Theorem hgmapfnN

Description: Functionality of scalar sigma map. (Contributed by NM, 7-Jun-2015) (New usage is discouraged.)

	Ref	Expression
Hypotheses	hgmapfn.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	hgmapfn.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	hgmapfn.r	⊢ 𝑅 = ( Scalar ‘ 𝑈 )
	hgmapfn.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
	hgmapfn.g	⊢ 𝐺 = ( ( HGMap ‘ 𝐾 ) ‘ 𝑊 )
	hgmapfn.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
Assertion	hgmapfnN	⊢ ( 𝜑 → 𝐺 Fn 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		hgmapfn.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		hgmapfn.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
3		hgmapfn.r	⊢ 𝑅 = ( Scalar ‘ 𝑈 )
4		hgmapfn.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
5		hgmapfn.g	⊢ 𝐺 = ( ( HGMap ‘ 𝐾 ) ‘ 𝑊 )
6		hgmapfn.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
7		riotaex	⊢ ( ℩ 𝑗 ∈ 𝐵 ∀ 𝑥 ∈ ( Base ‘ 𝑈 ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑘 ( ·𝑠 ‘ 𝑈 ) 𝑥 ) ) = ( 𝑗 ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑥 ) ) ) ∈ V
8		eqid	⊢ ( 𝑘 ∈ 𝐵 ↦ ( ℩ 𝑗 ∈ 𝐵 ∀ 𝑥 ∈ ( Base ‘ 𝑈 ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑘 ( ·𝑠 ‘ 𝑈 ) 𝑥 ) ) = ( 𝑗 ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑥 ) ) ) ) = ( 𝑘 ∈ 𝐵 ↦ ( ℩ 𝑗 ∈ 𝐵 ∀ 𝑥 ∈ ( Base ‘ 𝑈 ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑘 ( ·𝑠 ‘ 𝑈 ) 𝑥 ) ) = ( 𝑗 ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑥 ) ) ) )
9	7 8	fnmpti	⊢ ( 𝑘 ∈ 𝐵 ↦ ( ℩ 𝑗 ∈ 𝐵 ∀ 𝑥 ∈ ( Base ‘ 𝑈 ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑘 ( ·𝑠 ‘ 𝑈 ) 𝑥 ) ) = ( 𝑗 ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑥 ) ) ) ) Fn 𝐵
10		eqid	⊢ ( Base ‘ 𝑈 ) = ( Base ‘ 𝑈 )
11		eqid	⊢ ( ·𝑠 ‘ 𝑈 ) = ( ·𝑠 ‘ 𝑈 )
12		eqid	⊢ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
13		eqid	⊢ ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) = ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) )
14		eqid	⊢ ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) = ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 )
15	1 2 10 11 3 4 12 13 14 5 6	hgmapfval	⊢ ( 𝜑 → 𝐺 = ( 𝑘 ∈ 𝐵 ↦ ( ℩ 𝑗 ∈ 𝐵 ∀ 𝑥 ∈ ( Base ‘ 𝑈 ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑘 ( ·𝑠 ‘ 𝑈 ) 𝑥 ) ) = ( 𝑗 ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑥 ) ) ) ) )
16	15	fneq1d	⊢ ( 𝜑 → ( 𝐺 Fn 𝐵 ↔ ( 𝑘 ∈ 𝐵 ↦ ( ℩ 𝑗 ∈ 𝐵 ∀ 𝑥 ∈ ( Base ‘ 𝑈 ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑘 ( ·𝑠 ‘ 𝑈 ) 𝑥 ) ) = ( 𝑗 ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑥 ) ) ) ) Fn 𝐵 ) )
17	9 16	mpbiri	⊢ ( 𝜑 → 𝐺 Fn 𝐵 )