Theorem hgmapffval

Description: Map from the scalar division ring of the vector space to the scalar division ring of its closed kernel dual. (Contributed by NM, 25-Mar-2015)

	Ref	Expression
Hypothesis	hgmapval.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
Assertion	hgmapffval	⊢ ( 𝐾 ∈ 𝑋 → ( HGMap ‘ 𝐾 ) = ( 𝑤 ∈ 𝐻 ↦ { 𝑎 ∣ [ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) / 𝑢 ] [ ( Base ‘ ( Scalar ‘ 𝑢 ) ) / 𝑏 ] [ ( ( HDMap ‘ 𝐾 ) ‘ 𝑤 ) / 𝑚 ] 𝑎 ∈ ( 𝑥 ∈ 𝑏 ↦ ( ℩ 𝑦 ∈ 𝑏 ∀ 𝑣 ∈ ( Base ‘ 𝑢 ) ( 𝑚 ‘ ( 𝑥 ( ·𝑠 ‘ 𝑢 ) 𝑣 ) ) = ( 𝑦 ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑤 ) ) ( 𝑚 ‘ 𝑣 ) ) ) ) } ) )

Proof

Step	Hyp	Ref	Expression
1		hgmapval.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 → ( DVecH ‘ 𝑘 ) = ( DVecH ‘ 𝐾 ) )
6	5	fveq1d	⊢ ( 𝑘 = 𝐾 → ( ( DVecH ‘ 𝑘 ) ‘ 𝑤 ) = ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) )
7		fveq2	⊢ ( 𝑘 = 𝐾 → ( HDMap ‘ 𝑘 ) = ( HDMap ‘ 𝐾 ) )
8	7	fveq1d	⊢ ( 𝑘 = 𝐾 → ( ( HDMap ‘ 𝑘 ) ‘ 𝑤 ) = ( ( HDMap ‘ 𝐾 ) ‘ 𝑤 ) )
9		fveq2	⊢ ( 𝑘 = 𝐾 → ( LCDual ‘ 𝑘 ) = ( LCDual ‘ 𝐾 ) )
10	9	fveq1d	⊢ ( 𝑘 = 𝐾 → ( ( LCDual ‘ 𝑘 ) ‘ 𝑤 ) = ( ( LCDual ‘ 𝐾 ) ‘ 𝑤 ) )
11	10	fveq2d	⊢ ( 𝑘 = 𝐾 → ( ·𝑠 ‘ ( ( LCDual ‘ 𝑘 ) ‘ 𝑤 ) ) = ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑤 ) ) )
12	11	oveqd	⊢ ( 𝑘 = 𝐾 → ( 𝑦 ( ·𝑠 ‘ ( ( LCDual ‘ 𝑘 ) ‘ 𝑤 ) ) ( 𝑚 ‘ 𝑣 ) ) = ( 𝑦 ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑤 ) ) ( 𝑚 ‘ 𝑣 ) ) )
13	12	eqeq2d	⊢ ( 𝑘 = 𝐾 → ( ( 𝑚 ‘ ( 𝑥 ( ·𝑠 ‘ 𝑢 ) 𝑣 ) ) = ( 𝑦 ( ·𝑠 ‘ ( ( LCDual ‘ 𝑘 ) ‘ 𝑤 ) ) ( 𝑚 ‘ 𝑣 ) ) ↔ ( 𝑚 ‘ ( 𝑥 ( ·𝑠 ‘ 𝑢 ) 𝑣 ) ) = ( 𝑦 ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑤 ) ) ( 𝑚 ‘ 𝑣 ) ) ) )
14	13	ralbidv	⊢ ( 𝑘 = 𝐾 → ( ∀ 𝑣 ∈ ( Base ‘ 𝑢 ) ( 𝑚 ‘ ( 𝑥 ( ·𝑠 ‘ 𝑢 ) 𝑣 ) ) = ( 𝑦 ( ·𝑠 ‘ ( ( LCDual ‘ 𝑘 ) ‘ 𝑤 ) ) ( 𝑚 ‘ 𝑣 ) ) ↔ ∀ 𝑣 ∈ ( Base ‘ 𝑢 ) ( 𝑚 ‘ ( 𝑥 ( ·𝑠 ‘ 𝑢 ) 𝑣 ) ) = ( 𝑦 ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑤 ) ) ( 𝑚 ‘ 𝑣 ) ) ) )
15	14	riotabidv	⊢ ( 𝑘 = 𝐾 → ( ℩ 𝑦 ∈ 𝑏 ∀ 𝑣 ∈ ( Base ‘ 𝑢 ) ( 𝑚 ‘ ( 𝑥 ( ·𝑠 ‘ 𝑢 ) 𝑣 ) ) = ( 𝑦 ( ·𝑠 ‘ ( ( LCDual ‘ 𝑘 ) ‘ 𝑤 ) ) ( 𝑚 ‘ 𝑣 ) ) ) = ( ℩ 𝑦 ∈ 𝑏 ∀ 𝑣 ∈ ( Base ‘ 𝑢 ) ( 𝑚 ‘ ( 𝑥 ( ·𝑠 ‘ 𝑢 ) 𝑣 ) ) = ( 𝑦 ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑤 ) ) ( 𝑚 ‘ 𝑣 ) ) ) )
16	15	mpteq2dv	⊢ ( 𝑘 = 𝐾 → ( 𝑥 ∈ 𝑏 ↦ ( ℩ 𝑦 ∈ 𝑏 ∀ 𝑣 ∈ ( Base ‘ 𝑢 ) ( 𝑚 ‘ ( 𝑥 ( ·𝑠 ‘ 𝑢 ) 𝑣 ) ) = ( 𝑦 ( ·𝑠 ‘ ( ( LCDual ‘ 𝑘 ) ‘ 𝑤 ) ) ( 𝑚 ‘ 𝑣 ) ) ) ) = ( 𝑥 ∈ 𝑏 ↦ ( ℩ 𝑦 ∈ 𝑏 ∀ 𝑣 ∈ ( Base ‘ 𝑢 ) ( 𝑚 ‘ ( 𝑥 ( ·𝑠 ‘ 𝑢 ) 𝑣 ) ) = ( 𝑦 ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑤 ) ) ( 𝑚 ‘ 𝑣 ) ) ) ) )
17	16	eleq2d	⊢ ( 𝑘 = 𝐾 → ( 𝑎 ∈ ( 𝑥 ∈ 𝑏 ↦ ( ℩ 𝑦 ∈ 𝑏 ∀ 𝑣 ∈ ( Base ‘ 𝑢 ) ( 𝑚 ‘ ( 𝑥 ( ·𝑠 ‘ 𝑢 ) 𝑣 ) ) = ( 𝑦 ( ·𝑠 ‘ ( ( LCDual ‘ 𝑘 ) ‘ 𝑤 ) ) ( 𝑚 ‘ 𝑣 ) ) ) ) ↔ 𝑎 ∈ ( 𝑥 ∈ 𝑏 ↦ ( ℩ 𝑦 ∈ 𝑏 ∀ 𝑣 ∈ ( Base ‘ 𝑢 ) ( 𝑚 ‘ ( 𝑥 ( ·𝑠 ‘ 𝑢 ) 𝑣 ) ) = ( 𝑦 ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑤 ) ) ( 𝑚 ‘ 𝑣 ) ) ) ) ) )
18	8 17	sbceqbid	⊢ ( 𝑘 = 𝐾 → ( [ ( ( HDMap ‘ 𝑘 ) ‘ 𝑤 ) / 𝑚 ] 𝑎 ∈ ( 𝑥 ∈ 𝑏 ↦ ( ℩ 𝑦 ∈ 𝑏 ∀ 𝑣 ∈ ( Base ‘ 𝑢 ) ( 𝑚 ‘ ( 𝑥 ( ·𝑠 ‘ 𝑢 ) 𝑣 ) ) = ( 𝑦 ( ·𝑠 ‘ ( ( LCDual ‘ 𝑘 ) ‘ 𝑤 ) ) ( 𝑚 ‘ 𝑣 ) ) ) ) ↔ [ ( ( HDMap ‘ 𝐾 ) ‘ 𝑤 ) / 𝑚 ] 𝑎 ∈ ( 𝑥 ∈ 𝑏 ↦ ( ℩ 𝑦 ∈ 𝑏 ∀ 𝑣 ∈ ( Base ‘ 𝑢 ) ( 𝑚 ‘ ( 𝑥 ( ·𝑠 ‘ 𝑢 ) 𝑣 ) ) = ( 𝑦 ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑤 ) ) ( 𝑚 ‘ 𝑣 ) ) ) ) ) )
19	18	sbcbidv	⊢ ( 𝑘 = 𝐾 → ( [ ( Base ‘ ( Scalar ‘ 𝑢 ) ) / 𝑏 ] [ ( ( HDMap ‘ 𝑘 ) ‘ 𝑤 ) / 𝑚 ] 𝑎 ∈ ( 𝑥 ∈ 𝑏 ↦ ( ℩ 𝑦 ∈ 𝑏 ∀ 𝑣 ∈ ( Base ‘ 𝑢 ) ( 𝑚 ‘ ( 𝑥 ( ·𝑠 ‘ 𝑢 ) 𝑣 ) ) = ( 𝑦 ( ·𝑠 ‘ ( ( LCDual ‘ 𝑘 ) ‘ 𝑤 ) ) ( 𝑚 ‘ 𝑣 ) ) ) ) ↔ [ ( Base ‘ ( Scalar ‘ 𝑢 ) ) / 𝑏 ] [ ( ( HDMap ‘ 𝐾 ) ‘ 𝑤 ) / 𝑚 ] 𝑎 ∈ ( 𝑥 ∈ 𝑏 ↦ ( ℩ 𝑦 ∈ 𝑏 ∀ 𝑣 ∈ ( Base ‘ 𝑢 ) ( 𝑚 ‘ ( 𝑥 ( ·𝑠 ‘ 𝑢 ) 𝑣 ) ) = ( 𝑦 ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑤 ) ) ( 𝑚 ‘ 𝑣 ) ) ) ) ) )
20	6 19	sbceqbid	⊢ ( 𝑘 = 𝐾 → ( [ ( ( DVecH ‘ 𝑘 ) ‘ 𝑤 ) / 𝑢 ] [ ( Base ‘ ( Scalar ‘ 𝑢 ) ) / 𝑏 ] [ ( ( HDMap ‘ 𝑘 ) ‘ 𝑤 ) / 𝑚 ] 𝑎 ∈ ( 𝑥 ∈ 𝑏 ↦ ( ℩ 𝑦 ∈ 𝑏 ∀ 𝑣 ∈ ( Base ‘ 𝑢 ) ( 𝑚 ‘ ( 𝑥 ( ·𝑠 ‘ 𝑢 ) 𝑣 ) ) = ( 𝑦 ( ·𝑠 ‘ ( ( LCDual ‘ 𝑘 ) ‘ 𝑤 ) ) ( 𝑚 ‘ 𝑣 ) ) ) ) ↔ [ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) / 𝑢 ] [ ( Base ‘ ( Scalar ‘ 𝑢 ) ) / 𝑏 ] [ ( ( HDMap ‘ 𝐾 ) ‘ 𝑤 ) / 𝑚 ] 𝑎 ∈ ( 𝑥 ∈ 𝑏 ↦ ( ℩ 𝑦 ∈ 𝑏 ∀ 𝑣 ∈ ( Base ‘ 𝑢 ) ( 𝑚 ‘ ( 𝑥 ( ·𝑠 ‘ 𝑢 ) 𝑣 ) ) = ( 𝑦 ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑤 ) ) ( 𝑚 ‘ 𝑣 ) ) ) ) ) )
21	20	abbidv	⊢ ( 𝑘 = 𝐾 → { 𝑎 ∣ [ ( ( DVecH ‘ 𝑘 ) ‘ 𝑤 ) / 𝑢 ] [ ( Base ‘ ( Scalar ‘ 𝑢 ) ) / 𝑏 ] [ ( ( HDMap ‘ 𝑘 ) ‘ 𝑤 ) / 𝑚 ] 𝑎 ∈ ( 𝑥 ∈ 𝑏 ↦ ( ℩ 𝑦 ∈ 𝑏 ∀ 𝑣 ∈ ( Base ‘ 𝑢 ) ( 𝑚 ‘ ( 𝑥 ( ·𝑠 ‘ 𝑢 ) 𝑣 ) ) = ( 𝑦 ( ·𝑠 ‘ ( ( LCDual ‘ 𝑘 ) ‘ 𝑤 ) ) ( 𝑚 ‘ 𝑣 ) ) ) ) } = { 𝑎 ∣ [ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) / 𝑢 ] [ ( Base ‘ ( Scalar ‘ 𝑢 ) ) / 𝑏 ] [ ( ( HDMap ‘ 𝐾 ) ‘ 𝑤 ) / 𝑚 ] 𝑎 ∈ ( 𝑥 ∈ 𝑏 ↦ ( ℩ 𝑦 ∈ 𝑏 ∀ 𝑣 ∈ ( Base ‘ 𝑢 ) ( 𝑚 ‘ ( 𝑥 ( ·𝑠 ‘ 𝑢 ) 𝑣 ) ) = ( 𝑦 ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑤 ) ) ( 𝑚 ‘ 𝑣 ) ) ) ) } )
22	4 21	mpteq12dv	⊢ ( 𝑘 = 𝐾 → ( 𝑤 ∈ ( LHyp ‘ 𝑘 ) ↦ { 𝑎 ∣ [ ( ( DVecH ‘ 𝑘 ) ‘ 𝑤 ) / 𝑢 ] [ ( Base ‘ ( Scalar ‘ 𝑢 ) ) / 𝑏 ] [ ( ( HDMap ‘ 𝑘 ) ‘ 𝑤 ) / 𝑚 ] 𝑎 ∈ ( 𝑥 ∈ 𝑏 ↦ ( ℩ 𝑦 ∈ 𝑏 ∀ 𝑣 ∈ ( Base ‘ 𝑢 ) ( 𝑚 ‘ ( 𝑥 ( ·𝑠 ‘ 𝑢 ) 𝑣 ) ) = ( 𝑦 ( ·𝑠 ‘ ( ( LCDual ‘ 𝑘 ) ‘ 𝑤 ) ) ( 𝑚 ‘ 𝑣 ) ) ) ) } ) = ( 𝑤 ∈ 𝐻 ↦ { 𝑎 ∣ [ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) / 𝑢 ] [ ( Base ‘ ( Scalar ‘ 𝑢 ) ) / 𝑏 ] [ ( ( HDMap ‘ 𝐾 ) ‘ 𝑤 ) / 𝑚 ] 𝑎 ∈ ( 𝑥 ∈ 𝑏 ↦ ( ℩ 𝑦 ∈ 𝑏 ∀ 𝑣 ∈ ( Base ‘ 𝑢 ) ( 𝑚 ‘ ( 𝑥 ( ·𝑠 ‘ 𝑢 ) 𝑣 ) ) = ( 𝑦 ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑤 ) ) ( 𝑚 ‘ 𝑣 ) ) ) ) } ) )
23		df-hgmap	⊢ HGMap = ( 𝑘 ∈ V ↦ ( 𝑤 ∈ ( LHyp ‘ 𝑘 ) ↦ { 𝑎 ∣ [ ( ( DVecH ‘ 𝑘 ) ‘ 𝑤 ) / 𝑢 ] [ ( Base ‘ ( Scalar ‘ 𝑢 ) ) / 𝑏 ] [ ( ( HDMap ‘ 𝑘 ) ‘ 𝑤 ) / 𝑚 ] 𝑎 ∈ ( 𝑥 ∈ 𝑏 ↦ ( ℩ 𝑦 ∈ 𝑏 ∀ 𝑣 ∈ ( Base ‘ 𝑢 ) ( 𝑚 ‘ ( 𝑥 ( ·𝑠 ‘ 𝑢 ) 𝑣 ) ) = ( 𝑦 ( ·𝑠 ‘ ( ( LCDual ‘ 𝑘 ) ‘ 𝑤 ) ) ( 𝑚 ‘ 𝑣 ) ) ) ) } ) )
24	22 23 1	mptfvmpt	⊢ ( 𝐾 ∈ V → ( HGMap ‘ 𝐾 ) = ( 𝑤 ∈ 𝐻 ↦ { 𝑎 ∣ [ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) / 𝑢 ] [ ( Base ‘ ( Scalar ‘ 𝑢 ) ) / 𝑏 ] [ ( ( HDMap ‘ 𝐾 ) ‘ 𝑤 ) / 𝑚 ] 𝑎 ∈ ( 𝑥 ∈ 𝑏 ↦ ( ℩ 𝑦 ∈ 𝑏 ∀ 𝑣 ∈ ( Base ‘ 𝑢 ) ( 𝑚 ‘ ( 𝑥 ( ·𝑠 ‘ 𝑢 ) 𝑣 ) ) = ( 𝑦 ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑤 ) ) ( 𝑚 ‘ 𝑣 ) ) ) ) } ) )
25	2 24	syl	⊢ ( 𝐾 ∈ 𝑋 → ( HGMap ‘ 𝐾 ) = ( 𝑤 ∈ 𝐻 ↦ { 𝑎 ∣ [ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) / 𝑢 ] [ ( Base ‘ ( Scalar ‘ 𝑢 ) ) / 𝑏 ] [ ( ( HDMap ‘ 𝐾 ) ‘ 𝑤 ) / 𝑚 ] 𝑎 ∈ ( 𝑥 ∈ 𝑏 ↦ ( ℩ 𝑦 ∈ 𝑏 ∀ 𝑣 ∈ ( Base ‘ 𝑢 ) ( 𝑚 ‘ ( 𝑥 ( ·𝑠 ‘ 𝑢 ) 𝑣 ) ) = ( 𝑦 ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑤 ) ) ( 𝑚 ‘ 𝑣 ) ) ) ) } ) )