Metamath Proof Explorer

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 ) )