Theorem mapdpglem8

Description: Lemma for mapdpg . Baer p. 45, line 4: "...so that (F(x-y))* <= (Fy)*. This would imply that F(x-y) <= F(y)..." (Contributed by NM, 20-Mar-2015)

	Ref	Expression
Hypotheses	mapdpglem.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	mapdpglem.m	⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
	mapdpglem.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	mapdpglem.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
	mapdpglem.s	⊢ − = ( -g ‘ 𝑈 )
	mapdpglem.n	⊢ 𝑁 = ( LSpan ‘ 𝑈 )
	mapdpglem.c	⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
	mapdpglem.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
	mapdpglem.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
	mapdpglem.y	⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
	mapdpglem1.p	⊢ ⊕ = ( LSSum ‘ 𝐶 )
	mapdpglem2.j	⊢ 𝐽 = ( LSpan ‘ 𝐶 )
	mapdpglem3.f	⊢ 𝐹 = ( Base ‘ 𝐶 )
	mapdpglem3.te	⊢ ( 𝜑 → 𝑡 ∈ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ⊕ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) )
	mapdpglem3.a	⊢ 𝐴 = ( Scalar ‘ 𝑈 )
	mapdpglem3.b	⊢ 𝐵 = ( Base ‘ 𝐴 )
	mapdpglem3.t	⊢ · = ( ·𝑠 ‘ 𝐶 )
	mapdpglem3.r	⊢ 𝑅 = ( -g ‘ 𝐶 )
	mapdpglem3.g	⊢ ( 𝜑 → 𝐺 ∈ 𝐹 )
	mapdpglem3.e	⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐺 } ) )
	mapdpglem4.q	⊢ 𝑄 = ( 0g ‘ 𝑈 )
	mapdpglem.ne	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
	mapdpglem4.jt	⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { 𝑡 } ) )
	mapdpglem4.z	⊢ 0 = ( 0g ‘ 𝐴 )
	mapdpglem4.g4	⊢ ( 𝜑 → 𝑔 ∈ 𝐵 )
	mapdpglem4.z4	⊢ ( 𝜑 → 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) )
	mapdpglem4.t4	⊢ ( 𝜑 → 𝑡 = ( ( 𝑔 · 𝐺 ) 𝑅 𝑧 ) )
	mapdpglem4.xn	⊢ ( 𝜑 → 𝑋 ≠ 𝑄 )
	mapdpglem4.g0	⊢ ( 𝜑 → 𝑔 = 0 )
Assertion	mapdpglem8	⊢ ( 𝜑 → ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ⊆ ( 𝑁 ‘ { 𝑌 } ) )

Proof

Step	Hyp	Ref	Expression
1		mapdpglem.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		mapdpglem.m	⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
3		mapdpglem.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
4		mapdpglem.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
5		mapdpglem.s	⊢ − = ( -g ‘ 𝑈 )
6		mapdpglem.n	⊢ 𝑁 = ( LSpan ‘ 𝑈 )
7		mapdpglem.c	⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
8		mapdpglem.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
9		mapdpglem.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
10		mapdpglem.y	⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
11		mapdpglem1.p	⊢ ⊕ = ( LSSum ‘ 𝐶 )
12		mapdpglem2.j	⊢ 𝐽 = ( LSpan ‘ 𝐶 )
13		mapdpglem3.f	⊢ 𝐹 = ( Base ‘ 𝐶 )
14		mapdpglem3.te	⊢ ( 𝜑 → 𝑡 ∈ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ⊕ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) )
15		mapdpglem3.a	⊢ 𝐴 = ( Scalar ‘ 𝑈 )
16		mapdpglem3.b	⊢ 𝐵 = ( Base ‘ 𝐴 )
17		mapdpglem3.t	⊢ · = ( ·𝑠 ‘ 𝐶 )
18		mapdpglem3.r	⊢ 𝑅 = ( -g ‘ 𝐶 )
19		mapdpglem3.g	⊢ ( 𝜑 → 𝐺 ∈ 𝐹 )
20		mapdpglem3.e	⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐺 } ) )
21		mapdpglem4.q	⊢ 𝑄 = ( 0g ‘ 𝑈 )
22		mapdpglem.ne	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
23		mapdpglem4.jt	⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { 𝑡 } ) )
24		mapdpglem4.z	⊢ 0 = ( 0g ‘ 𝐴 )
25		mapdpglem4.g4	⊢ ( 𝜑 → 𝑔 ∈ 𝐵 )
26		mapdpglem4.z4	⊢ ( 𝜑 → 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) )
27		mapdpglem4.t4	⊢ ( 𝜑 → 𝑡 = ( ( 𝑔 · 𝐺 ) 𝑅 𝑧 ) )
28		mapdpglem4.xn	⊢ ( 𝜑 → 𝑋 ≠ 𝑄 )
29		mapdpglem4.g0	⊢ ( 𝜑 → 𝑔 = 0 )
30		eqid	⊢ ( LSubSp ‘ 𝐶 ) = ( LSubSp ‘ 𝐶 )
31	1 7 8	lcdlmod	⊢ ( 𝜑 → 𝐶 ∈ LMod )
32		eqid	⊢ ( LSubSp ‘ 𝑈 ) = ( LSubSp ‘ 𝑈 )
33	1 3 8	dvhlmod	⊢ ( 𝜑 → 𝑈 ∈ LMod )
34	4 32 6	lspsncl	⊢ ( ( 𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉 ) → ( 𝑁 ‘ { 𝑌 } ) ∈ ( LSubSp ‘ 𝑈 ) )
35	33 10 34	syl2anc	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) ∈ ( LSubSp ‘ 𝑈 ) )
36	1 2 3 32 7 30 8 35	mapdcl2	⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ∈ ( LSubSp ‘ 𝐶 ) )
37	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29	mapdpglem6	⊢ ( 𝜑 → 𝑡 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) )
38	30 12 31 36 37	lspsnel5a	⊢ ( 𝜑 → ( 𝐽 ‘ { 𝑡 } ) ⊆ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) )
39	23 38	eqsstrd	⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) ⊆ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) )
40	4 5	lmodvsubcl	⊢ ( ( 𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) → ( 𝑋 − 𝑌 ) ∈ 𝑉 )
41	33 9 10 40	syl3anc	⊢ ( 𝜑 → ( 𝑋 − 𝑌 ) ∈ 𝑉 )
42	4 32 6	lspsncl	⊢ ( ( 𝑈 ∈ LMod ∧ ( 𝑋 − 𝑌 ) ∈ 𝑉 ) → ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ∈ ( LSubSp ‘ 𝑈 ) )
43	33 41 42	syl2anc	⊢ ( 𝜑 → ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ∈ ( LSubSp ‘ 𝑈 ) )
44	1 3 32 2 8 43 35	mapdord	⊢ ( 𝜑 → ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) ⊆ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ↔ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ⊆ ( 𝑁 ‘ { 𝑌 } ) ) )
45	39 44	mpbid	⊢ ( 𝜑 → ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ⊆ ( 𝑁 ‘ { 𝑌 } ) )