Theorem hgmaprnlem1N

Description: Lemma for hgmaprnN . (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.k2	⊢ ( 𝜑 → 𝑘 ∈ 𝐵 )
	hgmaprnlem1.sk	⊢ ( 𝜑 → 𝑠 = ( 𝑘 · 𝑡 ) )
Assertion	hgmaprnlem1N	⊢ ( 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.k2	⊢ ( 𝜑 → 𝑘 ∈ 𝐵 )
22		hgmaprnlem1.sk	⊢ ( 𝜑 → 𝑠 = ( 𝑘 · 𝑡 ) )
23	22	fveq2d	⊢ ( 𝜑 → ( 𝑆 ‘ 𝑠 ) = ( 𝑆 ‘ ( 𝑘 · 𝑡 ) ) )
24	18	eldifad	⊢ ( 𝜑 → 𝑡 ∈ 𝑉 )
25	1 2 3 6 4 5 8 12 14 15 16 24 21	hgmapvs	⊢ ( 𝜑 → ( 𝑆 ‘ ( 𝑘 · 𝑡 ) ) = ( ( 𝐺 ‘ 𝑘 ) ∙ ( 𝑆 ‘ 𝑡 ) ) )
26	23 20 25	3eqtr3d	⊢ ( 𝜑 → ( 𝑧 ∙ ( 𝑆 ‘ 𝑡 ) ) = ( ( 𝐺 ‘ 𝑘 ) ∙ ( 𝑆 ‘ 𝑡 ) ) )
27	1 8 16	lcdlvec	⊢ ( 𝜑 → 𝐶 ∈ LVec )
28	1 2 4 5 8 10 11 15 16 21	hgmapdcl	⊢ ( 𝜑 → ( 𝐺 ‘ 𝑘 ) ∈ 𝐴 )
29	1 2 3 8 9 14 16 24	hdmapcl	⊢ ( 𝜑 → ( 𝑆 ‘ 𝑡 ) ∈ 𝐷 )
30		eldifsni	⊢ ( 𝑡 ∈ ( 𝑉 ∖ { 0 } ) → 𝑡 ≠ 0 )
31	18 30	syl	⊢ ( 𝜑 → 𝑡 ≠ 0 )
32	1 2 3 7 8 13 14 16 24	hdmapeq0	⊢ ( 𝜑 → ( ( 𝑆 ‘ 𝑡 ) = 𝑄 ↔ 𝑡 = 0 ) )
33	32	necon3bid	⊢ ( 𝜑 → ( ( 𝑆 ‘ 𝑡 ) ≠ 𝑄 ↔ 𝑡 ≠ 0 ) )
34	31 33	mpbird	⊢ ( 𝜑 → ( 𝑆 ‘ 𝑡 ) ≠ 𝑄 )
35	9 12 10 11 13 27 17 28 29 34	lvecvscan2	⊢ ( 𝜑 → ( ( 𝑧 ∙ ( 𝑆 ‘ 𝑡 ) ) = ( ( 𝐺 ‘ 𝑘 ) ∙ ( 𝑆 ‘ 𝑡 ) ) ↔ 𝑧 = ( 𝐺 ‘ 𝑘 ) ) )
36	26 35	mpbid	⊢ ( 𝜑 → 𝑧 = ( 𝐺 ‘ 𝑘 ) )
37	1 2 4 5 15 16	hgmapfnN	⊢ ( 𝜑 → 𝐺 Fn 𝐵 )
38		fnfvelrn	⊢ ( ( 𝐺 Fn 𝐵 ∧ 𝑘 ∈ 𝐵 ) → ( 𝐺 ‘ 𝑘 ) ∈ ran 𝐺 )
39	37 21 38	syl2anc	⊢ ( 𝜑 → ( 𝐺 ‘ 𝑘 ) ∈ ran 𝐺 )
40	36 39	eqeltrd	⊢ ( 𝜑 → 𝑧 ∈ ran 𝐺 )