Theorem hgmaprnlem2N

Description: Lemma for hgmaprnN . Part 15 of Baer p. 50 line 20. We only require a subset relation, rather than equality, so that the case of zero z is taken care of automatically. (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	hgmaprnlem2N	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑠 } ) ⊆ ( 𝑁 ‘ { 𝑡 } ) )

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 8 16	lcdlmod	⊢ ( 𝜑 → 𝐶 ∈ LMod )
25	18	eldifad	⊢ ( 𝜑 → 𝑡 ∈ 𝑉 )
26	1 2 3 8 9 14 16 25	hdmapcl	⊢ ( 𝜑 → ( 𝑆 ‘ 𝑡 ) ∈ 𝐷 )
27	10 11 9 12 23	lspsnvsi	⊢ ( ( 𝐶 ∈ LMod ∧ 𝑧 ∈ 𝐴 ∧ ( 𝑆 ‘ 𝑡 ) ∈ 𝐷 ) → ( 𝐿 ‘ { ( 𝑧 ∙ ( 𝑆 ‘ 𝑡 ) ) } ) ⊆ ( 𝐿 ‘ { ( 𝑆 ‘ 𝑡 ) } ) )
28	24 17 26 27	syl3anc	⊢ ( 𝜑 → ( 𝐿 ‘ { ( 𝑧 ∙ ( 𝑆 ‘ 𝑡 ) ) } ) ⊆ ( 𝐿 ‘ { ( 𝑆 ‘ 𝑡 ) } ) )
29	1 2 3 22 8 23 21 14 16 19	hdmap10	⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑠 } ) ) = ( 𝐿 ‘ { ( 𝑆 ‘ 𝑠 ) } ) )
30	20	sneqd	⊢ ( 𝜑 → { ( 𝑆 ‘ 𝑠 ) } = { ( 𝑧 ∙ ( 𝑆 ‘ 𝑡 ) ) } )
31	30	fveq2d	⊢ ( 𝜑 → ( 𝐿 ‘ { ( 𝑆 ‘ 𝑠 ) } ) = ( 𝐿 ‘ { ( 𝑧 ∙ ( 𝑆 ‘ 𝑡 ) ) } ) )
32	29 31	eqtrd	⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑠 } ) ) = ( 𝐿 ‘ { ( 𝑧 ∙ ( 𝑆 ‘ 𝑡 ) ) } ) )
33	1 2 3 22 8 23 21 14 16 25	hdmap10	⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑡 } ) ) = ( 𝐿 ‘ { ( 𝑆 ‘ 𝑡 ) } ) )
34	28 32 33	3sstr4d	⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑠 } ) ) ⊆ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑡 } ) ) )
35		eqid	⊢ ( LSubSp ‘ 𝑈 ) = ( LSubSp ‘ 𝑈 )
36	1 2 16	dvhlmod	⊢ ( 𝜑 → 𝑈 ∈ LMod )
37	3 35 22	lspsncl	⊢ ( ( 𝑈 ∈ LMod ∧ 𝑠 ∈ 𝑉 ) → ( 𝑁 ‘ { 𝑠 } ) ∈ ( LSubSp ‘ 𝑈 ) )
38	36 19 37	syl2anc	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑠 } ) ∈ ( LSubSp ‘ 𝑈 ) )
39	3 35 22	lspsncl	⊢ ( ( 𝑈 ∈ LMod ∧ 𝑡 ∈ 𝑉 ) → ( 𝑁 ‘ { 𝑡 } ) ∈ ( LSubSp ‘ 𝑈 ) )
40	36 25 39	syl2anc	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑡 } ) ∈ ( LSubSp ‘ 𝑈 ) )
41	1 2 35 21 16 38 40	mapdord	⊢ ( 𝜑 → ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑠 } ) ) ⊆ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑡 } ) ) ↔ ( 𝑁 ‘ { 𝑠 } ) ⊆ ( 𝑁 ‘ { 𝑡 } ) ) )
42	34 41	mpbid	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑠 } ) ⊆ ( 𝑁 ‘ { 𝑡 } ) )