Theorem hvmapcl2

Description: Closure of the vector to functional map. (Contributed by NM, 27-Mar-2015)

	Ref	Expression
Hypotheses	hvmap1o2.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	hvmap1o2.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	hvmap1o2.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
	hvmap1o2.z	⊢ 0 = ( 0g ‘ 𝑈 )
	hvmap1o2.c	⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
	hvmap1o2.f	⊢ 𝐹 = ( Base ‘ 𝐶 )
	hvmap1o2.o	⊢ 𝑂 = ( 0g ‘ 𝐶 )
	hvmap1o2.m	⊢ 𝑀 = ( ( HVMap ‘ 𝐾 ) ‘ 𝑊 )
	hvmap1o2.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
	hvmapcl2.x	⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
Assertion	hvmapcl2	⊢ ( 𝜑 → ( 𝑀 ‘ 𝑋 ) ∈ ( 𝐹 ∖ { 𝑂 } ) )

Proof

Step	Hyp	Ref	Expression
1		hvmap1o2.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		hvmap1o2.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
3		hvmap1o2.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
4		hvmap1o2.z	⊢ 0 = ( 0g ‘ 𝑈 )
5		hvmap1o2.c	⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
6		hvmap1o2.f	⊢ 𝐹 = ( Base ‘ 𝐶 )
7		hvmap1o2.o	⊢ 𝑂 = ( 0g ‘ 𝐶 )
8		hvmap1o2.m	⊢ 𝑀 = ( ( HVMap ‘ 𝐾 ) ‘ 𝑊 )
9		hvmap1o2.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
10		hvmapcl2.x	⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
11	1 2 3 4 5 6 7 8 9	hvmap1o2	⊢ ( 𝜑 → 𝑀 : ( 𝑉 ∖ { 0 } ) –1-1-onto→ ( 𝐹 ∖ { 𝑂 } ) )
12		f1of	⊢ ( 𝑀 : ( 𝑉 ∖ { 0 } ) –1-1-onto→ ( 𝐹 ∖ { 𝑂 } ) → 𝑀 : ( 𝑉 ∖ { 0 } ) ⟶ ( 𝐹 ∖ { 𝑂 } ) )
13	11 12	syl	⊢ ( 𝜑 → 𝑀 : ( 𝑉 ∖ { 0 } ) ⟶ ( 𝐹 ∖ { 𝑂 } ) )
14	13 10	ffvelcdmd	⊢ ( 𝜑 → ( 𝑀 ‘ 𝑋 ) ∈ ( 𝐹 ∖ { 𝑂 } ) )