Theorem lkrval

Description: Value of the kernel of a functional. (Contributed by NM, 15-Apr-2014)

	Ref	Expression
Hypotheses	lkrfval.d	⊢ 𝐷 = ( Scalar ‘ 𝑊 )
	lkrfval.o	⊢ 0 = ( 0g ‘ 𝐷 )
	lkrfval.f	⊢ 𝐹 = ( LFnl ‘ 𝑊 )
	lkrfval.k	⊢ 𝐾 = ( LKer ‘ 𝑊 )
Assertion	lkrval	⊢ ( ( 𝑊 ∈ 𝑋 ∧ 𝐺 ∈ 𝐹 ) → ( 𝐾 ‘ 𝐺 ) = ( ◡ 𝐺 “ { 0 } ) )

Proof

Step	Hyp	Ref	Expression
1		lkrfval.d	⊢ 𝐷 = ( Scalar ‘ 𝑊 )
2		lkrfval.o	⊢ 0 = ( 0g ‘ 𝐷 )
3		lkrfval.f	⊢ 𝐹 = ( LFnl ‘ 𝑊 )
4		lkrfval.k	⊢ 𝐾 = ( LKer ‘ 𝑊 )
5	1 2 3 4	lkrfval	⊢ ( 𝑊 ∈ 𝑋 → 𝐾 = ( 𝑓 ∈ 𝐹 ↦ ( ◡ 𝑓 “ { 0 } ) ) )
6	5	fveq1d	⊢ ( 𝑊 ∈ 𝑋 → ( 𝐾 ‘ 𝐺 ) = ( ( 𝑓 ∈ 𝐹 ↦ ( ◡ 𝑓 “ { 0 } ) ) ‘ 𝐺 ) )
7		cnvexg	⊢ ( 𝐺 ∈ 𝐹 → ◡ 𝐺 ∈ V )
8		imaexg	⊢ ( ◡ 𝐺 ∈ V → ( ◡ 𝐺 “ { 0 } ) ∈ V )
9	7 8	syl	⊢ ( 𝐺 ∈ 𝐹 → ( ◡ 𝐺 “ { 0 } ) ∈ V )
10		cnveq	⊢ ( 𝑓 = 𝐺 → ◡ 𝑓 = ◡ 𝐺 )
11	10	imaeq1d	⊢ ( 𝑓 = 𝐺 → ( ◡ 𝑓 “ { 0 } ) = ( ◡ 𝐺 “ { 0 } ) )
12		eqid	⊢ ( 𝑓 ∈ 𝐹 ↦ ( ◡ 𝑓 “ { 0 } ) ) = ( 𝑓 ∈ 𝐹 ↦ ( ◡ 𝑓 “ { 0 } ) )
13	11 12	fvmptg	⊢ ( ( 𝐺 ∈ 𝐹 ∧ ( ◡ 𝐺 “ { 0 } ) ∈ V ) → ( ( 𝑓 ∈ 𝐹 ↦ ( ◡ 𝑓 “ { 0 } ) ) ‘ 𝐺 ) = ( ◡ 𝐺 “ { 0 } ) )
14	9 13	mpdan	⊢ ( 𝐺 ∈ 𝐹 → ( ( 𝑓 ∈ 𝐹 ↦ ( ◡ 𝑓 “ { 0 } ) ) ‘ 𝐺 ) = ( ◡ 𝐺 “ { 0 } ) )
15	6 14	sylan9eq	⊢ ( ( 𝑊 ∈ 𝑋 ∧ 𝐺 ∈ 𝐹 ) → ( 𝐾 ‘ 𝐺 ) = ( ◡ 𝐺 “ { 0 } ) )