Theorem hgmaprnlem4N

Description: Lemma for hgmaprnN . Eliminate s . (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 } ) )
Assertion	hgmaprnlem4N	⊢ ( 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	1 8 16	lcdlmod	⊢ ( 𝜑 → 𝐶 ∈ LMod )
20	18	eldifad	⊢ ( 𝜑 → 𝑡 ∈ 𝑉 )
21	1 2 3 8 9 14 16 20	hdmapcl	⊢ ( 𝜑 → ( 𝑆 ‘ 𝑡 ) ∈ 𝐷 )
22	9 10 12 11	lmodvscl	⊢ ( ( 𝐶 ∈ LMod ∧ 𝑧 ∈ 𝐴 ∧ ( 𝑆 ‘ 𝑡 ) ∈ 𝐷 ) → ( 𝑧 ∙ ( 𝑆 ‘ 𝑡 ) ) ∈ 𝐷 )
23	19 17 21 22	syl3anc	⊢ ( 𝜑 → ( 𝑧 ∙ ( 𝑆 ‘ 𝑡 ) ) ∈ 𝐷 )
24	1 8 9 14 16	hdmaprnN	⊢ ( 𝜑 → ran 𝑆 = 𝐷 )
25	23 24	eleqtrrd	⊢ ( 𝜑 → ( 𝑧 ∙ ( 𝑆 ‘ 𝑡 ) ) ∈ ran 𝑆 )
26	1 2 3 14 16	hdmapfnN	⊢ ( 𝜑 → 𝑆 Fn 𝑉 )
27		fvelrnb	⊢ ( 𝑆 Fn 𝑉 → ( ( 𝑧 ∙ ( 𝑆 ‘ 𝑡 ) ) ∈ ran 𝑆 ↔ ∃ 𝑠 ∈ 𝑉 ( 𝑆 ‘ 𝑠 ) = ( 𝑧 ∙ ( 𝑆 ‘ 𝑡 ) ) ) )
28	26 27	syl	⊢ ( 𝜑 → ( ( 𝑧 ∙ ( 𝑆 ‘ 𝑡 ) ) ∈ ran 𝑆 ↔ ∃ 𝑠 ∈ 𝑉 ( 𝑆 ‘ 𝑠 ) = ( 𝑧 ∙ ( 𝑆 ‘ 𝑡 ) ) ) )
29	25 28	mpbid	⊢ ( 𝜑 → ∃ 𝑠 ∈ 𝑉 ( 𝑆 ‘ 𝑠 ) = ( 𝑧 ∙ ( 𝑆 ‘ 𝑡 ) ) )
30	16	3ad2ant1	⊢ ( ( 𝜑 ∧ 𝑠 ∈ 𝑉 ∧ ( 𝑆 ‘ 𝑠 ) = ( 𝑧 ∙ ( 𝑆 ‘ 𝑡 ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
31	17	3ad2ant1	⊢ ( ( 𝜑 ∧ 𝑠 ∈ 𝑉 ∧ ( 𝑆 ‘ 𝑠 ) = ( 𝑧 ∙ ( 𝑆 ‘ 𝑡 ) ) ) → 𝑧 ∈ 𝐴 )
32	18	3ad2ant1	⊢ ( ( 𝜑 ∧ 𝑠 ∈ 𝑉 ∧ ( 𝑆 ‘ 𝑠 ) = ( 𝑧 ∙ ( 𝑆 ‘ 𝑡 ) ) ) → 𝑡 ∈ ( 𝑉 ∖ { 0 } ) )
33		simp2	⊢ ( ( 𝜑 ∧ 𝑠 ∈ 𝑉 ∧ ( 𝑆 ‘ 𝑠 ) = ( 𝑧 ∙ ( 𝑆 ‘ 𝑡 ) ) ) → 𝑠 ∈ 𝑉 )
34		simp3	⊢ ( ( 𝜑 ∧ 𝑠 ∈ 𝑉 ∧ ( 𝑆 ‘ 𝑠 ) = ( 𝑧 ∙ ( 𝑆 ‘ 𝑡 ) ) ) → ( 𝑆 ‘ 𝑠 ) = ( 𝑧 ∙ ( 𝑆 ‘ 𝑡 ) ) )
35		eqid	⊢ ( ( mapd ‘ 𝐾 ) ‘ 𝑊 ) = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
36		eqid	⊢ ( LSpan ‘ 𝑈 ) = ( LSpan ‘ 𝑈 )
37		eqid	⊢ ( LSpan ‘ 𝐶 ) = ( LSpan ‘ 𝐶 )
38	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 30 31 32 33 34 35 36 37	hgmaprnlem3N	⊢ ( ( 𝜑 ∧ 𝑠 ∈ 𝑉 ∧ ( 𝑆 ‘ 𝑠 ) = ( 𝑧 ∙ ( 𝑆 ‘ 𝑡 ) ) ) → 𝑧 ∈ ran 𝐺 )
39	38	rexlimdv3a	⊢ ( 𝜑 → ( ∃ 𝑠 ∈ 𝑉 ( 𝑆 ‘ 𝑠 ) = ( 𝑧 ∙ ( 𝑆 ‘ 𝑡 ) ) → 𝑧 ∈ ran 𝐺 ) )
40	29 39	mpd	⊢ ( 𝜑 → 𝑧 ∈ ran 𝐺 )