Theorem hvmaplkr

Description: Kernel of the vector to functional map. TODO: make this become lcfrlem11 . (Contributed by NM, 29-Mar-2015)

	Ref	Expression
Hypotheses	hvmaplkr.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	hvmaplkr.o	⊢ 𝑂 = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
	hvmaplkr.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	hvmaplkr.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
	hvmaplkr.z	⊢ 0 = ( 0g ‘ 𝑈 )
	hvmaplkr.l	⊢ 𝐿 = ( LKer ‘ 𝑈 )
	hvmaplkr.m	⊢ 𝑀 = ( ( HVMap ‘ 𝐾 ) ‘ 𝑊 )
	hvmaplkr.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
	hvmaplkr.x	⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
Assertion	hvmaplkr	⊢ ( 𝜑 → ( 𝐿 ‘ ( 𝑀 ‘ 𝑋 ) ) = ( 𝑂 ‘ { 𝑋 } ) )

Proof

Step	Hyp	Ref	Expression
1		hvmaplkr.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		hvmaplkr.o	⊢ 𝑂 = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
3		hvmaplkr.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
4		hvmaplkr.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
5		hvmaplkr.z	⊢ 0 = ( 0g ‘ 𝑈 )
6		hvmaplkr.l	⊢ 𝐿 = ( LKer ‘ 𝑈 )
7		hvmaplkr.m	⊢ 𝑀 = ( ( HVMap ‘ 𝐾 ) ‘ 𝑊 )
8		hvmaplkr.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
9		hvmaplkr.x	⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
10		eqid	⊢ ( +g ‘ 𝑈 ) = ( +g ‘ 𝑈 )
11		eqid	⊢ ( ·𝑠 ‘ 𝑈 ) = ( ·𝑠 ‘ 𝑈 )
12		eqid	⊢ ( Scalar ‘ 𝑈 ) = ( Scalar ‘ 𝑈 )
13		eqid	⊢ ( Base ‘ ( Scalar ‘ 𝑈 ) ) = ( Base ‘ ( Scalar ‘ 𝑈 ) )
14	1 3 2 4 10 11 5 12 13 7 8	hvmapfval	⊢ ( 𝜑 → 𝑀 = ( 𝑥 ∈ ( 𝑉 ∖ { 0 } ) ↦ ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑗 ∈ ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∃ 𝑡 ∈ ( 𝑂 ‘ { 𝑥 } ) 𝑣 = ( 𝑡 ( +g ‘ 𝑈 ) ( 𝑗 ( ·𝑠 ‘ 𝑈 ) 𝑥 ) ) ) ) ) )
15	14	fveq1d	⊢ ( 𝜑 → ( 𝑀 ‘ 𝑋 ) = ( ( 𝑥 ∈ ( 𝑉 ∖ { 0 } ) ↦ ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑗 ∈ ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∃ 𝑡 ∈ ( 𝑂 ‘ { 𝑥 } ) 𝑣 = ( 𝑡 ( +g ‘ 𝑈 ) ( 𝑗 ( ·𝑠 ‘ 𝑈 ) 𝑥 ) ) ) ) ) ‘ 𝑋 ) )
16	15	fveq2d	⊢ ( 𝜑 → ( 𝐿 ‘ ( 𝑀 ‘ 𝑋 ) ) = ( 𝐿 ‘ ( ( 𝑥 ∈ ( 𝑉 ∖ { 0 } ) ↦ ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑗 ∈ ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∃ 𝑡 ∈ ( 𝑂 ‘ { 𝑥 } ) 𝑣 = ( 𝑡 ( +g ‘ 𝑈 ) ( 𝑗 ( ·𝑠 ‘ 𝑈 ) 𝑥 ) ) ) ) ) ‘ 𝑋 ) ) )
17		eqid	⊢ ( LFnl ‘ 𝑈 ) = ( LFnl ‘ 𝑈 )
18		eqid	⊢ ( LDual ‘ 𝑈 ) = ( LDual ‘ 𝑈 )
19		eqid	⊢ ( 0g ‘ ( LDual ‘ 𝑈 ) ) = ( 0g ‘ ( LDual ‘ 𝑈 ) )
20		eqid	⊢ { 𝑓 ∈ ( LFnl ‘ 𝑈 ) ∣ ( 𝑂 ‘ ( 𝑂 ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) } = { 𝑓 ∈ ( LFnl ‘ 𝑈 ) ∣ ( 𝑂 ‘ ( 𝑂 ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) }
21		eqid	⊢ ( 𝑥 ∈ ( 𝑉 ∖ { 0 } ) ↦ ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑗 ∈ ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∃ 𝑡 ∈ ( 𝑂 ‘ { 𝑥 } ) 𝑣 = ( 𝑡 ( +g ‘ 𝑈 ) ( 𝑗 ( ·𝑠 ‘ 𝑈 ) 𝑥 ) ) ) ) ) = ( 𝑥 ∈ ( 𝑉 ∖ { 0 } ) ↦ ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑗 ∈ ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∃ 𝑡 ∈ ( 𝑂 ‘ { 𝑥 } ) 𝑣 = ( 𝑡 ( +g ‘ 𝑈 ) ( 𝑗 ( ·𝑠 ‘ 𝑈 ) 𝑥 ) ) ) ) )
22	1 2 3 4 10 11 12 13 5 17 6 18 19 20 21 8 9	lcfrlem11	⊢ ( 𝜑 → ( 𝐿 ‘ ( ( 𝑥 ∈ ( 𝑉 ∖ { 0 } ) ↦ ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑗 ∈ ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∃ 𝑡 ∈ ( 𝑂 ‘ { 𝑥 } ) 𝑣 = ( 𝑡 ( +g ‘ 𝑈 ) ( 𝑗 ( ·𝑠 ‘ 𝑈 ) 𝑥 ) ) ) ) ) ‘ 𝑋 ) ) = ( 𝑂 ‘ { 𝑋 } ) )
23	16 22	eqtrd	⊢ ( 𝜑 → ( 𝐿 ‘ ( 𝑀 ‘ 𝑋 ) ) = ( 𝑂 ‘ { 𝑋 } ) )