Theorem hgmaprnlem3N

Description: Lemma for hgmaprnN . Eliminate k . (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 )
	hgmaprnlem1.t2	⊢ ( 𝜑 → 𝑡 ∈ ( 𝑉 ∖ { 0 } ) )
	hgmaprnlem1.s2	⊢ ( 𝜑 → 𝑠 ∈ 𝑉 )
	hgmaprnlem1.sz	⊢ ( 𝜑 → ( 𝑆 ‘ 𝑠 ) = ( 𝑧 ∙ ( 𝑆 ‘ 𝑡 ) ) )
	hgmaprnlem1.m	⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
	hgmaprnlem1.n	⊢ 𝑁 = ( LSpan ‘ 𝑈 )
	hgmaprnlem1.l	⊢ 𝐿 = ( LSpan ‘ 𝐶 )
Assertion	hgmaprnlem3N	⊢ ( 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		hgmaprnlem1.t2	⊢ ( 𝜑 → 𝑡 ∈ ( 𝑉 ∖ { 0 } ) )
19		hgmaprnlem1.s2	⊢ ( 𝜑 → 𝑠 ∈ 𝑉 )
20		hgmaprnlem1.sz	⊢ ( 𝜑 → ( 𝑆 ‘ 𝑠 ) = ( 𝑧 ∙ ( 𝑆 ‘ 𝑡 ) ) )
21		hgmaprnlem1.m	⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
22		hgmaprnlem1.n	⊢ 𝑁 = ( LSpan ‘ 𝑈 )
23		hgmaprnlem1.l	⊢ 𝐿 = ( LSpan ‘ 𝐶 )
24	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23	hgmaprnlem2N	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑠 } ) ⊆ ( 𝑁 ‘ { 𝑡 } ) )
25	1 2 16	dvhlmod	⊢ ( 𝜑 → 𝑈 ∈ LMod )
26	18	eldifad	⊢ ( 𝜑 → 𝑡 ∈ 𝑉 )
27	3 4 5 6 22 25 19 26	lspsnss2	⊢ ( 𝜑 → ( ( 𝑁 ‘ { 𝑠 } ) ⊆ ( 𝑁 ‘ { 𝑡 } ) ↔ ∃ 𝑘 ∈ 𝐵 𝑠 = ( 𝑘 · 𝑡 ) ) )
28	24 27	mpbid	⊢ ( 𝜑 → ∃ 𝑘 ∈ 𝐵 𝑠 = ( 𝑘 · 𝑡 ) )
29	16	3ad2ant1	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐵 ∧ 𝑠 = ( 𝑘 · 𝑡 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
30	17	3ad2ant1	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐵 ∧ 𝑠 = ( 𝑘 · 𝑡 ) ) → 𝑧 ∈ 𝐴 )
31	18	3ad2ant1	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐵 ∧ 𝑠 = ( 𝑘 · 𝑡 ) ) → 𝑡 ∈ ( 𝑉 ∖ { 0 } ) )
32	19	3ad2ant1	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐵 ∧ 𝑠 = ( 𝑘 · 𝑡 ) ) → 𝑠 ∈ 𝑉 )
33	20	3ad2ant1	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐵 ∧ 𝑠 = ( 𝑘 · 𝑡 ) ) → ( 𝑆 ‘ 𝑠 ) = ( 𝑧 ∙ ( 𝑆 ‘ 𝑡 ) ) )
34		simp2	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐵 ∧ 𝑠 = ( 𝑘 · 𝑡 ) ) → 𝑘 ∈ 𝐵 )
35		simp3	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐵 ∧ 𝑠 = ( 𝑘 · 𝑡 ) ) → 𝑠 = ( 𝑘 · 𝑡 ) )
36	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 29 30 31 32 33 34 35	hgmaprnlem1N	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐵 ∧ 𝑠 = ( 𝑘 · 𝑡 ) ) → 𝑧 ∈ ran 𝐺 )
37	36	rexlimdv3a	⊢ ( 𝜑 → ( ∃ 𝑘 ∈ 𝐵 𝑠 = ( 𝑘 · 𝑡 ) → 𝑧 ∈ ran 𝐺 ) )
38	28 37	mpd	⊢ ( 𝜑 → 𝑧 ∈ ran 𝐺 )