Theorem mapdpglem5N

Description: Lemma for mapdpg . (Contributed by NM, 20-Mar-2015) (New usage is discouraged.)

	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 } ) )
Assertion	mapdpglem5N	⊢ ( 𝜑 → 𝑡 ≠ ( 0g ‘ 𝐶 ) )

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		eqid	⊢ ( LSAtoms ‘ 𝑈 ) = ( LSAtoms ‘ 𝑈 )
25		eqid	⊢ ( LSAtoms ‘ 𝐶 ) = ( LSAtoms ‘ 𝐶 )
26	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22	mapdpglem4N	⊢ ( 𝜑 → ( 𝑋 − 𝑌 ) ≠ 𝑄 )
27	1 3 8	dvhlmod	⊢ ( 𝜑 → 𝑈 ∈ LMod )
28	4 5	lmodvsubcl	⊢ ( ( 𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) → ( 𝑋 − 𝑌 ) ∈ 𝑉 )
29	27 9 10 28	syl3anc	⊢ ( 𝜑 → ( 𝑋 − 𝑌 ) ∈ 𝑉 )
30	4 6 21 24 27 29	lsatspn0	⊢ ( 𝜑 → ( ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ∈ ( LSAtoms ‘ 𝑈 ) ↔ ( 𝑋 − 𝑌 ) ≠ 𝑄 ) )
31	26 30	mpbird	⊢ ( 𝜑 → ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ∈ ( LSAtoms ‘ 𝑈 ) )
32	1 2 3 24 7 25 8 31	mapdat	⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) ∈ ( LSAtoms ‘ 𝐶 ) )
33	23 32	eqeltrrd	⊢ ( 𝜑 → ( 𝐽 ‘ { 𝑡 } ) ∈ ( LSAtoms ‘ 𝐶 ) )
34		eqid	⊢ ( 0g ‘ 𝐶 ) = ( 0g ‘ 𝐶 )
35	1 7 8	lcdlmod	⊢ ( 𝜑 → 𝐶 ∈ LMod )
36	1 2 3 4 5 6 7 8 9 10 11 12 13 14	mapdpglem2a	⊢ ( 𝜑 → 𝑡 ∈ 𝐹 )
37	13 12 34 25 35 36	lsatspn0	⊢ ( 𝜑 → ( ( 𝐽 ‘ { 𝑡 } ) ∈ ( LSAtoms ‘ 𝐶 ) ↔ 𝑡 ≠ ( 0g ‘ 𝐶 ) ) )
38	33 37	mpbid	⊢ ( 𝜑 → 𝑡 ≠ ( 0g ‘ 𝐶 ) )