Theorem lcfrlem35

Description: Lemma for lcfr . (Contributed by NM, 2-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 ) ) )
Assertion	lcfrlem35	⊢ ( 𝜑 → ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) = ( 𝐿 ‘ 𝐶 ) )

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		eqid	⊢ ( LSSum ‘ 𝑈 ) = ( LSSum ‘ 𝑈 )
27	1 2 3 4 5 6 7 8 9 10 11 12 13 26	lcfrlem23	⊢ ( 𝜑 → ( ( ⊥ ‘ { 𝑋 , 𝑌 } ) ( LSSum ‘ 𝑈 ) 𝐵 ) = ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) )
28	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20	lcfrlem24	⊢ ( 𝜑 → ( ⊥ ‘ { 𝑋 , 𝑌 } ) = ( ( 𝐿 ‘ ( 𝐽 ‘ 𝑋 ) ) ∩ ( 𝐿 ‘ ( 𝐽 ‘ 𝑌 ) ) ) )
29		eqid	⊢ ( .r ‘ 𝑆 ) = ( .r ‘ 𝑆 )
30		eqid	⊢ ( LFnl ‘ 𝑈 ) = ( LFnl ‘ 𝑈 )
31		eqid	⊢ ( ·𝑠 ‘ 𝐷 ) = ( ·𝑠 ‘ 𝐷 )
32	1 3 9	dvhlvec	⊢ ( 𝜑 → 𝑈 ∈ LVec )
33		eqid	⊢ ( 0g ‘ 𝐷 ) = ( 0g ‘ 𝐷 )
34		eqid	⊢ { 𝑓 ∈ ( LFnl ‘ 𝑈 ) ∣ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) } = { 𝑓 ∈ ( LFnl ‘ 𝑈 ) ∣ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) }
35	1 2 3 4 5 14 15 17 6 30 20 21 33 34 18 9 10	lcfrlem10	⊢ ( 𝜑 → ( 𝐽 ‘ 𝑋 ) ∈ ( LFnl ‘ 𝑈 ) )
36	1 2 3 4 5 14 15 17 6 30 20 21 33 34 18 9 11	lcfrlem10	⊢ ( 𝜑 → ( 𝐽 ‘ 𝑌 ) ∈ ( LFnl ‘ 𝑈 ) )
37		eqid	⊢ ( LSubSp ‘ 𝑈 ) = ( LSubSp ‘ 𝑈 )
38	1 3 9	dvhlmod	⊢ ( 𝜑 → 𝑈 ∈ LMod )
39	1 2 3 4 5 6 7 8 9 10 11 12 13	lcfrlem22	⊢ ( 𝜑 → 𝐵 ∈ 𝐴 )
40	37 8 38 39	lsatlssel	⊢ ( 𝜑 → 𝐵 ∈ ( LSubSp ‘ 𝑈 ) )
41	4 37	lssel	⊢ ( ( 𝐵 ∈ ( LSubSp ‘ 𝑈 ) ∧ 𝐼 ∈ 𝐵 ) → 𝐼 ∈ 𝑉 )
42	40 19 41	syl2anc	⊢ ( 𝜑 → 𝐼 ∈ 𝑉 )
43	4 15 29 16 23 30 21 31 24 32 35 36 42 22 25 20	lcfrlem2	⊢ ( 𝜑 → ( ( 𝐿 ‘ ( 𝐽 ‘ 𝑋 ) ) ∩ ( 𝐿 ‘ ( 𝐽 ‘ 𝑌 ) ) ) ⊆ ( 𝐿 ‘ 𝐶 ) )
44	28 43	eqsstrd	⊢ ( 𝜑 → ( ⊥ ‘ { 𝑋 , 𝑌 } ) ⊆ ( 𝐿 ‘ 𝐶 ) )
45	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22	lcfrlem28	⊢ ( 𝜑 → 𝐼 ≠ 0 )
46	6 7 8 32 39 19 45	lsatel	⊢ ( 𝜑 → 𝐵 = ( 𝑁 ‘ { 𝐼 } ) )
47	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	lcfrlem30	⊢ ( 𝜑 → 𝐶 ∈ ( LFnl ‘ 𝑈 ) )
48	30 20 37	lkrlss	⊢ ( ( 𝑈 ∈ LMod ∧ 𝐶 ∈ ( LFnl ‘ 𝑈 ) ) → ( 𝐿 ‘ 𝐶 ) ∈ ( LSubSp ‘ 𝑈 ) )
49	38 47 48	syl2anc	⊢ ( 𝜑 → ( 𝐿 ‘ 𝐶 ) ∈ ( LSubSp ‘ 𝑈 ) )
50	4 15 29 16 23 30 21 31 24 32 35 36 42 22 25 20	lcfrlem3	⊢ ( 𝜑 → 𝐼 ∈ ( 𝐿 ‘ 𝐶 ) )
51	37 7 38 49 50	lspsnel5a	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝐼 } ) ⊆ ( 𝐿 ‘ 𝐶 ) )
52	46 51	eqsstrd	⊢ ( 𝜑 → 𝐵 ⊆ ( 𝐿 ‘ 𝐶 ) )
53	37	lsssssubg	⊢ ( 𝑈 ∈ LMod → ( LSubSp ‘ 𝑈 ) ⊆ ( SubGrp ‘ 𝑈 ) )
54	38 53	syl	⊢ ( 𝜑 → ( LSubSp ‘ 𝑈 ) ⊆ ( SubGrp ‘ 𝑈 ) )
55	10	eldifad	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
56	11	eldifad	⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
57		prssi	⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) → { 𝑋 , 𝑌 } ⊆ 𝑉 )
58	55 56 57	syl2anc	⊢ ( 𝜑 → { 𝑋 , 𝑌 } ⊆ 𝑉 )
59	1 3 4 37 2	dochlss	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ { 𝑋 , 𝑌 } ⊆ 𝑉 ) → ( ⊥ ‘ { 𝑋 , 𝑌 } ) ∈ ( LSubSp ‘ 𝑈 ) )
60	9 58 59	syl2anc	⊢ ( 𝜑 → ( ⊥ ‘ { 𝑋 , 𝑌 } ) ∈ ( LSubSp ‘ 𝑈 ) )
61	54 60	sseldd	⊢ ( 𝜑 → ( ⊥ ‘ { 𝑋 , 𝑌 } ) ∈ ( SubGrp ‘ 𝑈 ) )
62	54 40	sseldd	⊢ ( 𝜑 → 𝐵 ∈ ( SubGrp ‘ 𝑈 ) )
63	54 49	sseldd	⊢ ( 𝜑 → ( 𝐿 ‘ 𝐶 ) ∈ ( SubGrp ‘ 𝑈 ) )
64	26	lsmlub	⊢ ( ( ( ⊥ ‘ { 𝑋 , 𝑌 } ) ∈ ( SubGrp ‘ 𝑈 ) ∧ 𝐵 ∈ ( SubGrp ‘ 𝑈 ) ∧ ( 𝐿 ‘ 𝐶 ) ∈ ( SubGrp ‘ 𝑈 ) ) → ( ( ( ⊥ ‘ { 𝑋 , 𝑌 } ) ⊆ ( 𝐿 ‘ 𝐶 ) ∧ 𝐵 ⊆ ( 𝐿 ‘ 𝐶 ) ) ↔ ( ( ⊥ ‘ { 𝑋 , 𝑌 } ) ( LSSum ‘ 𝑈 ) 𝐵 ) ⊆ ( 𝐿 ‘ 𝐶 ) ) )
65	61 62 63 64	syl3anc	⊢ ( 𝜑 → ( ( ( ⊥ ‘ { 𝑋 , 𝑌 } ) ⊆ ( 𝐿 ‘ 𝐶 ) ∧ 𝐵 ⊆ ( 𝐿 ‘ 𝐶 ) ) ↔ ( ( ⊥ ‘ { 𝑋 , 𝑌 } ) ( LSSum ‘ 𝑈 ) 𝐵 ) ⊆ ( 𝐿 ‘ 𝐶 ) ) )
66	44 52 65	mpbi2and	⊢ ( 𝜑 → ( ( ⊥ ‘ { 𝑋 , 𝑌 } ) ( LSSum ‘ 𝑈 ) 𝐵 ) ⊆ ( 𝐿 ‘ 𝐶 ) )
67	27 66	eqsstrrd	⊢ ( 𝜑 → ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) ⊆ ( 𝐿 ‘ 𝐶 ) )
68		eqid	⊢ ( LSHyp ‘ 𝑈 ) = ( LSHyp ‘ 𝑈 )
69	1 2 3 4 5 6 7 8 9 10 11 12	lcfrlem17	⊢ ( 𝜑 → ( 𝑋 + 𝑌 ) ∈ ( 𝑉 ∖ { 0 } ) )
70	1 2 3 4 6 68 9 69	dochsnshp	⊢ ( 𝜑 → ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) ∈ ( LSHyp ‘ 𝑈 ) )
71	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	lcfrlem34	⊢ ( 𝜑 → 𝐶 ≠ ( 0g ‘ 𝐷 ) )
72	68 30 20 21 33 32 47	lduallkr3	⊢ ( 𝜑 → ( ( 𝐿 ‘ 𝐶 ) ∈ ( LSHyp ‘ 𝑈 ) ↔ 𝐶 ≠ ( 0g ‘ 𝐷 ) ) )
73	71 72	mpbird	⊢ ( 𝜑 → ( 𝐿 ‘ 𝐶 ) ∈ ( LSHyp ‘ 𝑈 ) )
74	68 32 70 73	lshpcmp	⊢ ( 𝜑 → ( ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) ⊆ ( 𝐿 ‘ 𝐶 ) ↔ ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) = ( 𝐿 ‘ 𝐶 ) ) )
75	67 74	mpbid	⊢ ( 𝜑 → ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) = ( 𝐿 ‘ 𝐶 ) )