Theorem lcfrlem25

Description: Lemma for lcfr . Special case of lcfrlem35 when ( ( JY )I ) is zero. (Contributed by NM, 11-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 )
	lcfrlem25.jz	⊢ ( 𝜑 → ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) = 𝑄 )
	lcfrlem25.in	⊢ ( 𝜑 → 𝐼 ≠ 0 )
Assertion	lcfrlem25	⊢ ( 𝜑 → ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) = ( 𝐿 ‘ ( 𝐽 ‘ 𝑌 ) ) )

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		lcfrlem25.jz	⊢ ( 𝜑 → ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) = 𝑄 )
23		lcfrlem25.in	⊢ ( 𝜑 → 𝐼 ≠ 0 )
24		eqid	⊢ ( LSSum ‘ 𝑈 ) = ( LSSum ‘ 𝑈 )
25	1 2 3 4 5 6 7 8 9 10 11 12 13 24	lcfrlem23	⊢ ( 𝜑 → ( ( ⊥ ‘ { 𝑋 , 𝑌 } ) ( LSSum ‘ 𝑈 ) 𝐵 ) = ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) )
26	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 ) ) ) )
27		inss2	⊢ ( ( 𝐿 ‘ ( 𝐽 ‘ 𝑋 ) ) ∩ ( 𝐿 ‘ ( 𝐽 ‘ 𝑌 ) ) ) ⊆ ( 𝐿 ‘ ( 𝐽 ‘ 𝑌 ) )
28	26 27	eqsstrdi	⊢ ( 𝜑 → ( ⊥ ‘ { 𝑋 , 𝑌 } ) ⊆ ( 𝐿 ‘ ( 𝐽 ‘ 𝑌 ) ) )
29	1 3 9	dvhlvec	⊢ ( 𝜑 → 𝑈 ∈ LVec )
30	1 2 3 4 5 6 7 8 9 10 11 12 13	lcfrlem22	⊢ ( 𝜑 → 𝐵 ∈ 𝐴 )
31	6 7 8 29 30 19 23	lsatel	⊢ ( 𝜑 → 𝐵 = ( 𝑁 ‘ { 𝐼 } ) )
32		eqid	⊢ ( LSubSp ‘ 𝑈 ) = ( LSubSp ‘ 𝑈 )
33	1 3 9	dvhlmod	⊢ ( 𝜑 → 𝑈 ∈ LMod )
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	34 20 32	lkrlss	⊢ ( ( 𝑈 ∈ LMod ∧ ( 𝐽 ‘ 𝑌 ) ∈ ( LFnl ‘ 𝑈 ) ) → ( 𝐿 ‘ ( 𝐽 ‘ 𝑌 ) ) ∈ ( LSubSp ‘ 𝑈 ) )
39	33 37 38	syl2anc	⊢ ( 𝜑 → ( 𝐿 ‘ ( 𝐽 ‘ 𝑌 ) ) ∈ ( LSubSp ‘ 𝑈 ) )
40	4 8 33 30	lsatssv	⊢ ( 𝜑 → 𝐵 ⊆ 𝑉 )
41	40 19	sseldd	⊢ ( 𝜑 → 𝐼 ∈ 𝑉 )
42	4 15 16 34 20 33 37 41	ellkr2	⊢ ( 𝜑 → ( 𝐼 ∈ ( 𝐿 ‘ ( 𝐽 ‘ 𝑌 ) ) ↔ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) = 𝑄 ) )
43	22 42	mpbird	⊢ ( 𝜑 → 𝐼 ∈ ( 𝐿 ‘ ( 𝐽 ‘ 𝑌 ) ) )
44	32 7 33 39 43	lspsnel5a	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝐼 } ) ⊆ ( 𝐿 ‘ ( 𝐽 ‘ 𝑌 ) ) )
45	31 44	eqsstrd	⊢ ( 𝜑 → 𝐵 ⊆ ( 𝐿 ‘ ( 𝐽 ‘ 𝑌 ) ) )
46	32	lsssssubg	⊢ ( 𝑈 ∈ LMod → ( LSubSp ‘ 𝑈 ) ⊆ ( SubGrp ‘ 𝑈 ) )
47	33 46	syl	⊢ ( 𝜑 → ( LSubSp ‘ 𝑈 ) ⊆ ( SubGrp ‘ 𝑈 ) )
48	10	eldifad	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
49	11	eldifad	⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
50		prssi	⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) → { 𝑋 , 𝑌 } ⊆ 𝑉 )
51	48 49 50	syl2anc	⊢ ( 𝜑 → { 𝑋 , 𝑌 } ⊆ 𝑉 )
52	1 3 4 32 2	dochlss	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ { 𝑋 , 𝑌 } ⊆ 𝑉 ) → ( ⊥ ‘ { 𝑋 , 𝑌 } ) ∈ ( LSubSp ‘ 𝑈 ) )
53	9 51 52	syl2anc	⊢ ( 𝜑 → ( ⊥ ‘ { 𝑋 , 𝑌 } ) ∈ ( LSubSp ‘ 𝑈 ) )
54	47 53	sseldd	⊢ ( 𝜑 → ( ⊥ ‘ { 𝑋 , 𝑌 } ) ∈ ( SubGrp ‘ 𝑈 ) )
55	4 32 7 33 48 49	lspprcl	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ∈ ( LSubSp ‘ 𝑈 ) )
56	1 2 3 4 5 6 7 8 9 10 11 12	lcfrlem17	⊢ ( 𝜑 → ( 𝑋 + 𝑌 ) ∈ ( 𝑉 ∖ { 0 } ) )
57	56	eldifad	⊢ ( 𝜑 → ( 𝑋 + 𝑌 ) ∈ 𝑉 )
58	57	snssd	⊢ ( 𝜑 → { ( 𝑋 + 𝑌 ) } ⊆ 𝑉 )
59	1 3 4 32 2	dochlss	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ { ( 𝑋 + 𝑌 ) } ⊆ 𝑉 ) → ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) ∈ ( LSubSp ‘ 𝑈 ) )
60	9 58 59	syl2anc	⊢ ( 𝜑 → ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) ∈ ( LSubSp ‘ 𝑈 ) )
61	32	lssincl	⊢ ( ( 𝑈 ∈ LMod ∧ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ∈ ( LSubSp ‘ 𝑈 ) ∧ ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) ∈ ( LSubSp ‘ 𝑈 ) ) → ( ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ∩ ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) ) ∈ ( LSubSp ‘ 𝑈 ) )
62	33 55 60 61	syl3anc	⊢ ( 𝜑 → ( ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ∩ ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) ) ∈ ( LSubSp ‘ 𝑈 ) )
63	13 62	eqeltrid	⊢ ( 𝜑 → 𝐵 ∈ ( LSubSp ‘ 𝑈 ) )
64	47 63	sseldd	⊢ ( 𝜑 → 𝐵 ∈ ( SubGrp ‘ 𝑈 ) )
65	47 39	sseldd	⊢ ( 𝜑 → ( 𝐿 ‘ ( 𝐽 ‘ 𝑌 ) ) ∈ ( SubGrp ‘ 𝑈 ) )
66	24	lsmlub	⊢ ( ( ( ⊥ ‘ { 𝑋 , 𝑌 } ) ∈ ( SubGrp ‘ 𝑈 ) ∧ 𝐵 ∈ ( SubGrp ‘ 𝑈 ) ∧ ( 𝐿 ‘ ( 𝐽 ‘ 𝑌 ) ) ∈ ( SubGrp ‘ 𝑈 ) ) → ( ( ( ⊥ ‘ { 𝑋 , 𝑌 } ) ⊆ ( 𝐿 ‘ ( 𝐽 ‘ 𝑌 ) ) ∧ 𝐵 ⊆ ( 𝐿 ‘ ( 𝐽 ‘ 𝑌 ) ) ) ↔ ( ( ⊥ ‘ { 𝑋 , 𝑌 } ) ( LSSum ‘ 𝑈 ) 𝐵 ) ⊆ ( 𝐿 ‘ ( 𝐽 ‘ 𝑌 ) ) ) )
67	54 64 65 66	syl3anc	⊢ ( 𝜑 → ( ( ( ⊥ ‘ { 𝑋 , 𝑌 } ) ⊆ ( 𝐿 ‘ ( 𝐽 ‘ 𝑌 ) ) ∧ 𝐵 ⊆ ( 𝐿 ‘ ( 𝐽 ‘ 𝑌 ) ) ) ↔ ( ( ⊥ ‘ { 𝑋 , 𝑌 } ) ( LSSum ‘ 𝑈 ) 𝐵 ) ⊆ ( 𝐿 ‘ ( 𝐽 ‘ 𝑌 ) ) ) )
68	28 45 67	mpbi2and	⊢ ( 𝜑 → ( ( ⊥ ‘ { 𝑋 , 𝑌 } ) ( LSSum ‘ 𝑈 ) 𝐵 ) ⊆ ( 𝐿 ‘ ( 𝐽 ‘ 𝑌 ) ) )
69	25 68	eqsstrrd	⊢ ( 𝜑 → ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) ⊆ ( 𝐿 ‘ ( 𝐽 ‘ 𝑌 ) ) )
70		eqid	⊢ ( LSHyp ‘ 𝑈 ) = ( LSHyp ‘ 𝑈 )
71	1 2 3 4 6 70 9 56	dochsnshp	⊢ ( 𝜑 → ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) ∈ ( LSHyp ‘ 𝑈 ) )
72	1 2 3 4 5 14 15 17 6 34 20 21 35 36 18 9 11	lcfrlem13	⊢ ( 𝜑 → ( 𝐽 ‘ 𝑌 ) ∈ ( { 𝑓 ∈ ( LFnl ‘ 𝑈 ) ∣ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) } ∖ { ( 0g ‘ 𝐷 ) } ) )
73		eldifsni	⊢ ( ( 𝐽 ‘ 𝑌 ) ∈ ( { 𝑓 ∈ ( LFnl ‘ 𝑈 ) ∣ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) } ∖ { ( 0g ‘ 𝐷 ) } ) → ( 𝐽 ‘ 𝑌 ) ≠ ( 0g ‘ 𝐷 ) )
74	72 73	syl	⊢ ( 𝜑 → ( 𝐽 ‘ 𝑌 ) ≠ ( 0g ‘ 𝐷 ) )
75	70 34 20 21 35 29 37	lduallkr3	⊢ ( 𝜑 → ( ( 𝐿 ‘ ( 𝐽 ‘ 𝑌 ) ) ∈ ( LSHyp ‘ 𝑈 ) ↔ ( 𝐽 ‘ 𝑌 ) ≠ ( 0g ‘ 𝐷 ) ) )
76	74 75	mpbird	⊢ ( 𝜑 → ( 𝐿 ‘ ( 𝐽 ‘ 𝑌 ) ) ∈ ( LSHyp ‘ 𝑈 ) )
77	70 29 71 76	lshpcmp	⊢ ( 𝜑 → ( ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) ⊆ ( 𝐿 ‘ ( 𝐽 ‘ 𝑌 ) ) ↔ ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) = ( 𝐿 ‘ ( 𝐽 ‘ 𝑌 ) ) ) )
78	69 77	mpbid	⊢ ( 𝜑 → ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) = ( 𝐿 ‘ ( 𝐽 ‘ 𝑌 ) ) )