Theorem hvmaplfl

Description: The vector to functional map value is a functional. (Contributed by NM, 28-Mar-2015)

	Ref	Expression
Hypotheses	hvmaplfl.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	hvmaplfl.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	hvmaplfl.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
	hvmaplfl.z	⊢ 0 = ( 0g ‘ 𝑈 )
	hvmaplfl.f	⊢ 𝐹 = ( LFnl ‘ 𝑈 )
	hvmaplfl.m	⊢ 𝑀 = ( ( HVMap ‘ 𝐾 ) ‘ 𝑊 )
	hvmaplfl.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
	hvmaplfl.x	⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
Assertion	hvmaplfl	⊢ ( 𝜑 → ( 𝑀 ‘ 𝑋 ) ∈ 𝐹 )

Proof

Step	Hyp	Ref	Expression
1		hvmaplfl.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		hvmaplfl.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
3		hvmaplfl.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
4		hvmaplfl.z	⊢ 0 = ( 0g ‘ 𝑈 )
5		hvmaplfl.f	⊢ 𝐹 = ( LFnl ‘ 𝑈 )
6		hvmaplfl.m	⊢ 𝑀 = ( ( HVMap ‘ 𝐾 ) ‘ 𝑊 )
7		hvmaplfl.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
8		hvmaplfl.x	⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
9		eqid	⊢ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
10		eqid	⊢ ( Base ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) = ( Base ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) )
11		eqid	⊢ ( 0g ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) = ( 0g ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) )
12	1 2 3 4 9 10 11 6 7 8	hvmapcl2	⊢ ( 𝜑 → ( 𝑀 ‘ 𝑋 ) ∈ ( ( Base ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) } ) )
13	12	eldifad	⊢ ( 𝜑 → ( 𝑀 ‘ 𝑋 ) ∈ ( Base ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) )
14	1 9 10 2 5 7 13	lcdvbaselfl	⊢ ( 𝜑 → ( 𝑀 ‘ 𝑋 ) ∈ 𝐹 )