Metamath Proof Explorer

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 = ( 0g𝑈 )
hvmap1o2.c 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
hvmap1o2.f 𝐹 = ( Base ‘ 𝐶 )
hvmap1o2.o 𝑂 = ( 0g𝐶 )
hvmap1o2.m 𝑀 = ( ( HVMap ‘ 𝐾 ) ‘ 𝑊 )
hvmap1o2.k ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
hvmapcl2.x ( 𝜑𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
Assertion hvmapcl2 ( 𝜑 → ( 𝑀𝑋 ) ∈ ( 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 = ( 0g𝑈 )
5 hvmap1o2.c 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
6 hvmap1o2.f 𝐹 = ( Base ‘ 𝐶 )
7 hvmap1o2.o 𝑂 = ( 0g𝐶 )
8 hvmap1o2.m 𝑀 = ( ( HVMap ‘ 𝐾 ) ‘ 𝑊 )
9 hvmap1o2.k ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
10 hvmapcl2.x ( 𝜑𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
11 1 2 3 4 5 6 7 8 9 hvmap1o2 ( 𝜑𝑀 : ( 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 ( 𝜑𝑀 : ( 𝑉 ∖ { 0 } ) ⟶ ( 𝐹 ∖ { 𝑂 } ) )
14 13 10 ffvelrnd ( 𝜑 → ( 𝑀𝑋 ) ∈ ( 𝐹 ∖ { 𝑂 } ) )