Theorem lcfrlem33

Description: Lemma for lcfr . (Contributed by NM, 10-Mar-2015)

	Ref	Expression
Hypotheses	lcfrlem17.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	lcfrlem17.o	⊢ ⊥ = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
	lcfrlem17.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	lcfrlem17.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
	lcfrlem17.p	⊢ + = ( +g ‘ 𝑈 )
	lcfrlem17.z	⊢ 0 = ( 0g ‘ 𝑈 )
	lcfrlem17.n	⊢ 𝑁 = ( LSpan ‘ 𝑈 )
	lcfrlem17.a	⊢ 𝐴 = ( LSAtoms ‘ 𝑈 )
	lcfrlem17.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
	lcfrlem17.x	⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
	lcfrlem17.y	⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
	lcfrlem17.ne	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
	lcfrlem22.b	⊢ 𝐵 = ( ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ∩ ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) )
	lcfrlem24.t	⊢ · = ( ·𝑠 ‘ 𝑈 )
	lcfrlem24.s	⊢ 𝑆 = ( Scalar ‘ 𝑈 )
	lcfrlem24.q	⊢ 𝑄 = ( 0g ‘ 𝑆 )
	lcfrlem24.r	⊢ 𝑅 = ( Base ‘ 𝑆 )
	lcfrlem24.j	⊢ 𝐽 = ( 𝑥 ∈ ( 𝑉 ∖ { 0 } ) ↦ ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑘 ∈ 𝑅 ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑥 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑥 ) ) ) ) )
	lcfrlem24.ib	⊢ ( 𝜑 → 𝐼 ∈ 𝐵 )
	lcfrlem24.l	⊢ 𝐿 = ( LKer ‘ 𝑈 )
	lcfrlem25.d	⊢ 𝐷 = ( LDual ‘ 𝑈 )
	lcfrlem28.jn	⊢ ( 𝜑 → ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ≠ 𝑄 )
	lcfrlem29.i	⊢ 𝐹 = ( invr ‘ 𝑆 )
	lcfrlem30.m	⊢ − = ( -g ‘ 𝐷 )
	lcfrlem30.c	⊢ 𝐶 = ( ( 𝐽 ‘ 𝑋 ) − ( ( ( 𝐹 ‘ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ) ( .r ‘ 𝑆 ) ( ( 𝐽 ‘ 𝑋 ) ‘ 𝐼 ) ) ( ·𝑠 ‘ 𝐷 ) ( 𝐽 ‘ 𝑌 ) ) )
	lcfrlem33.xi	⊢ ( 𝜑 → ( ( 𝐽 ‘ 𝑋 ) ‘ 𝐼 ) = 𝑄 )
Assertion	lcfrlem33	⊢ ( 𝜑 → 𝐶 ≠ ( 0g ‘ 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		lcfrlem17.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		lcfrlem17.o	⊢ ⊥ = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
3		lcfrlem17.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
4		lcfrlem17.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
5		lcfrlem17.p	⊢ + = ( +g ‘ 𝑈 )
6		lcfrlem17.z	⊢ 0 = ( 0g ‘ 𝑈 )
7		lcfrlem17.n	⊢ 𝑁 = ( LSpan ‘ 𝑈 )
8		lcfrlem17.a	⊢ 𝐴 = ( LSAtoms ‘ 𝑈 )
9		lcfrlem17.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
10		lcfrlem17.x	⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
11		lcfrlem17.y	⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
12		lcfrlem17.ne	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
13		lcfrlem22.b	⊢ 𝐵 = ( ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ∩ ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) )
14		lcfrlem24.t	⊢ · = ( ·𝑠 ‘ 𝑈 )
15		lcfrlem24.s	⊢ 𝑆 = ( Scalar ‘ 𝑈 )
16		lcfrlem24.q	⊢ 𝑄 = ( 0g ‘ 𝑆 )
17		lcfrlem24.r	⊢ 𝑅 = ( Base ‘ 𝑆 )
18		lcfrlem24.j	⊢ 𝐽 = ( 𝑥 ∈ ( 𝑉 ∖ { 0 } ) ↦ ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑘 ∈ 𝑅 ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑥 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑥 ) ) ) ) )
19		lcfrlem24.ib	⊢ ( 𝜑 → 𝐼 ∈ 𝐵 )
20		lcfrlem24.l	⊢ 𝐿 = ( LKer ‘ 𝑈 )
21		lcfrlem25.d	⊢ 𝐷 = ( LDual ‘ 𝑈 )
22		lcfrlem28.jn	⊢ ( 𝜑 → ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ≠ 𝑄 )
23		lcfrlem29.i	⊢ 𝐹 = ( invr ‘ 𝑆 )
24		lcfrlem30.m	⊢ − = ( -g ‘ 𝐷 )
25		lcfrlem30.c	⊢ 𝐶 = ( ( 𝐽 ‘ 𝑋 ) − ( ( ( 𝐹 ‘ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ) ( .r ‘ 𝑆 ) ( ( 𝐽 ‘ 𝑋 ) ‘ 𝐼 ) ) ( ·𝑠 ‘ 𝐷 ) ( 𝐽 ‘ 𝑌 ) ) )
26		lcfrlem33.xi	⊢ ( 𝜑 → ( ( 𝐽 ‘ 𝑋 ) ‘ 𝐼 ) = 𝑄 )
27	26	oveq2d	⊢ ( 𝜑 → ( ( 𝐹 ‘ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ) ( .r ‘ 𝑆 ) ( ( 𝐽 ‘ 𝑋 ) ‘ 𝐼 ) ) = ( ( 𝐹 ‘ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ) ( .r ‘ 𝑆 ) 𝑄 ) )
28	1 3 9	dvhlmod	⊢ ( 𝜑 → 𝑈 ∈ LMod )
29	15	lmodring	⊢ ( 𝑈 ∈ LMod → 𝑆 ∈ Ring )
30	28 29	syl	⊢ ( 𝜑 → 𝑆 ∈ Ring )
31	1 3 9	dvhlvec	⊢ ( 𝜑 → 𝑈 ∈ LVec )
32	15	lvecdrng	⊢ ( 𝑈 ∈ LVec → 𝑆 ∈ DivRing )
33	31 32	syl	⊢ ( 𝜑 → 𝑆 ∈ DivRing )
34		eqid	⊢ ( LFnl ‘ 𝑈 ) = ( LFnl ‘ 𝑈 )
35		eqid	⊢ ( 0g ‘ 𝐷 ) = ( 0g ‘ 𝐷 )
36		eqid	⊢ { 𝑓 ∈ ( LFnl ‘ 𝑈 ) ∣ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) } = { 𝑓 ∈ ( LFnl ‘ 𝑈 ) ∣ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) }
37	1 2 3 4 5 14 15 17 6 34 20 21 35 36 18 9 11	lcfrlem10	⊢ ( 𝜑 → ( 𝐽 ‘ 𝑌 ) ∈ ( LFnl ‘ 𝑈 ) )
38	1 2 3 4 5 6 7 8 9 10 11 12 13	lcfrlem22	⊢ ( 𝜑 → 𝐵 ∈ 𝐴 )
39	4 8 28 38	lsatssv	⊢ ( 𝜑 → 𝐵 ⊆ 𝑉 )
40	39 19	sseldd	⊢ ( 𝜑 → 𝐼 ∈ 𝑉 )
41	15 17 4 34	lflcl	⊢ ( ( 𝑈 ∈ LMod ∧ ( 𝐽 ‘ 𝑌 ) ∈ ( LFnl ‘ 𝑈 ) ∧ 𝐼 ∈ 𝑉 ) → ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ∈ 𝑅 )
42	28 37 40 41	syl3anc	⊢ ( 𝜑 → ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ∈ 𝑅 )
43	17 16 23	drnginvrcl	⊢ ( ( 𝑆 ∈ DivRing ∧ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ∈ 𝑅 ∧ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ≠ 𝑄 ) → ( 𝐹 ‘ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ) ∈ 𝑅 )
44	33 42 22 43	syl3anc	⊢ ( 𝜑 → ( 𝐹 ‘ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ) ∈ 𝑅 )
45		eqid	⊢ ( .r ‘ 𝑆 ) = ( .r ‘ 𝑆 )
46	17 45 16	ringrz	⊢ ( ( 𝑆 ∈ Ring ∧ ( 𝐹 ‘ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ) ∈ 𝑅 ) → ( ( 𝐹 ‘ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ) ( .r ‘ 𝑆 ) 𝑄 ) = 𝑄 )
47	30 44 46	syl2anc	⊢ ( 𝜑 → ( ( 𝐹 ‘ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ) ( .r ‘ 𝑆 ) 𝑄 ) = 𝑄 )
48	27 47	eqtrd	⊢ ( 𝜑 → ( ( 𝐹 ‘ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ) ( .r ‘ 𝑆 ) ( ( 𝐽 ‘ 𝑋 ) ‘ 𝐼 ) ) = 𝑄 )
49	48	oveq1d	⊢ ( 𝜑 → ( ( ( 𝐹 ‘ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ) ( .r ‘ 𝑆 ) ( ( 𝐽 ‘ 𝑋 ) ‘ 𝐼 ) ) ( ·𝑠 ‘ 𝐷 ) ( 𝐽 ‘ 𝑌 ) ) = ( 𝑄 ( ·𝑠 ‘ 𝐷 ) ( 𝐽 ‘ 𝑌 ) ) )
50		eqid	⊢ ( ·𝑠 ‘ 𝐷 ) = ( ·𝑠 ‘ 𝐷 )
51	34 15 16 21 50 35 28 37	ldual0vs	⊢ ( 𝜑 → ( 𝑄 ( ·𝑠 ‘ 𝐷 ) ( 𝐽 ‘ 𝑌 ) ) = ( 0g ‘ 𝐷 ) )
52	49 51	eqtrd	⊢ ( 𝜑 → ( ( ( 𝐹 ‘ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ) ( .r ‘ 𝑆 ) ( ( 𝐽 ‘ 𝑋 ) ‘ 𝐼 ) ) ( ·𝑠 ‘ 𝐷 ) ( 𝐽 ‘ 𝑌 ) ) = ( 0g ‘ 𝐷 ) )
53	52	oveq2d	⊢ ( 𝜑 → ( ( 𝐽 ‘ 𝑋 ) − ( ( ( 𝐹 ‘ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ) ( .r ‘ 𝑆 ) ( ( 𝐽 ‘ 𝑋 ) ‘ 𝐼 ) ) ( ·𝑠 ‘ 𝐷 ) ( 𝐽 ‘ 𝑌 ) ) ) = ( ( 𝐽 ‘ 𝑋 ) − ( 0g ‘ 𝐷 ) ) )
54	21 28	ldualgrp	⊢ ( 𝜑 → 𝐷 ∈ Grp )
55		eqid	⊢ ( Base ‘ 𝐷 ) = ( Base ‘ 𝐷 )
56	1 2 3 4 5 14 15 17 6 34 20 21 35 36 18 9 10	lcfrlem10	⊢ ( 𝜑 → ( 𝐽 ‘ 𝑋 ) ∈ ( LFnl ‘ 𝑈 ) )
57	34 21 55 28 56	ldualelvbase	⊢ ( 𝜑 → ( 𝐽 ‘ 𝑋 ) ∈ ( Base ‘ 𝐷 ) )
58	55 35 24	grpsubid1	⊢ ( ( 𝐷 ∈ Grp ∧ ( 𝐽 ‘ 𝑋 ) ∈ ( Base ‘ 𝐷 ) ) → ( ( 𝐽 ‘ 𝑋 ) − ( 0g ‘ 𝐷 ) ) = ( 𝐽 ‘ 𝑋 ) )
59	54 57 58	syl2anc	⊢ ( 𝜑 → ( ( 𝐽 ‘ 𝑋 ) − ( 0g ‘ 𝐷 ) ) = ( 𝐽 ‘ 𝑋 ) )
60	53 59	eqtrd	⊢ ( 𝜑 → ( ( 𝐽 ‘ 𝑋 ) − ( ( ( 𝐹 ‘ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ) ( .r ‘ 𝑆 ) ( ( 𝐽 ‘ 𝑋 ) ‘ 𝐼 ) ) ( ·𝑠 ‘ 𝐷 ) ( 𝐽 ‘ 𝑌 ) ) ) = ( 𝐽 ‘ 𝑋 ) )
61	25 60	eqtrid	⊢ ( 𝜑 → 𝐶 = ( 𝐽 ‘ 𝑋 ) )
62	1 2 3 4 5 14 15 17 6 34 20 21 35 36 18 9 10	lcfrlem13	⊢ ( 𝜑 → ( 𝐽 ‘ 𝑋 ) ∈ ( { 𝑓 ∈ ( LFnl ‘ 𝑈 ) ∣ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) } ∖ { ( 0g ‘ 𝐷 ) } ) )
63		eldifsni	⊢ ( ( 𝐽 ‘ 𝑋 ) ∈ ( { 𝑓 ∈ ( LFnl ‘ 𝑈 ) ∣ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) } ∖ { ( 0g ‘ 𝐷 ) } ) → ( 𝐽 ‘ 𝑋 ) ≠ ( 0g ‘ 𝐷 ) )
64	62 63	syl	⊢ ( 𝜑 → ( 𝐽 ‘ 𝑋 ) ≠ ( 0g ‘ 𝐷 ) )
65	61 64	eqnetrd	⊢ ( 𝜑 → 𝐶 ≠ ( 0g ‘ 𝐷 ) )