Theorem hgmapcl

Description: Closure of scalar sigma map i.e. the map from the vector space scalar base to the dual space scalar base. (Contributed by NM, 6-Jun-2015)

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

Proof

Step	Hyp	Ref	Expression
1		hgmapcl.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		hgmapcl.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
3		hgmapcl.r	⊢ 𝑅 = ( Scalar ‘ 𝑈 )
4		hgmapcl.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
5		hgmapcl.g	⊢ 𝐺 = ( ( HGMap ‘ 𝐾 ) ‘ 𝑊 )
6		hgmapcl.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
7		hgmapcl.f	⊢ ( 𝜑 → 𝐹 ∈ 𝐵 )
8		eqid	⊢ ( Base ‘ 𝑈 ) = ( Base ‘ 𝑈 )
9		eqid	⊢ ( ·𝑠 ‘ 𝑈 ) = ( ·𝑠 ‘ 𝑈 )
10		eqid	⊢ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
11		eqid	⊢ ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) = ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) )
12		eqid	⊢ ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) = ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 )
13	1 2 8 9 3 4 10 11 12 5 6 7	hgmapval	⊢ ( 𝜑 → ( 𝐺 ‘ 𝐹 ) = ( ℩ 𝑔 ∈ 𝐵 ∀ 𝑥 ∈ ( Base ‘ 𝑈 ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝐹 ( ·𝑠 ‘ 𝑈 ) 𝑥 ) ) = ( 𝑔 ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑥 ) ) ) )
14	1 2 8 9 3 4 10 11 12 6 7	hdmap14lem15	⊢ ( 𝜑 → ∃! 𝑔 ∈ 𝐵 ∀ 𝑥 ∈ ( Base ‘ 𝑈 ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝐹 ( ·𝑠 ‘ 𝑈 ) 𝑥 ) ) = ( 𝑔 ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑥 ) ) )
15		riotacl	⊢ ( ∃! 𝑔 ∈ 𝐵 ∀ 𝑥 ∈ ( Base ‘ 𝑈 ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝐹 ( ·𝑠 ‘ 𝑈 ) 𝑥 ) ) = ( 𝑔 ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑥 ) ) → ( ℩ 𝑔 ∈ 𝐵 ∀ 𝑥 ∈ ( Base ‘ 𝑈 ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝐹 ( ·𝑠 ‘ 𝑈 ) 𝑥 ) ) = ( 𝑔 ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑥 ) ) ) ∈ 𝐵 )
16	14 15	syl	⊢ ( 𝜑 → ( ℩ 𝑔 ∈ 𝐵 ∀ 𝑥 ∈ ( Base ‘ 𝑈 ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝐹 ( ·𝑠 ‘ 𝑈 ) 𝑥 ) ) = ( 𝑔 ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑥 ) ) ) ∈ 𝐵 )
17	13 16	eqeltrd	⊢ ( 𝜑 → ( 𝐺 ‘ 𝐹 ) ∈ 𝐵 )