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 ∧ 𝑊𝐻 ) )
hdmapval2.t ( 𝜑𝑇𝑉 )
hdmapval2.f ( 𝜑𝐹𝐷 )
Assertion hdmapval2lem ( 𝜑 → ( ( 𝑆𝑇 ) = 𝐹 ↔ ∀ 𝑧𝑉 ( ¬ 𝑧 ∈ ( ( 𝑁 ‘ { 𝐸 } ) ∪ ( 𝑁 ‘ { 𝑇 } ) ) → 𝐹 = ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝐸 , ( 𝐽𝐸 ) , 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 ∧ 𝑊𝐻 ) )
12 hdmapval2.t ( 𝜑𝑇𝑉 )
13 hdmapval2.f ( 𝜑𝐹𝐷 )
14 1 2 3 4 5 6 7 8 9 10 11 12 hdmapval ( 𝜑 → ( 𝑆𝑇 ) = ( 𝐷𝑧𝑉 ( ¬ 𝑧 ∈ ( ( 𝑁 ‘ { 𝐸 } ) ∪ ( 𝑁 ‘ { 𝑇 } ) ) → = ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝐸 , ( 𝐽𝐸 ) , 𝑧 ⟩ ) , 𝑇 ⟩ ) ) ) )
15 14 eqeq1d ( 𝜑 → ( ( 𝑆𝑇 ) = 𝐹 ↔ ( 𝐷𝑧𝑉 ( ¬ 𝑧 ∈ ( ( 𝑁 ‘ { 𝐸 } ) ∪ ( 𝑁 ‘ { 𝑇 } ) ) → = ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝐸 , ( 𝐽𝐸 ) , 𝑧 ⟩ ) , 𝑇 ⟩ ) ) ) = 𝐹 ) )
16 eqid ( 0g𝑈 ) = ( 0g𝑈 )
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 ( 𝜑𝐸 ∈ ( 𝑉 ∖ { ( 0g𝑈 ) } ) )
22 1 3 4 16 5 6 17 18 8 11 21 mapdhvmap ( 𝜑 → ( ( ( mapd ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑁 ‘ { 𝐸 } ) ) = ( ( LSpan ‘ 𝐶 ) ‘ { ( 𝐽𝐸 ) } ) )
23 eqid ( 0g𝐶 ) = ( 0g𝐶 )
24 1 3 4 16 6 7 23 8 11 21 hvmapcl2 ( 𝜑 → ( 𝐽𝐸 ) ∈ ( 𝐷 ∖ { ( 0g𝐶 ) } ) )
25 24 eldifad ( 𝜑 → ( 𝐽𝐸 ) ∈ 𝐷 )
26 1 3 4 16 5 6 7 17 18 9 11 22 21 25 12 hdmap1eu ( 𝜑 → ∃! 𝐷𝑧𝑉 ( ¬ 𝑧 ∈ ( ( 𝑁 ‘ { 𝐸 } ) ∪ ( 𝑁 ‘ { 𝑇 } ) ) → = ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝐸 , ( 𝐽𝐸 ) , 𝑧 ⟩ ) , 𝑇 ⟩ ) ) )
27 nfv 𝜑
28 nfcvd ( 𝜑 𝐹 )
29 nfvd ( 𝜑 → Ⅎ 𝑧𝑉 ( ¬ 𝑧 ∈ ( ( 𝑁 ‘ { 𝐸 } ) ∪ ( 𝑁 ‘ { 𝑇 } ) ) → 𝐹 = ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝐸 , ( 𝐽𝐸 ) , 𝑧 ⟩ ) , 𝑇 ⟩ ) ) )
30 eqeq1 ( = 𝐹 → ( = ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝐸 , ( 𝐽𝐸 ) , 𝑧 ⟩ ) , 𝑇 ⟩ ) ↔ 𝐹 = ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝐸 , ( 𝐽𝐸 ) , 𝑧 ⟩ ) , 𝑇 ⟩ ) ) )
31 30 imbi2d ( = 𝐹 → ( ( ¬ 𝑧 ∈ ( ( 𝑁 ‘ { 𝐸 } ) ∪ ( 𝑁 ‘ { 𝑇 } ) ) → = ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝐸 , ( 𝐽𝐸 ) , 𝑧 ⟩ ) , 𝑇 ⟩ ) ) ↔ ( ¬ 𝑧 ∈ ( ( 𝑁 ‘ { 𝐸 } ) ∪ ( 𝑁 ‘ { 𝑇 } ) ) → 𝐹 = ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝐸 , ( 𝐽𝐸 ) , 𝑧 ⟩ ) , 𝑇 ⟩ ) ) ) )
32 31 ralbidv ( = 𝐹 → ( ∀ 𝑧𝑉 ( ¬ 𝑧 ∈ ( ( 𝑁 ‘ { 𝐸 } ) ∪ ( 𝑁 ‘ { 𝑇 } ) ) → = ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝐸 , ( 𝐽𝐸 ) , 𝑧 ⟩ ) , 𝑇 ⟩ ) ) ↔ ∀ 𝑧𝑉 ( ¬ 𝑧 ∈ ( ( 𝑁 ‘ { 𝐸 } ) ∪ ( 𝑁 ‘ { 𝑇 } ) ) → 𝐹 = ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝐸 , ( 𝐽𝐸 ) , 𝑧 ⟩ ) , 𝑇 ⟩ ) ) ) )
33 32 adantl ( ( 𝜑 = 𝐹 ) → ( ∀ 𝑧𝑉 ( ¬ 𝑧 ∈ ( ( 𝑁 ‘ { 𝐸 } ) ∪ ( 𝑁 ‘ { 𝑇 } ) ) → = ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝐸 , ( 𝐽𝐸 ) , 𝑧 ⟩ ) , 𝑇 ⟩ ) ) ↔ ∀ 𝑧𝑉 ( ¬ 𝑧 ∈ ( ( 𝑁 ‘ { 𝐸 } ) ∪ ( 𝑁 ‘ { 𝑇 } ) ) → 𝐹 = ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝐸 , ( 𝐽𝐸 ) , 𝑧 ⟩ ) , 𝑇 ⟩ ) ) ) )
34 27 28 29 13 33 riota2df ( ( 𝜑 ∧ ∃! 𝐷𝑧𝑉 ( ¬ 𝑧 ∈ ( ( 𝑁 ‘ { 𝐸 } ) ∪ ( 𝑁 ‘ { 𝑇 } ) ) → = ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝐸 , ( 𝐽𝐸 ) , 𝑧 ⟩ ) , 𝑇 ⟩ ) ) ) → ( ∀ 𝑧𝑉 ( ¬ 𝑧 ∈ ( ( 𝑁 ‘ { 𝐸 } ) ∪ ( 𝑁 ‘ { 𝑇 } ) ) → 𝐹 = ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝐸 , ( 𝐽𝐸 ) , 𝑧 ⟩ ) , 𝑇 ⟩ ) ) ↔ ( 𝐷𝑧𝑉 ( ¬ 𝑧 ∈ ( ( 𝑁 ‘ { 𝐸 } ) ∪ ( 𝑁 ‘ { 𝑇 } ) ) → = ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝐸 , ( 𝐽𝐸 ) , 𝑧 ⟩ ) , 𝑇 ⟩ ) ) ) = 𝐹 ) )
35 26 34 mpdan ( 𝜑 → ( ∀ 𝑧𝑉 ( ¬ 𝑧 ∈ ( ( 𝑁 ‘ { 𝐸 } ) ∪ ( 𝑁 ‘ { 𝑇 } ) ) → 𝐹 = ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝐸 , ( 𝐽𝐸 ) , 𝑧 ⟩ ) , 𝑇 ⟩ ) ) ↔ ( 𝐷𝑧𝑉 ( ¬ 𝑧 ∈ ( ( 𝑁 ‘ { 𝐸 } ) ∪ ( 𝑁 ‘ { 𝑇 } ) ) → = ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝐸 , ( 𝐽𝐸 ) , 𝑧 ⟩ ) , 𝑇 ⟩ ) ) ) = 𝐹 ) )
36 15 35 bitr4d ( 𝜑 → ( ( 𝑆𝑇 ) = 𝐹 ↔ ∀ 𝑧𝑉 ( ¬ 𝑧 ∈ ( ( 𝑁 ‘ { 𝐸 } ) ∪ ( 𝑁 ‘ { 𝑇 } ) ) → 𝐹 = ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝐸 , ( 𝐽𝐸 ) , 𝑧 ⟩ ) , 𝑇 ⟩ ) ) ) )