# Metamath Proof Explorer

## Theorem hdmapfnN

Description: Functionality of map from vectors to functionals with closed kernels. (Contributed by NM, 30-May-2015) (New usage is discouraged.)

Ref Expression
Hypotheses hdmapfn.h 𝐻 = ( LHyp ‘ 𝐾 )
hdmapfn.u 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
hdmapfn.v 𝑉 = ( Base ‘ 𝑈 )
hdmapfn.s 𝑆 = ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 )
hdmapfn.k ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
Assertion hdmapfnN ( 𝜑𝑆 Fn 𝑉 )

### Proof

Step Hyp Ref Expression
1 hdmapfn.h 𝐻 = ( LHyp ‘ 𝐾 )
2 hdmapfn.u 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
3 hdmapfn.v 𝑉 = ( Base ‘ 𝑈 )
4 hdmapfn.s 𝑆 = ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 )
5 hdmapfn.k ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
6 riotaex ( 𝑦 ∈ ( Base ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ∀ 𝑧𝑉 ( ¬ 𝑧 ∈ ( ( ( LSpan ‘ 𝑈 ) ‘ { ⟨ ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) ⟩ } ) ∪ ( ( LSpan ‘ 𝑈 ) ‘ { 𝑡 } ) ) → 𝑦 = ( ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑊 ) ‘ ⟨ 𝑧 , ( ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑊 ) ‘ ⟨ ⟨ ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) ⟩ , ( ( ( HVMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ ⟨ ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) ⟩ ) , 𝑧 ⟩ ) , 𝑡 ⟩ ) ) ) ∈ V
7 eqid ( 𝑡𝑉 ↦ ( 𝑦 ∈ ( Base ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ∀ 𝑧𝑉 ( ¬ 𝑧 ∈ ( ( ( LSpan ‘ 𝑈 ) ‘ { ⟨ ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) ⟩ } ) ∪ ( ( LSpan ‘ 𝑈 ) ‘ { 𝑡 } ) ) → 𝑦 = ( ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑊 ) ‘ ⟨ 𝑧 , ( ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑊 ) ‘ ⟨ ⟨ ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) ⟩ , ( ( ( HVMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ ⟨ ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) ⟩ ) , 𝑧 ⟩ ) , 𝑡 ⟩ ) ) ) ) = ( 𝑡𝑉 ↦ ( 𝑦 ∈ ( Base ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ∀ 𝑧𝑉 ( ¬ 𝑧 ∈ ( ( ( LSpan ‘ 𝑈 ) ‘ { ⟨ ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) ⟩ } ) ∪ ( ( LSpan ‘ 𝑈 ) ‘ { 𝑡 } ) ) → 𝑦 = ( ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑊 ) ‘ ⟨ 𝑧 , ( ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑊 ) ‘ ⟨ ⟨ ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) ⟩ , ( ( ( HVMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ ⟨ ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) ⟩ ) , 𝑧 ⟩ ) , 𝑡 ⟩ ) ) ) )
8 6 7 fnmpti ( 𝑡𝑉 ↦ ( 𝑦 ∈ ( Base ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ∀ 𝑧𝑉 ( ¬ 𝑧 ∈ ( ( ( LSpan ‘ 𝑈 ) ‘ { ⟨ ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) ⟩ } ) ∪ ( ( LSpan ‘ 𝑈 ) ‘ { 𝑡 } ) ) → 𝑦 = ( ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑊 ) ‘ ⟨ 𝑧 , ( ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑊 ) ‘ ⟨ ⟨ ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) ⟩ , ( ( ( HVMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ ⟨ ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) ⟩ ) , 𝑧 ⟩ ) , 𝑡 ⟩ ) ) ) ) Fn 𝑉
9 eqid ⟨ ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) ⟩ = ⟨ ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) ⟩
10 eqid ( LSpan ‘ 𝑈 ) = ( LSpan ‘ 𝑈 )
11 eqid ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
12 eqid ( Base ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) = ( Base ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) )
13 eqid ( ( HVMap ‘ 𝐾 ) ‘ 𝑊 ) = ( ( HVMap ‘ 𝐾 ) ‘ 𝑊 )
14 eqid ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑊 ) = ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑊 )
15 1 9 2 3 10 11 12 13 14 4 5 hdmapfval ( 𝜑𝑆 = ( 𝑡𝑉 ↦ ( 𝑦 ∈ ( Base ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ∀ 𝑧𝑉 ( ¬ 𝑧 ∈ ( ( ( LSpan ‘ 𝑈 ) ‘ { ⟨ ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) ⟩ } ) ∪ ( ( LSpan ‘ 𝑈 ) ‘ { 𝑡 } ) ) → 𝑦 = ( ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑊 ) ‘ ⟨ 𝑧 , ( ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑊 ) ‘ ⟨ ⟨ ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) ⟩ , ( ( ( HVMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ ⟨ ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) ⟩ ) , 𝑧 ⟩ ) , 𝑡 ⟩ ) ) ) ) )
16 15 fneq1d ( 𝜑 → ( 𝑆 Fn 𝑉 ↔ ( 𝑡𝑉 ↦ ( 𝑦 ∈ ( Base ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ∀ 𝑧𝑉 ( ¬ 𝑧 ∈ ( ( ( LSpan ‘ 𝑈 ) ‘ { ⟨ ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) ⟩ } ) ∪ ( ( LSpan ‘ 𝑈 ) ‘ { 𝑡 } ) ) → 𝑦 = ( ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑊 ) ‘ ⟨ 𝑧 , ( ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑊 ) ‘ ⟨ ⟨ ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) ⟩ , ( ( ( HVMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ ⟨ ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) ⟩ ) , 𝑧 ⟩ ) , 𝑡 ⟩ ) ) ) ) Fn 𝑉 ) )
17 8 16 mpbiri ( 𝜑𝑆 Fn 𝑉 )