Theorem hgmaprnlem5N

Description: Lemma for hgmaprnN . Eliminate t . (Contributed by NM, 7-Jun-2015) (New usage is discouraged.)

	Ref	Expression
Hypotheses	hgmaprnlem1.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	hgmaprnlem1.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	hgmaprnlem1.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
	hgmaprnlem1.r	⊢ 𝑅 = ( Scalar ‘ 𝑈 )
	hgmaprnlem1.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
	hgmaprnlem1.t	⊢ · = ( ·𝑠 ‘ 𝑈 )
	hgmaprnlem1.o	⊢ 0 = ( 0g ‘ 𝑈 )
	hgmaprnlem1.c	⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
	hgmaprnlem1.d	⊢ 𝐷 = ( Base ‘ 𝐶 )
	hgmaprnlem1.p	⊢ 𝑃 = ( Scalar ‘ 𝐶 )
	hgmaprnlem1.a	⊢ 𝐴 = ( Base ‘ 𝑃 )
	hgmaprnlem1.e	⊢ ∙ = ( ·𝑠 ‘ 𝐶 )
	hgmaprnlem1.q	⊢ 𝑄 = ( 0g ‘ 𝐶 )
	hgmaprnlem1.s	⊢ 𝑆 = ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 )
	hgmaprnlem1.g	⊢ 𝐺 = ( ( HGMap ‘ 𝐾 ) ‘ 𝑊 )
	hgmaprnlem1.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
	hgmaprnlem1.z	⊢ ( 𝜑 → 𝑧 ∈ 𝐴 )
Assertion	hgmaprnlem5N	⊢ ( 𝜑 → 𝑧 ∈ ran 𝐺 )

Proof

Step	Hyp	Ref	Expression
1		hgmaprnlem1.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		hgmaprnlem1.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
3		hgmaprnlem1.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
4		hgmaprnlem1.r	⊢ 𝑅 = ( Scalar ‘ 𝑈 )
5		hgmaprnlem1.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
6		hgmaprnlem1.t	⊢ · = ( ·𝑠 ‘ 𝑈 )
7		hgmaprnlem1.o	⊢ 0 = ( 0g ‘ 𝑈 )
8		hgmaprnlem1.c	⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
9		hgmaprnlem1.d	⊢ 𝐷 = ( Base ‘ 𝐶 )
10		hgmaprnlem1.p	⊢ 𝑃 = ( Scalar ‘ 𝐶 )
11		hgmaprnlem1.a	⊢ 𝐴 = ( Base ‘ 𝑃 )
12		hgmaprnlem1.e	⊢ ∙ = ( ·𝑠 ‘ 𝐶 )
13		hgmaprnlem1.q	⊢ 𝑄 = ( 0g ‘ 𝐶 )
14		hgmaprnlem1.s	⊢ 𝑆 = ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 )
15		hgmaprnlem1.g	⊢ 𝐺 = ( ( HGMap ‘ 𝐾 ) ‘ 𝑊 )
16		hgmaprnlem1.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
17		hgmaprnlem1.z	⊢ ( 𝜑 → 𝑧 ∈ 𝐴 )
18	1 2 3 7 16	dvh1dim	⊢ ( 𝜑 → ∃ 𝑡 ∈ 𝑉 𝑡 ≠ 0 )
19		eldifsn	⊢ ( 𝑡 ∈ ( 𝑉 ∖ { 0 } ) ↔ ( 𝑡 ∈ 𝑉 ∧ 𝑡 ≠ 0 ) )
20	16	adantr	⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( 𝑉 ∖ { 0 } ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
21	17	adantr	⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( 𝑉 ∖ { 0 } ) ) → 𝑧 ∈ 𝐴 )
22		simpr	⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( 𝑉 ∖ { 0 } ) ) → 𝑡 ∈ ( 𝑉 ∖ { 0 } ) )
23	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 20 21 22	hgmaprnlem4N	⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( 𝑉 ∖ { 0 } ) ) → 𝑧 ∈ ran 𝐺 )
24	19 23	sylan2br	⊢ ( ( 𝜑 ∧ ( 𝑡 ∈ 𝑉 ∧ 𝑡 ≠ 0 ) ) → 𝑧 ∈ ran 𝐺 )
25	18 24	rexlimddv	⊢ ( 𝜑 → 𝑧 ∈ ran 𝐺 )