Metamath Proof Explorer

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 ) ) )