsc.l	⊢ 𝐿 = ( LKer ‘ 𝑊 )
6		ldualkrsc.d	⊢ 𝐷 = ( LDual ‘ 𝑊 )
7		ldualkrsc.s	⊢ · = ( ·𝑠 ‘ 𝐷 )
8		ldualkrsc.w	⊢ ( 𝜑 → 𝑊 ∈ LVec )
9		ldualkrsc.g	⊢ ( 𝜑 → 𝐺 ∈ 𝐹 )
10		ldualkrsc.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐾 )
11		ldualkrsc.e	⊢ ( 𝜑 → 𝑋 ≠ 0 )
12		eqid	⊢ ( Base ‘ 𝑊 ) = ( Base ‘ 𝑊 )
13		eqid	⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 )
14	4 12 1 2 13 6 7 8 10 9	ldualvs	⊢ ( 𝜑 → ( 𝑋 · 𝐺 ) = ( 𝐺 ∘f ( .r ‘ 𝑅 ) ( ( Base ‘ 𝑊 ) × { 𝑋 } ) ) )
15	14	fveq2d	⊢ ( 𝜑 → ( 𝐿 ‘ ( 𝑋 · 𝐺 ) ) = ( 𝐿 ‘ ( 𝐺 ∘f ( .r ‘ 𝑅 ) ( ( Base ‘ 𝑊 ) × { 𝑋 } ) ) ) )
16	12 1 2 13 4 5 8 9 10 3 11	lkrsc	⊢ ( 𝜑 → ( 𝐿 ‘ ( 𝐺 ∘f ( .r ‘ 𝑅 ) ( ( Base ‘ 𝑊 ) × { 𝑋 } ) ) ) = ( 𝐿 ‘ 𝐺 ) )
17	15 16	eqtrd	⊢ ( 𝜑 → ( 𝐿 ‘ ( 𝑋 · 𝐺 ) ) = ( 𝐿 ‘ 𝐺 ) )