Theorem lcfrlem37

Description: Lemma for lcfr . (Contributed by NM, 8-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 ) ) )
	lcfrlem37.g	⊢ ( 𝜑 → 𝐺 ∈ ( LSubSp ‘ 𝐷 ) )
	lcfrlem37.gs	⊢ ( 𝜑 → 𝐺 ⊆ { 𝑓 ∈ ( LFnl ‘ 𝑈 ) ∣ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) } )
	lcfrlem37.e	⊢ 𝐸 = ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘ ( 𝐿 ‘ 𝑔 ) )
	lcfrlem37.xe	⊢ ( 𝜑 → 𝑋 ∈ 𝐸 )
	lcfrlem37.ye	⊢ ( 𝜑 → 𝑌 ∈ 𝐸 )
Assertion	lcfrlem37	⊢ ( 𝜑 → ( 𝑋 + 𝑌 ) ∈ 𝐸 )

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		lcfrlem37.g	⊢ ( 𝜑 → 𝐺 ∈ ( LSubSp ‘ 𝐷 ) )
27		lcfrlem37.gs	⊢ ( 𝜑 → 𝐺 ⊆ { 𝑓 ∈ ( LFnl ‘ 𝑈 ) ∣ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) } )
28		lcfrlem37.e	⊢ 𝐸 = ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘ ( 𝐿 ‘ 𝑔 ) )
29		lcfrlem37.xe	⊢ ( 𝜑 → 𝑋 ∈ 𝐸 )
30		lcfrlem37.ye	⊢ ( 𝜑 → 𝑌 ∈ 𝐸 )
31		eqid	⊢ ( LSubSp ‘ 𝐷 ) = ( LSubSp ‘ 𝐷 )
32	1 3 9	dvhlmod	⊢ ( 𝜑 → 𝑈 ∈ LMod )
33		eqid	⊢ ( LFnl ‘ 𝑈 ) = ( LFnl ‘ 𝑈 )
34		eqid	⊢ ( 0g ‘ 𝐷 ) = ( 0g ‘ 𝐷 )
35		eqid	⊢ { 𝑓 ∈ ( LFnl ‘ 𝑈 ) ∣ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) } = { 𝑓 ∈ ( LFnl ‘ 𝑈 ) ∣ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) }
36		eldifsni	⊢ ( 𝑋 ∈ ( 𝑉 ∖ { 0 } ) → 𝑋 ≠ 0 )
37	10 36	syl	⊢ ( 𝜑 → 𝑋 ≠ 0 )
38		eldifsn	⊢ ( 𝑋 ∈ ( 𝐸 ∖ { 0 } ) ↔ ( 𝑋 ∈ 𝐸 ∧ 𝑋 ≠ 0 ) )
39	29 37 38	sylanbrc	⊢ ( 𝜑 → 𝑋 ∈ ( 𝐸 ∖ { 0 } ) )
40	1 2 3 4 5 14 15 17 6 33 20 21 34 35 18 9 31 26 27 28 39	lcfrlem16	⊢ ( 𝜑 → ( 𝐽 ‘ 𝑋 ) ∈ 𝐺 )
41		eqid	⊢ ( ·𝑠 ‘ 𝐷 ) = ( ·𝑠 ‘ 𝐷 )
42	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23	lcfrlem29	⊢ ( 𝜑 → ( ( 𝐹 ‘ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ) ( .r ‘ 𝑆 ) ( ( 𝐽 ‘ 𝑋 ) ‘ 𝐼 ) ) ∈ 𝑅 )
43		eldifsni	⊢ ( 𝑌 ∈ ( 𝑉 ∖ { 0 } ) → 𝑌 ≠ 0 )
44	11 43	syl	⊢ ( 𝜑 → 𝑌 ≠ 0 )
45		eldifsn	⊢ ( 𝑌 ∈ ( 𝐸 ∖ { 0 } ) ↔ ( 𝑌 ∈ 𝐸 ∧ 𝑌 ≠ 0 ) )
46	30 44 45	sylanbrc	⊢ ( 𝜑 → 𝑌 ∈ ( 𝐸 ∖ { 0 } ) )
47	1 2 3 4 5 14 15 17 6 33 20 21 34 35 18 9 31 26 27 28 46	lcfrlem16	⊢ ( 𝜑 → ( 𝐽 ‘ 𝑌 ) ∈ 𝐺 )
48	15 17 21 41 31 32 26 42 47	ldualssvscl	⊢ ( 𝜑 → ( ( ( 𝐹 ‘ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ) ( .r ‘ 𝑆 ) ( ( 𝐽 ‘ 𝑋 ) ‘ 𝐼 ) ) ( ·𝑠 ‘ 𝐷 ) ( 𝐽 ‘ 𝑌 ) ) ∈ 𝐺 )
49	21 24 31 32 26 40 48	ldualssvsubcl	⊢ ( 𝜑 → ( ( 𝐽 ‘ 𝑋 ) − ( ( ( 𝐹 ‘ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ) ( .r ‘ 𝑆 ) ( ( 𝐽 ‘ 𝑋 ) ‘ 𝐼 ) ) ( ·𝑠 ‘ 𝐷 ) ( 𝐽 ‘ 𝑌 ) ) ) ∈ 𝐺 )
50	25 49	eqeltrid	⊢ ( 𝜑 → 𝐶 ∈ 𝐺 )
51	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	lcfrlem36	⊢ ( 𝜑 → ( 𝑋 + 𝑌 ) ∈ ( ⊥ ‘ ( 𝐿 ‘ 𝐶 ) ) )
52		2fveq3	⊢ ( 𝑔 = 𝐶 → ( ⊥ ‘ ( 𝐿 ‘ 𝑔 ) ) = ( ⊥ ‘ ( 𝐿 ‘ 𝐶 ) ) )
53	52	eleq2d	⊢ ( 𝑔 = 𝐶 → ( ( 𝑋 + 𝑌 ) ∈ ( ⊥ ‘ ( 𝐿 ‘ 𝑔 ) ) ↔ ( 𝑋 + 𝑌 ) ∈ ( ⊥ ‘ ( 𝐿 ‘ 𝐶 ) ) ) )
54	53	rspcev	⊢ ( ( 𝐶 ∈ 𝐺 ∧ ( 𝑋 + 𝑌 ) ∈ ( ⊥ ‘ ( 𝐿 ‘ 𝐶 ) ) ) → ∃ 𝑔 ∈ 𝐺 ( 𝑋 + 𝑌 ) ∈ ( ⊥ ‘ ( 𝐿 ‘ 𝑔 ) ) )
55	50 51 54	syl2anc	⊢ ( 𝜑 → ∃ 𝑔 ∈ 𝐺 ( 𝑋 + 𝑌 ) ∈ ( ⊥ ‘ ( 𝐿 ‘ 𝑔 ) ) )
56		eliun	⊢ ( ( 𝑋 + 𝑌 ) ∈ ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘ ( 𝐿 ‘ 𝑔 ) ) ↔ ∃ 𝑔 ∈ 𝐺 ( 𝑋 + 𝑌 ) ∈ ( ⊥ ‘ ( 𝐿 ‘ 𝑔 ) ) )
57	55 56	sylibr	⊢ ( 𝜑 → ( 𝑋 + 𝑌 ) ∈ ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘ ( 𝐿 ‘ 𝑔 ) ) )
58	57 28	eleqtrrdi	⊢ ( 𝜑 → ( 𝑋 + 𝑌 ) ∈ 𝐸 )