Theorem hdmapffval

Description: Map from vectors to functionals in the closed kernel dual space. (Contributed by NM, 15-May-2015)

	Ref	Expression
Hypothesis	hdmapval.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
Assertion	hdmapffval	⊢ ( 𝐾 ∈ 𝑋 → ( HDMap ‘ 𝐾 ) = ( 𝑤 ∈ 𝐻 ↦ { 𝑎 ∣ [ ⟨ ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ) ⟩ / 𝑒 ] [ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) / 𝑢 ] [ ( Base ‘ 𝑢 ) / 𝑣 ] [ ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑤 ) / 𝑖 ] 𝑎 ∈ ( 𝑡 ∈ 𝑣 ↦ ( ℩ 𝑦 ∈ ( Base ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑤 ) ) ∀ 𝑧 ∈ 𝑣 ( ¬ 𝑧 ∈ ( ( ( LSpan ‘ 𝑢 ) ‘ { 𝑒 } ) ∪ ( ( LSpan ‘ 𝑢 ) ‘ { 𝑡 } ) ) → 𝑦 = ( 𝑖 ‘ ⟨ 𝑧 , ( 𝑖 ‘ ⟨ 𝑒 , ( ( ( HVMap ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑒 ) , 𝑧 ⟩ ) , 𝑡 ⟩ ) ) ) ) } ) )

Proof

Step	Hyp	Ref	Expression
1		hdmapval.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		elex	⊢ ( 𝐾 ∈ 𝑋 → 𝐾 ∈ V )
3		fveq2	⊢ ( 𝑘 = 𝐾 → ( LHyp ‘ 𝑘 ) = ( LHyp ‘ 𝐾 ) )
4	3 1	eqtr4di	⊢ ( 𝑘 = 𝐾 → ( LHyp ‘ 𝑘 ) = 𝐻 )
5		fveq2	⊢ ( 𝑘 = 𝐾 → ( Base ‘ 𝑘 ) = ( Base ‘ 𝐾 ) )
6	5	reseq2d	⊢ ( 𝑘 = 𝐾 → ( I ↾ ( Base ‘ 𝑘 ) ) = ( I ↾ ( Base ‘ 𝐾 ) ) )
7		fveq2	⊢ ( 𝑘 = 𝐾 → ( LTrn ‘ 𝑘 ) = ( LTrn ‘ 𝐾 ) )
8	7	fveq1d	⊢ ( 𝑘 = 𝐾 → ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) = ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) )
9	8	reseq2d	⊢ ( 𝑘 = 𝐾 → ( I ↾ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ) = ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ) )
10	6 9	opeq12d	⊢ ( 𝑘 = 𝐾 → ⟨ ( I ↾ ( Base ‘ 𝑘 ) ) , ( I ↾ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ) ⟩ = ⟨ ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ) ⟩ )
11		fveq2	⊢ ( 𝑘 = 𝐾 → ( DVecH ‘ 𝑘 ) = ( DVecH ‘ 𝐾 ) )
12	11	fveq1d	⊢ ( 𝑘 = 𝐾 → ( ( DVecH ‘ 𝑘 ) ‘ 𝑤 ) = ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) )
13		fveq2	⊢ ( 𝑘 = 𝐾 → ( HDMap1 ‘ 𝑘 ) = ( HDMap1 ‘ 𝐾 ) )
14	13	fveq1d	⊢ ( 𝑘 = 𝐾 → ( ( HDMap1 ‘ 𝑘 ) ‘ 𝑤 ) = ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑤 ) )
15		fveq2	⊢ ( 𝑘 = 𝐾 → ( LCDual ‘ 𝑘 ) = ( LCDual ‘ 𝐾 ) )
16	15	fveq1d	⊢ ( 𝑘 = 𝐾 → ( ( LCDual ‘ 𝑘 ) ‘ 𝑤 ) = ( ( LCDual ‘ 𝐾 ) ‘ 𝑤 ) )
17	16	fveq2d	⊢ ( 𝑘 = 𝐾 → ( Base ‘ ( ( LCDual ‘ 𝑘 ) ‘ 𝑤 ) ) = ( Base ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑤 ) ) )
18		fveq2	⊢ ( 𝑘 = 𝐾 → ( HVMap ‘ 𝑘 ) = ( HVMap ‘ 𝐾 ) )
19	18	fveq1d	⊢ ( 𝑘 = 𝐾 → ( ( HVMap ‘ 𝑘 ) ‘ 𝑤 ) = ( ( HVMap ‘ 𝐾 ) ‘ 𝑤 ) )
20	19	fveq1d	⊢ ( 𝑘 = 𝐾 → ( ( ( HVMap ‘ 𝑘 ) ‘ 𝑤 ) ‘ 𝑒 ) = ( ( ( HVMap ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑒 ) )
21	20	oteq2d	⊢ ( 𝑘 = 𝐾 → ⟨ 𝑒 , ( ( ( HVMap ‘ 𝑘 ) ‘ 𝑤 ) ‘ 𝑒 ) , 𝑧 ⟩ = ⟨ 𝑒 , ( ( ( HVMap ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑒 ) , 𝑧 ⟩ )
22	21	fveq2d	⊢ ( 𝑘 = 𝐾 → ( 𝑖 ‘ ⟨ 𝑒 , ( ( ( HVMap ‘ 𝑘 ) ‘ 𝑤 ) ‘ 𝑒 ) , 𝑧 ⟩ ) = ( 𝑖 ‘ ⟨ 𝑒 , ( ( ( HVMap ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑒 ) , 𝑧 ⟩ ) )
23	22	oteq2d	⊢ ( 𝑘 = 𝐾 → ⟨ 𝑧 , ( 𝑖 ‘ ⟨ 𝑒 , ( ( ( HVMap ‘ 𝑘 ) ‘ 𝑤 ) ‘ 𝑒 ) , 𝑧 ⟩ ) , 𝑡 ⟩ = ⟨ 𝑧 , ( 𝑖 ‘ ⟨ 𝑒 , ( ( ( HVMap ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑒 ) , 𝑧 ⟩ ) , 𝑡 ⟩ )
24	23	fveq2d	⊢ ( 𝑘 = 𝐾 → ( 𝑖 ‘ ⟨ 𝑧 , ( 𝑖 ‘ ⟨ 𝑒 , ( ( ( HVMap ‘ 𝑘 ) ‘ 𝑤 ) ‘ 𝑒 ) , 𝑧 ⟩ ) , 𝑡 ⟩ ) = ( 𝑖 ‘ ⟨ 𝑧 , ( 𝑖 ‘ ⟨ 𝑒 , ( ( ( HVMap ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑒 ) , 𝑧 ⟩ ) , 𝑡 ⟩ ) )
25	24	eqeq2d	⊢ ( 𝑘 = 𝐾 → ( 𝑦 = ( 𝑖 ‘ ⟨ 𝑧 , ( 𝑖 ‘ ⟨ 𝑒 , ( ( ( HVMap ‘ 𝑘 ) ‘ 𝑤 ) ‘ 𝑒 ) , 𝑧 ⟩ ) , 𝑡 ⟩ ) ↔ 𝑦 = ( 𝑖 ‘ ⟨ 𝑧 , ( 𝑖 ‘ ⟨ 𝑒 , ( ( ( HVMap ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑒 ) , 𝑧 ⟩ ) , 𝑡 ⟩ ) ) )
26	25	imbi2d	⊢ ( 𝑘 = 𝐾 → ( ( ¬ 𝑧 ∈ ( ( ( LSpan ‘ 𝑢 ) ‘ { 𝑒 } ) ∪ ( ( LSpan ‘ 𝑢 ) ‘ { 𝑡 } ) ) → 𝑦 = ( 𝑖 ‘ ⟨ 𝑧 , ( 𝑖 ‘ ⟨ 𝑒 , ( ( ( HVMap ‘ 𝑘 ) ‘ 𝑤 ) ‘ 𝑒 ) , 𝑧 ⟩ ) , 𝑡 ⟩ ) ) ↔ ( ¬ 𝑧 ∈ ( ( ( LSpan ‘ 𝑢 ) ‘ { 𝑒 } ) ∪ ( ( LSpan ‘ 𝑢 ) ‘ { 𝑡 } ) ) → 𝑦 = ( 𝑖 ‘ ⟨ 𝑧 , ( 𝑖 ‘ ⟨ 𝑒 , ( ( ( HVMap ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑒 ) , 𝑧 ⟩ ) , 𝑡 ⟩ ) ) ) )
27	26	ralbidv	⊢ ( 𝑘 = 𝐾 → ( ∀ 𝑧 ∈ 𝑣 ( ¬ 𝑧 ∈ ( ( ( LSpan ‘ 𝑢 ) ‘ { 𝑒 } ) ∪ ( ( LSpan ‘ 𝑢 ) ‘ { 𝑡 } ) ) → 𝑦 = ( 𝑖 ‘ ⟨ 𝑧 , ( 𝑖 ‘ ⟨ 𝑒 , ( ( ( HVMap ‘ 𝑘 ) ‘ 𝑤 ) ‘ 𝑒 ) , 𝑧 ⟩ ) , 𝑡 ⟩ ) ) ↔ ∀ 𝑧 ∈ 𝑣 ( ¬ 𝑧 ∈ ( ( ( LSpan ‘ 𝑢 ) ‘ { 𝑒 } ) ∪ ( ( LSpan ‘ 𝑢 ) ‘ { 𝑡 } ) ) → 𝑦 = ( 𝑖 ‘ ⟨ 𝑧 , ( 𝑖 ‘ ⟨ 𝑒 , ( ( ( HVMap ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑒 ) , 𝑧 ⟩ ) , 𝑡 ⟩ ) ) ) )
28	17 27	riotaeqbidv	⊢ ( 𝑘 = 𝐾 → ( ℩ 𝑦 ∈ ( Base ‘ ( ( LCDual ‘ 𝑘 ) ‘ 𝑤 ) ) ∀ 𝑧 ∈ 𝑣 ( ¬ 𝑧 ∈ ( ( ( LSpan ‘ 𝑢 ) ‘ { 𝑒 } ) ∪ ( ( LSpan ‘ 𝑢 ) ‘ { 𝑡 } ) ) → 𝑦 = ( 𝑖 ‘ ⟨ 𝑧 , ( 𝑖 ‘ ⟨ 𝑒 , ( ( ( HVMap ‘ 𝑘 ) ‘ 𝑤 ) ‘ 𝑒 ) , 𝑧 ⟩ ) , 𝑡 ⟩ ) ) ) = ( ℩ 𝑦 ∈ ( Base ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑤 ) ) ∀ 𝑧 ∈ 𝑣 ( ¬ 𝑧 ∈ ( ( ( LSpan ‘ 𝑢 ) ‘ { 𝑒 } ) ∪ ( ( LSpan ‘ 𝑢 ) ‘ { 𝑡 } ) ) → 𝑦 = ( 𝑖 ‘ ⟨ 𝑧 , ( 𝑖 ‘ ⟨ 𝑒 , ( ( ( HVMap ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑒 ) , 𝑧 ⟩ ) , 𝑡 ⟩ ) ) ) )
29	28	mpteq2dv	⊢ ( 𝑘 = 𝐾 → ( 𝑡 ∈ 𝑣 ↦ ( ℩ 𝑦 ∈ ( Base ‘ ( ( LCDual ‘ 𝑘 ) ‘ 𝑤 ) ) ∀ 𝑧 ∈ 𝑣 ( ¬ 𝑧 ∈ ( ( ( LSpan ‘ 𝑢 ) ‘ { 𝑒 } ) ∪ ( ( LSpan ‘ 𝑢 ) ‘ { 𝑡 } ) ) → 𝑦 = ( 𝑖 ‘ ⟨ 𝑧 , ( 𝑖 ‘ ⟨ 𝑒 , ( ( ( HVMap ‘ 𝑘 ) ‘ 𝑤 ) ‘ 𝑒 ) , 𝑧 ⟩ ) , 𝑡 ⟩ ) ) ) ) = ( 𝑡 ∈ 𝑣 ↦ ( ℩ 𝑦 ∈ ( Base ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑤 ) ) ∀ 𝑧 ∈ 𝑣 ( ¬ 𝑧 ∈ ( ( ( LSpan ‘ 𝑢 ) ‘ { 𝑒 } ) ∪ ( ( LSpan ‘ 𝑢 ) ‘ { 𝑡 } ) ) → 𝑦 = ( 𝑖 ‘ ⟨ 𝑧 , ( 𝑖 ‘ ⟨ 𝑒 , ( ( ( HVMap ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑒 ) , 𝑧 ⟩ ) , 𝑡 ⟩ ) ) ) ) )
30	29	eleq2d	⊢ ( 𝑘 = 𝐾 → ( 𝑎 ∈ ( 𝑡 ∈ 𝑣 ↦ ( ℩ 𝑦 ∈ ( Base ‘ ( ( LCDual ‘ 𝑘 ) ‘ 𝑤 ) ) ∀ 𝑧 ∈ 𝑣 ( ¬ 𝑧 ∈ ( ( ( LSpan ‘ 𝑢 ) ‘ { 𝑒 } ) ∪ ( ( LSpan ‘ 𝑢 ) ‘ { 𝑡 } ) ) → 𝑦 = ( 𝑖 ‘ ⟨ 𝑧 , ( 𝑖 ‘ ⟨ 𝑒 , ( ( ( HVMap ‘ 𝑘 ) ‘ 𝑤 ) ‘ 𝑒 ) , 𝑧 ⟩ ) , 𝑡 ⟩ ) ) ) ) ↔ 𝑎 ∈ ( 𝑡 ∈ 𝑣 ↦ ( ℩ 𝑦 ∈ ( Base ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑤 ) ) ∀ 𝑧 ∈ 𝑣 ( ¬ 𝑧 ∈ ( ( ( LSpan ‘ 𝑢 ) ‘ { 𝑒 } ) ∪ ( ( LSpan ‘ 𝑢 ) ‘ { 𝑡 } ) ) → 𝑦 = ( 𝑖 ‘ ⟨ 𝑧 , ( 𝑖 ‘ ⟨ 𝑒 , ( ( ( HVMap ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑒 ) , 𝑧 ⟩ ) , 𝑡 ⟩ ) ) ) ) ) )
31	14 30	sbceqbid	⊢ ( 𝑘 = 𝐾 → ( [ ( ( HDMap1 ‘ 𝑘 ) ‘ 𝑤 ) / 𝑖 ] 𝑎 ∈ ( 𝑡 ∈ 𝑣 ↦ ( ℩ 𝑦 ∈ ( Base ‘ ( ( LCDual ‘ 𝑘 ) ‘ 𝑤 ) ) ∀ 𝑧 ∈ 𝑣 ( ¬ 𝑧 ∈ ( ( ( LSpan ‘ 𝑢 ) ‘ { 𝑒 } ) ∪ ( ( LSpan ‘ 𝑢 ) ‘ { 𝑡 } ) ) → 𝑦 = ( 𝑖 ‘ ⟨ 𝑧 , ( 𝑖 ‘ ⟨ 𝑒 , ( ( ( HVMap ‘ 𝑘 ) ‘ 𝑤 ) ‘ 𝑒 ) , 𝑧 ⟩ ) , 𝑡 ⟩ ) ) ) ) ↔ [ ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑤 ) / 𝑖 ] 𝑎 ∈ ( 𝑡 ∈ 𝑣 ↦ ( ℩ 𝑦 ∈ ( Base ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑤 ) ) ∀ 𝑧 ∈ 𝑣 ( ¬ 𝑧 ∈ ( ( ( LSpan ‘ 𝑢 ) ‘ { 𝑒 } ) ∪ ( ( LSpan ‘ 𝑢 ) ‘ { 𝑡 } ) ) → 𝑦 = ( 𝑖 ‘ ⟨ 𝑧 , ( 𝑖 ‘ ⟨ 𝑒 , ( ( ( HVMap ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑒 ) , 𝑧 ⟩ ) , 𝑡 ⟩ ) ) ) ) ) )
32	31	sbcbidv	⊢ ( 𝑘 = 𝐾 → ( [ ( Base ‘ 𝑢 ) / 𝑣 ] [ ( ( HDMap1 ‘ 𝑘 ) ‘ 𝑤 ) / 𝑖 ] 𝑎 ∈ ( 𝑡 ∈ 𝑣 ↦ ( ℩ 𝑦 ∈ ( Base ‘ ( ( LCDual ‘ 𝑘 ) ‘ 𝑤 ) ) ∀ 𝑧 ∈ 𝑣 ( ¬ 𝑧 ∈ ( ( ( LSpan ‘ 𝑢 ) ‘ { 𝑒 } ) ∪ ( ( LSpan ‘ 𝑢 ) ‘ { 𝑡 } ) ) → 𝑦 = ( 𝑖 ‘ ⟨ 𝑧 , ( 𝑖 ‘ ⟨ 𝑒 , ( ( ( HVMap ‘ 𝑘 ) ‘ 𝑤 ) ‘ 𝑒 ) , 𝑧 ⟩ ) , 𝑡 ⟩ ) ) ) ) ↔ [ ( Base ‘ 𝑢 ) / 𝑣 ] [ ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑤 ) / 𝑖 ] 𝑎 ∈ ( 𝑡 ∈ 𝑣 ↦ ( ℩ 𝑦 ∈ ( Base ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑤 ) ) ∀ 𝑧 ∈ 𝑣 ( ¬ 𝑧 ∈ ( ( ( LSpan ‘ 𝑢 ) ‘ { 𝑒 } ) ∪ ( ( LSpan ‘ 𝑢 ) ‘ { 𝑡 } ) ) → 𝑦 = ( 𝑖 ‘ ⟨ 𝑧 , ( 𝑖 ‘ ⟨ 𝑒 , ( ( ( HVMap ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑒 ) , 𝑧 ⟩ ) , 𝑡 ⟩ ) ) ) ) ) )
33	12 32	sbceqbid	⊢ ( 𝑘 = 𝐾 → ( [ ( ( DVecH ‘ 𝑘 ) ‘ 𝑤 ) / 𝑢 ] [ ( Base ‘ 𝑢 ) / 𝑣 ] [ ( ( HDMap1 ‘ 𝑘 ) ‘ 𝑤 ) / 𝑖 ] 𝑎 ∈ ( 𝑡 ∈ 𝑣 ↦ ( ℩ 𝑦 ∈ ( Base ‘ ( ( LCDual ‘ 𝑘 ) ‘ 𝑤 ) ) ∀ 𝑧 ∈ 𝑣 ( ¬ 𝑧 ∈ ( ( ( LSpan ‘ 𝑢 ) ‘ { 𝑒 } ) ∪ ( ( LSpan ‘ 𝑢 ) ‘ { 𝑡 } ) ) → 𝑦 = ( 𝑖 ‘ ⟨ 𝑧 , ( 𝑖 ‘ ⟨ 𝑒 , ( ( ( HVMap ‘ 𝑘 ) ‘ 𝑤 ) ‘ 𝑒 ) , 𝑧 ⟩ ) , 𝑡 ⟩ ) ) ) ) ↔ [ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) / 𝑢 ] [ ( Base ‘ 𝑢 ) / 𝑣 ] [ ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑤 ) / 𝑖 ] 𝑎 ∈ ( 𝑡 ∈ 𝑣 ↦ ( ℩ 𝑦 ∈ ( Base ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑤 ) ) ∀ 𝑧 ∈ 𝑣 ( ¬ 𝑧 ∈ ( ( ( LSpan ‘ 𝑢 ) ‘ { 𝑒 } ) ∪ ( ( LSpan ‘ 𝑢 ) ‘ { 𝑡 } ) ) → 𝑦 = ( 𝑖 ‘ ⟨ 𝑧 , ( 𝑖 ‘ ⟨ 𝑒 , ( ( ( HVMap ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑒 ) , 𝑧 ⟩ ) , 𝑡 ⟩ ) ) ) ) ) )
34	10 33	sbceqbid	⊢ ( 𝑘 = 𝐾 → ( [ ⟨ ( I ↾ ( Base ‘ 𝑘 ) ) , ( I ↾ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ) ⟩ / 𝑒 ] [ ( ( DVecH ‘ 𝑘 ) ‘ 𝑤 ) / 𝑢 ] [ ( Base ‘ 𝑢 ) / 𝑣 ] [ ( ( HDMap1 ‘ 𝑘 ) ‘ 𝑤 ) / 𝑖 ] 𝑎 ∈ ( 𝑡 ∈ 𝑣 ↦ ( ℩ 𝑦 ∈ ( Base ‘ ( ( LCDual ‘ 𝑘 ) ‘ 𝑤 ) ) ∀ 𝑧 ∈ 𝑣 ( ¬ 𝑧 ∈ ( ( ( LSpan ‘ 𝑢 ) ‘ { 𝑒 } ) ∪ ( ( LSpan ‘ 𝑢 ) ‘ { 𝑡 } ) ) → 𝑦 = ( 𝑖 ‘ ⟨ 𝑧 , ( 𝑖 ‘ ⟨ 𝑒 , ( ( ( HVMap ‘ 𝑘 ) ‘ 𝑤 ) ‘ 𝑒 ) , 𝑧 ⟩ ) , 𝑡 ⟩ ) ) ) ) ↔ [ ⟨ ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ) ⟩ / 𝑒 ] [ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) / 𝑢 ] [ ( Base ‘ 𝑢 ) / 𝑣 ] [ ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑤 ) / 𝑖 ] 𝑎 ∈ ( 𝑡 ∈ 𝑣 ↦ ( ℩ 𝑦 ∈ ( Base ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑤 ) ) ∀ 𝑧 ∈ 𝑣 ( ¬ 𝑧 ∈ ( ( ( LSpan ‘ 𝑢 ) ‘ { 𝑒 } ) ∪ ( ( LSpan ‘ 𝑢 ) ‘ { 𝑡 } ) ) → 𝑦 = ( 𝑖 ‘ ⟨ 𝑧 , ( 𝑖 ‘ ⟨ 𝑒 , ( ( ( HVMap ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑒 ) , 𝑧 ⟩ ) , 𝑡 ⟩ ) ) ) ) ) )
35	34	abbidv	⊢ ( 𝑘 = 𝐾 → { 𝑎 ∣ [ ⟨ ( I ↾ ( Base ‘ 𝑘 ) ) , ( I ↾ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ) ⟩ / 𝑒 ] [ ( ( DVecH ‘ 𝑘 ) ‘ 𝑤 ) / 𝑢 ] [ ( Base ‘ 𝑢 ) / 𝑣 ] [ ( ( HDMap1 ‘ 𝑘 ) ‘ 𝑤 ) / 𝑖 ] 𝑎 ∈ ( 𝑡 ∈ 𝑣 ↦ ( ℩ 𝑦 ∈ ( Base ‘ ( ( LCDual ‘ 𝑘 ) ‘ 𝑤 ) ) ∀ 𝑧 ∈ 𝑣 ( ¬ 𝑧 ∈ ( ( ( LSpan ‘ 𝑢 ) ‘ { 𝑒 } ) ∪ ( ( LSpan ‘ 𝑢 ) ‘ { 𝑡 } ) ) → 𝑦 = ( 𝑖 ‘ ⟨ 𝑧 , ( 𝑖 ‘ ⟨ 𝑒 , ( ( ( HVMap ‘ 𝑘 ) ‘ 𝑤 ) ‘ 𝑒 ) , 𝑧 ⟩ ) , 𝑡 ⟩ ) ) ) ) } = { 𝑎 ∣ [ ⟨ ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ) ⟩ / 𝑒 ] [ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) / 𝑢 ] [ ( Base ‘ 𝑢 ) / 𝑣 ] [ ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑤 ) / 𝑖 ] 𝑎 ∈ ( 𝑡 ∈ 𝑣 ↦ ( ℩ 𝑦 ∈ ( Base ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑤 ) ) ∀ 𝑧 ∈ 𝑣 ( ¬ 𝑧 ∈ ( ( ( LSpan ‘ 𝑢 ) ‘ { 𝑒 } ) ∪ ( ( LSpan ‘ 𝑢 ) ‘ { 𝑡 } ) ) → 𝑦 = ( 𝑖 ‘ ⟨ 𝑧 , ( 𝑖 ‘ ⟨ 𝑒 , ( ( ( HVMap ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑒 ) , 𝑧 ⟩ ) , 𝑡 ⟩ ) ) ) ) } )
36	4 35	mpteq12dv	⊢ ( 𝑘 = 𝐾 → ( 𝑤 ∈ ( LHyp ‘ 𝑘 ) ↦ { 𝑎 ∣ [ ⟨ ( I ↾ ( Base ‘ 𝑘 ) ) , ( I ↾ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ) ⟩ / 𝑒 ] [ ( ( DVecH ‘ 𝑘 ) ‘ 𝑤 ) / 𝑢 ] [ ( Base ‘ 𝑢 ) / 𝑣 ] [ ( ( HDMap1 ‘ 𝑘 ) ‘ 𝑤 ) / 𝑖 ] 𝑎 ∈ ( 𝑡 ∈ 𝑣 ↦ ( ℩ 𝑦 ∈ ( Base ‘ ( ( LCDual ‘ 𝑘 ) ‘ 𝑤 ) ) ∀ 𝑧 ∈ 𝑣 ( ¬ 𝑧 ∈ ( ( ( LSpan ‘ 𝑢 ) ‘ { 𝑒 } ) ∪ ( ( LSpan ‘ 𝑢 ) ‘ { 𝑡 } ) ) → 𝑦 = ( 𝑖 ‘ ⟨ 𝑧 , ( 𝑖 ‘ ⟨ 𝑒 , ( ( ( HVMap ‘ 𝑘 ) ‘ 𝑤 ) ‘ 𝑒 ) , 𝑧 ⟩ ) , 𝑡 ⟩ ) ) ) ) } ) = ( 𝑤 ∈ 𝐻 ↦ { 𝑎 ∣ [ ⟨ ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ) ⟩ / 𝑒 ] [ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) / 𝑢 ] [ ( Base ‘ 𝑢 ) / 𝑣 ] [ ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑤 ) / 𝑖 ] 𝑎 ∈ ( 𝑡 ∈ 𝑣 ↦ ( ℩ 𝑦 ∈ ( Base ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑤 ) ) ∀ 𝑧 ∈ 𝑣 ( ¬ 𝑧 ∈ ( ( ( LSpan ‘ 𝑢 ) ‘ { 𝑒 } ) ∪ ( ( LSpan ‘ 𝑢 ) ‘ { 𝑡 } ) ) → 𝑦 = ( 𝑖 ‘ ⟨ 𝑧 , ( 𝑖 ‘ ⟨ 𝑒 , ( ( ( HVMap ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑒 ) , 𝑧 ⟩ ) , 𝑡 ⟩ ) ) ) ) } ) )
37		df-hdmap	⊢ HDMap = ( 𝑘 ∈ V ↦ ( 𝑤 ∈ ( LHyp ‘ 𝑘 ) ↦ { 𝑎 ∣ [ ⟨ ( I ↾ ( Base ‘ 𝑘 ) ) , ( I ↾ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ) ⟩ / 𝑒 ] [ ( ( DVecH ‘ 𝑘 ) ‘ 𝑤 ) / 𝑢 ] [ ( Base ‘ 𝑢 ) / 𝑣 ] [ ( ( HDMap1 ‘ 𝑘 ) ‘ 𝑤 ) / 𝑖 ] 𝑎 ∈ ( 𝑡 ∈ 𝑣 ↦ ( ℩ 𝑦 ∈ ( Base ‘ ( ( LCDual ‘ 𝑘 ) ‘ 𝑤 ) ) ∀ 𝑧 ∈ 𝑣 ( ¬ 𝑧 ∈ ( ( ( LSpan ‘ 𝑢 ) ‘ { 𝑒 } ) ∪ ( ( LSpan ‘ 𝑢 ) ‘ { 𝑡 } ) ) → 𝑦 = ( 𝑖 ‘ ⟨ 𝑧 , ( 𝑖 ‘ ⟨ 𝑒 , ( ( ( HVMap ‘ 𝑘 ) ‘ 𝑤 ) ‘ 𝑒 ) , 𝑧 ⟩ ) , 𝑡 ⟩ ) ) ) ) } ) )
38	36 37 1	mptfvmpt	⊢ ( 𝐾 ∈ V → ( HDMap ‘ 𝐾 ) = ( 𝑤 ∈ 𝐻 ↦ { 𝑎 ∣ [ ⟨ ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ) ⟩ / 𝑒 ] [ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) / 𝑢 ] [ ( Base ‘ 𝑢 ) / 𝑣 ] [ ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑤 ) / 𝑖 ] 𝑎 ∈ ( 𝑡 ∈ 𝑣 ↦ ( ℩ 𝑦 ∈ ( Base ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑤 ) ) ∀ 𝑧 ∈ 𝑣 ( ¬ 𝑧 ∈ ( ( ( LSpan ‘ 𝑢 ) ‘ { 𝑒 } ) ∪ ( ( LSpan ‘ 𝑢 ) ‘ { 𝑡 } ) ) → 𝑦 = ( 𝑖 ‘ ⟨ 𝑧 , ( 𝑖 ‘ ⟨ 𝑒 , ( ( ( HVMap ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑒 ) , 𝑧 ⟩ ) , 𝑡 ⟩ ) ) ) ) } ) )
39	2 38	syl	⊢ ( 𝐾 ∈ 𝑋 → ( HDMap ‘ 𝐾 ) = ( 𝑤 ∈ 𝐻 ↦ { 𝑎 ∣ [ ⟨ ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ) ⟩ / 𝑒 ] [ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) / 𝑢 ] [ ( Base ‘ 𝑢 ) / 𝑣 ] [ ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑤 ) / 𝑖 ] 𝑎 ∈ ( 𝑡 ∈ 𝑣 ↦ ( ℩ 𝑦 ∈ ( Base ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑤 ) ) ∀ 𝑧 ∈ 𝑣 ( ¬ 𝑧 ∈ ( ( ( LSpan ‘ 𝑢 ) ‘ { 𝑒 } ) ∪ ( ( LSpan ‘ 𝑢 ) ‘ { 𝑡 } ) ) → 𝑦 = ( 𝑖 ‘ ⟨ 𝑧 , ( 𝑖 ‘ ⟨ 𝑒 , ( ( ( HVMap ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑒 ) , 𝑧 ⟩ ) , 𝑡 ⟩ ) ) ) ) } ) )