Theorem hgmaprnN

Description: Part of proof of part 16 in Baer p. 50 line 23, Fs=G, except that we use the original vector space scalars for the range. (Contributed by NM, 7-Jun-2015) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		hgmaprn.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		hgmaprn.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
3		hgmaprn.r	⊢ 𝑅 = ( Scalar ‘ 𝑈 )
4		hgmaprn.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
5		hgmaprn.g	⊢ 𝐺 = ( ( HGMap ‘ 𝐾 ) ‘ 𝑊 )
6		hgmaprn.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
7	1 2 3 4 5 6	hgmapfnN	⊢ ( 𝜑 → 𝐺 Fn 𝐵 )
8		eqid	⊢ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
9		eqid	⊢ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) = ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) )
10		eqid	⊢ ( Base ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) = ( Base ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) )
11	6	adantr	⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐵 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
12		simpr	⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐵 ) → 𝑧 ∈ 𝐵 )
13	1 2 3 4 8 9 10 5 11 12	hgmapdcl	⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐵 ) → ( 𝐺 ‘ 𝑧 ) ∈ ( Base ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) )
14	13	ralrimiva	⊢ ( 𝜑 → ∀ 𝑧 ∈ 𝐵 ( 𝐺 ‘ 𝑧 ) ∈ ( Base ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) )
15		fnfvrnss	⊢ ( ( 𝐺 Fn 𝐵 ∧ ∀ 𝑧 ∈ 𝐵 ( 𝐺 ‘ 𝑧 ) ∈ ( Base ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) ) → ran 𝐺 ⊆ ( Base ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) )
16	7 14 15	syl2anc	⊢ ( 𝜑 → ran 𝐺 ⊆ ( Base ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) )
17		eqid	⊢ ( Base ‘ 𝑈 ) = ( Base ‘ 𝑈 )
18		eqid	⊢ ( ·𝑠 ‘ 𝑈 ) = ( ·𝑠 ‘ 𝑈 )
19		eqid	⊢ ( 0g ‘ 𝑈 ) = ( 0g ‘ 𝑈 )
20		eqid	⊢ ( Base ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) = ( Base ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) )
21		eqid	⊢ ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) = ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) )
22		eqid	⊢ ( 0g ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) = ( 0g ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) )
23		eqid	⊢ ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) = ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 )
24	6	adantr	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( Base ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
25		simpr	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( Base ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) ) → 𝑧 ∈ ( Base ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) )
26	1 2 17 3 4 18 19 8 20 9 10 21 22 23 5 24 25	hgmaprnlem5N	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( Base ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) ) → 𝑧 ∈ ran 𝐺 )
27	16 26	eqelssd	⊢ ( 𝜑 → ran 𝐺 = ( Base ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) )
28	1 2 3 4 8 9 10 6	lcdsbase	⊢ ( 𝜑 → ( Base ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) = 𝐵 )
29	27 28	eqtrd	⊢ ( 𝜑 → ran 𝐺 = 𝐵 )