Metamath Proof Explorer

Theorem hdmapval2lem

Description: Lemma for hdmapval2 . (Contributed by NM, 15-May-2015)

Ref Expression
Hypotheses hdmapval2.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
hdmapval2.e ⊢ 𝐸 = ⟨ ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) ⟩
hdmapval2.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
hdmapval2.v ⊢ 𝑉 = ( Base ‘ 𝑈 )
hdmapval2.n ⊢ 𝑁 = ( LSpan ‘ 𝑈 )
hdmapval2.c ⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
hdmapval2.d ⊢ 𝐷 = ( Base ‘ 𝐶 )
hdmapval2.j ⊢ 𝐽 = ( ( HVMap ‘ 𝐾 ) ‘ 𝑊 )
hdmapval2.i ⊢ 𝐼 = ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑊 )
hdmapval2.s ⊢ 𝑆 = ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 )
hdmapval2.k ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
hdmapval2.t ⊢ ( 𝜑 → 𝑇 ∈ 𝑉 )
hdmapval2.f ⊢ ( 𝜑 → 𝐹 ∈ 𝐷 )
Assertion hdmapval2lem ( 𝜑 → ( ( 𝑆 ‘ 𝑇 ) = 𝐹 ↔ ∀ 𝑧 ∈ 𝑉 ( ¬ 𝑧 ∈ ( ( 𝑁 ‘ { 𝐸 } ) ∪ ( 𝑁 ‘ { 𝑇 } ) ) → 𝐹 = ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝐸 , ( 𝐽 ‘ 𝐸 ) , 𝑧 ⟩ ) , 𝑇 ⟩ ) ) ) )

Proof

Step Hyp Ref Expression
1 hdmapval2.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2 hdmapval2.e ⊢ 𝐸 = ⟨ ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) ⟩
3 hdmapval2.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
4 hdmapval2.v ⊢ 𝑉 = ( Base ‘ 𝑈 )
5 hdmapval2.n ⊢ 𝑁 = ( LSpan ‘ 𝑈 )
6 hdmapval2.c ⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
7 hdmapval2.d ⊢ 𝐷 = ( Base ‘ 𝐶 )
8 hdmapval2.j ⊢ 𝐽 = ( ( HVMap ‘ 𝐾 ) ‘ 𝑊 )
9 hdmapval2.i ⊢ 𝐼 = ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑊 )
10 hdmapval2.s ⊢ 𝑆 = ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 )
11 hdmapval2.k ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
12 hdmapval2.t ⊢ ( 𝜑 → 𝑇 ∈ 𝑉 )
13 hdmapval2.f ⊢ ( 𝜑 → 𝐹 ∈ 𝐷 )
14 1 2 3 4 5 6 7 8 9 10 11 12 hdmapval ⊢ ( 𝜑 → ( 𝑆 ‘ 𝑇 ) = ( ℩ ℎ ∈ 𝐷 ∀ 𝑧 ∈ 𝑉 ( ¬ 𝑧 ∈ ( ( 𝑁 ‘ { 𝐸 } ) ∪ ( 𝑁 ‘ { 𝑇 } ) ) → ℎ = ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝐸 , ( 𝐽 ‘ 𝐸 ) , 𝑧 ⟩ ) , 𝑇 ⟩ ) ) ) )
15 14 eqeq1d ⊢ ( 𝜑 → ( ( 𝑆 ‘ 𝑇 ) = 𝐹 ↔ ( ℩ ℎ ∈ 𝐷 ∀ 𝑧 ∈ 𝑉 ( ¬ 𝑧 ∈ ( ( 𝑁 ‘ { 𝐸 } ) ∪ ( 𝑁 ‘ { 𝑇 } ) ) → ℎ = ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝐸 , ( 𝐽 ‘ 𝐸 ) , 𝑧 ⟩ ) , 𝑇 ⟩ ) ) ) = 𝐹 ) )
16 eqid ⊢ ( 0g ‘ 𝑈 ) = ( 0g ‘ 𝑈 )
17 eqid ⊢ ( LSpan ‘ 𝐶 ) = ( LSpan ‘ 𝐶 )
18 eqid ⊢ ( ( mapd ‘ 𝐾 ) ‘ 𝑊 ) = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
19 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
20 eqid ⊢ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
21 1 19 20 3 4 16 2 11 dvheveccl ⊢ ( 𝜑 → 𝐸 ∈ ( 𝑉 ∖ { ( 0g ‘ 𝑈 ) } ) )
22 1 3 4 16 5 6 17 18 8 11 21 mapdhvmap ⊢ ( 𝜑 → ( ( ( mapd ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑁 ‘ { 𝐸 } ) ) = ( ( LSpan ‘ 𝐶 ) ‘ { ( 𝐽 ‘ 𝐸 ) } ) )
23 eqid ⊢ ( 0g ‘ 𝐶 ) = ( 0g ‘ 𝐶 )
24 1 3 4 16 6 7 23 8 11 21 hvmapcl2 ⊢ ( 𝜑 → ( 𝐽 ‘ 𝐸 ) ∈ ( 𝐷 ∖ { ( 0g ‘ 𝐶 ) } ) )
25 24 eldifad ⊢ ( 𝜑 → ( 𝐽 ‘ 𝐸 ) ∈ 𝐷 )
26 1 3 4 16 5 6 7 17 18 9 11 22 21 25 12 hdmap1eu ⊢ ( 𝜑 → ∃! ℎ ∈ 𝐷 ∀ 𝑧 ∈ 𝑉 ( ¬ 𝑧 ∈ ( ( 𝑁 ‘ { 𝐸 } ) ∪ ( 𝑁 ‘ { 𝑇 } ) ) → ℎ = ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝐸 , ( 𝐽 ‘ 𝐸 ) , 𝑧 ⟩ ) , 𝑇 ⟩ ) ) )
27 nfv ⊢ Ⅎ ℎ 𝜑
28 nfcvd ⊢ ( 𝜑 → Ⅎ ℎ 𝐹 )
29 nfvd ⊢ ( 𝜑 → Ⅎ ℎ ∀ 𝑧 ∈ 𝑉 ( ¬ 𝑧 ∈ ( ( 𝑁 ‘ { 𝐸 } ) ∪ ( 𝑁 ‘ { 𝑇 } ) ) → 𝐹 = ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝐸 , ( 𝐽 ‘ 𝐸 ) , 𝑧 ⟩ ) , 𝑇 ⟩ ) ) )
30 eqeq1 ⊢ ( ℎ = 𝐹 → ( ℎ = ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝐸 , ( 𝐽 ‘ 𝐸 ) , 𝑧 ⟩ ) , 𝑇 ⟩ ) ↔ 𝐹 = ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝐸 , ( 𝐽 ‘ 𝐸 ) , 𝑧 ⟩ ) , 𝑇 ⟩ ) ) )
31 30 imbi2d ⊢ ( ℎ = 𝐹 → ( ( ¬ 𝑧 ∈ ( ( 𝑁 ‘ { 𝐸 } ) ∪ ( 𝑁 ‘ { 𝑇 } ) ) → ℎ = ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝐸 , ( 𝐽 ‘ 𝐸 ) , 𝑧 ⟩ ) , 𝑇 ⟩ ) ) ↔ ( ¬ 𝑧 ∈ ( ( 𝑁 ‘ { 𝐸 } ) ∪ ( 𝑁 ‘ { 𝑇 } ) ) → 𝐹 = ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝐸 , ( 𝐽 ‘ 𝐸 ) , 𝑧 ⟩ ) , 𝑇 ⟩ ) ) ) )
32 31 ralbidv ⊢ ( ℎ = 𝐹 → ( ∀ 𝑧 ∈ 𝑉 ( ¬ 𝑧 ∈ ( ( 𝑁 ‘ { 𝐸 } ) ∪ ( 𝑁 ‘ { 𝑇 } ) ) → ℎ = ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝐸 , ( 𝐽 ‘ 𝐸 ) , 𝑧 ⟩ ) , 𝑇 ⟩ ) ) ↔ ∀ 𝑧 ∈ 𝑉 ( ¬ 𝑧 ∈ ( ( 𝑁 ‘ { 𝐸 } ) ∪ ( 𝑁 ‘ { 𝑇 } ) ) → 𝐹 = ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝐸 , ( 𝐽 ‘ 𝐸 ) , 𝑧 ⟩ ) , 𝑇 ⟩ ) ) ) )
33 32 adantl ⊢ ( ( 𝜑 ∧ ℎ = 𝐹 ) → ( ∀ 𝑧 ∈ 𝑉 ( ¬ 𝑧 ∈ ( ( 𝑁 ‘ { 𝐸 } ) ∪ ( 𝑁 ‘ { 𝑇 } ) ) → ℎ = ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝐸 , ( 𝐽 ‘ 𝐸 ) , 𝑧 ⟩ ) , 𝑇 ⟩ ) ) ↔ ∀ 𝑧 ∈ 𝑉 ( ¬ 𝑧 ∈ ( ( 𝑁 ‘ { 𝐸 } ) ∪ ( 𝑁 ‘ { 𝑇 } ) ) → 𝐹 = ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝐸 , ( 𝐽 ‘ 𝐸 ) , 𝑧 ⟩ ) , 𝑇 ⟩ ) ) ) )
34 27 28 29 13 33 riota2df ⊢ ( ( 𝜑 ∧ ∃! ℎ ∈ 𝐷 ∀ 𝑧 ∈ 𝑉 ( ¬ 𝑧 ∈ ( ( 𝑁 ‘ { 𝐸 } ) ∪ ( 𝑁 ‘ { 𝑇 } ) ) → ℎ = ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝐸 , ( 𝐽 ‘ 𝐸 ) , 𝑧 ⟩ ) , 𝑇 ⟩ ) ) ) → ( ∀ 𝑧 ∈ 𝑉 ( ¬ 𝑧 ∈ ( ( 𝑁 ‘ { 𝐸 } ) ∪ ( 𝑁 ‘ { 𝑇 } ) ) → 𝐹 = ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝐸 , ( 𝐽 ‘ 𝐸 ) , 𝑧 ⟩ ) , 𝑇 ⟩ ) ) ↔ ( ℩ ℎ ∈ 𝐷 ∀ 𝑧 ∈ 𝑉 ( ¬ 𝑧 ∈ ( ( 𝑁 ‘ { 𝐸 } ) ∪ ( 𝑁 ‘ { 𝑇 } ) ) → ℎ = ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝐸 , ( 𝐽 ‘ 𝐸 ) , 𝑧 ⟩ ) , 𝑇 ⟩ ) ) ) = 𝐹 ) )
35 26 34 mpdan ⊢ ( 𝜑 → ( ∀ 𝑧 ∈ 𝑉 ( ¬ 𝑧 ∈ ( ( 𝑁 ‘ { 𝐸 } ) ∪ ( 𝑁 ‘ { 𝑇 } ) ) → 𝐹 = ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝐸 , ( 𝐽 ‘ 𝐸 ) , 𝑧 ⟩ ) , 𝑇 ⟩ ) ) ↔ ( ℩ ℎ ∈ 𝐷 ∀ 𝑧 ∈ 𝑉 ( ¬ 𝑧 ∈ ( ( 𝑁 ‘ { 𝐸 } ) ∪ ( 𝑁 ‘ { 𝑇 } ) ) → ℎ = ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝐸 , ( 𝐽 ‘ 𝐸 ) , 𝑧 ⟩ ) , 𝑇 ⟩ ) ) ) = 𝐹 ) )
36 15 35 bitr4d ⊢ ( 𝜑 → ( ( 𝑆 ‘ 𝑇 ) = 𝐹 ↔ ∀ 𝑧 ∈ 𝑉 ( ¬ 𝑧 ∈ ( ( 𝑁 ‘ { 𝐸 } ) ∪ ( 𝑁 ‘ { 𝑇 } ) ) → 𝐹 = ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝐸 , ( 𝐽 ‘ 𝐸 ) , 𝑧 ⟩ ) , 𝑇 ⟩ ) ) ) )