Theorem hvmapclN

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

	Ref	Expression
Hypotheses	hvmap1o.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	hvmap1o.o	⊢ 𝑂 = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
	hvmap1o.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	hvmap1o.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
	hvmap1o.z	⊢ 0 = ( 0g ‘ 𝑈 )
	hvmap1o.f	⊢ 𝐹 = ( LFnl ‘ 𝑈 )
	hvmap1o.l	⊢ 𝐿 = ( LKer ‘ 𝑈 )
	hvmap1o.d	⊢ 𝐷 = ( LDual ‘ 𝑈 )
	hvmap1o.q	⊢ 𝑄 = ( 0g ‘ 𝐷 )
	hvmap1o.c	⊢ 𝐶 = { 𝑓 ∈ 𝐹 ∣ ( 𝑂 ‘ ( 𝑂 ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) }
	hvmap1o.m	⊢ 𝑀 = ( ( HVMap ‘ 𝐾 ) ‘ 𝑊 )
	hvmap1o.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
	hvmapcl.x	⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
Assertion	hvmapclN	⊢ ( 𝜑 → ( 𝑀 ‘ 𝑋 ) ∈ ( 𝐶 ∖ { 𝑄 } ) )

Proof

Step	Hyp	Ref	Expression
1		hvmap1o.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		hvmap1o.o	⊢ 𝑂 = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
3		hvmap1o.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
4		hvmap1o.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
5		hvmap1o.z	⊢ 0 = ( 0g ‘ 𝑈 )
6		hvmap1o.f	⊢ 𝐹 = ( LFnl ‘ 𝑈 )
7		hvmap1o.l	⊢ 𝐿 = ( LKer ‘ 𝑈 )
8		hvmap1o.d	⊢ 𝐷 = ( LDual ‘ 𝑈 )
9		hvmap1o.q	⊢ 𝑄 = ( 0g ‘ 𝐷 )
10		hvmap1o.c	⊢ 𝐶 = { 𝑓 ∈ 𝐹 ∣ ( 𝑂 ‘ ( 𝑂 ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) }
11		hvmap1o.m	⊢ 𝑀 = ( ( HVMap ‘ 𝐾 ) ‘ 𝑊 )
12		hvmap1o.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
13		hvmapcl.x	⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
14	1 2 3 4 5 6 7 8 9 10 11 12	hvmap1o	⊢ ( 𝜑 → 𝑀 : ( 𝑉 ∖ { 0 } ) –1-1-onto→ ( 𝐶 ∖ { 𝑄 } ) )
15		f1of	⊢ ( 𝑀 : ( 𝑉 ∖ { 0 } ) –1-1-onto→ ( 𝐶 ∖ { 𝑄 } ) → 𝑀 : ( 𝑉 ∖ { 0 } ) ⟶ ( 𝐶 ∖ { 𝑄 } ) )
16	14 15	syl	⊢ ( 𝜑 → 𝑀 : ( 𝑉 ∖ { 0 } ) ⟶ ( 𝐶 ∖ { 𝑄 } ) )
17	16 13	ffvelcdmd	⊢ ( 𝜑 → ( 𝑀 ‘ 𝑋 ) ∈ ( 𝐶 ∖ { 𝑄 } ) )