Theorem hdmap14lem15

Description: Part of proof of part 14 in Baer p. 50 line 3. Convert scalar base of dual to scalar base of vector space. (Contributed by NM, 6-Jun-2015)

	Ref	Expression
Hypotheses	hdmap14lem12.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	hdmap14lem12.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	hdmap14lem12.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
	hdmap14lem12.t	⊢ · = ( ·𝑠 ‘ 𝑈 )
	hdmap14lem12.r	⊢ 𝑅 = ( Scalar ‘ 𝑈 )
	hdmap14lem12.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
	hdmap14lem12.c	⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
	hdmap14lem12.e	⊢ ∙ = ( ·𝑠 ‘ 𝐶 )
	hdmap14lem12.s	⊢ 𝑆 = ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 )
	hdmap14lem12.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
	hdmap14lem12.f	⊢ ( 𝜑 → 𝐹 ∈ 𝐵 )
Assertion	hdmap14lem15	⊢ ( 𝜑 → ∃! 𝑔 ∈ 𝐵 ∀ 𝑥 ∈ 𝑉 ( 𝑆 ‘ ( 𝐹 · 𝑥 ) ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑥 ) ) )

Proof

Step	Hyp	Ref	Expression
1		hdmap14lem12.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		hdmap14lem12.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
3		hdmap14lem12.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
4		hdmap14lem12.t	⊢ · = ( ·𝑠 ‘ 𝑈 )
5		hdmap14lem12.r	⊢ 𝑅 = ( Scalar ‘ 𝑈 )
6		hdmap14lem12.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
7		hdmap14lem12.c	⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
8		hdmap14lem12.e	⊢ ∙ = ( ·𝑠 ‘ 𝐶 )
9		hdmap14lem12.s	⊢ 𝑆 = ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 )
10		hdmap14lem12.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
11		hdmap14lem12.f	⊢ ( 𝜑 → 𝐹 ∈ 𝐵 )
12		eqid	⊢ ( Scalar ‘ 𝐶 ) = ( Scalar ‘ 𝐶 )
13		eqid	⊢ ( Base ‘ ( Scalar ‘ 𝐶 ) ) = ( Base ‘ ( Scalar ‘ 𝐶 ) )
14	1 2 3 4 5 6 7 8 9 10 11 12 13	hdmap14lem14	⊢ ( 𝜑 → ∃! 𝑔 ∈ ( Base ‘ ( Scalar ‘ 𝐶 ) ) ∀ 𝑥 ∈ 𝑉 ( 𝑆 ‘ ( 𝐹 · 𝑥 ) ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑥 ) ) )
15	1 2 5 6 7 12 13 10	lcdsbase	⊢ ( 𝜑 → ( Base ‘ ( Scalar ‘ 𝐶 ) ) = 𝐵 )
16		reueq1	⊢ ( ( Base ‘ ( Scalar ‘ 𝐶 ) ) = 𝐵 → ( ∃! 𝑔 ∈ ( Base ‘ ( Scalar ‘ 𝐶 ) ) ∀ 𝑥 ∈ 𝑉 ( 𝑆 ‘ ( 𝐹 · 𝑥 ) ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑥 ) ) ↔ ∃! 𝑔 ∈ 𝐵 ∀ 𝑥 ∈ 𝑉 ( 𝑆 ‘ ( 𝐹 · 𝑥 ) ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑥 ) ) ) )
17	15 16	syl	⊢ ( 𝜑 → ( ∃! 𝑔 ∈ ( Base ‘ ( Scalar ‘ 𝐶 ) ) ∀ 𝑥 ∈ 𝑉 ( 𝑆 ‘ ( 𝐹 · 𝑥 ) ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑥 ) ) ↔ ∃! 𝑔 ∈ 𝐵 ∀ 𝑥 ∈ 𝑉 ( 𝑆 ‘ ( 𝐹 · 𝑥 ) ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑥 ) ) ) )
18	14 17	mpbid	⊢ ( 𝜑 → ∃! 𝑔 ∈ 𝐵 ∀ 𝑥 ∈ 𝑉 ( 𝑆 ‘ ( 𝐹 · 𝑥 ) ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑥 ) ) )