Metamath Proof Explorer

Theorem lcfrlem31

Description: Lemma for lcfr . (Contributed by NM, 10-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 ) ) )
lcfrlem31.xi ⊢ ( 𝜑 → ( ( 𝐽 ‘ 𝑋 ) ‘ 𝐼 ) ≠ 𝑄 )
lcfrlem31.cn ⊢ ( 𝜑 → 𝐶 = ( 0g ‘ 𝐷 ) )
Assertion lcfrlem31 ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) = ( 𝑁 ‘ { 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 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 lcfrlem31.xi ⊢ ( 𝜑 → ( ( 𝐽 ‘ 𝑋 ) ‘ 𝐼 ) ≠ 𝑄 )
27 lcfrlem31.cn ⊢ ( 𝜑 → 𝐶 = ( 0g ‘ 𝐷 ) )
28 25 27 syl5eqr ⊢ ( 𝜑 → ( ( 𝐽 ‘ 𝑋 ) − ( ( ( 𝐹 ‘ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ) ( .r ‘ 𝑆 ) ( ( 𝐽 ‘ 𝑋 ) ‘ 𝐼 ) ) ( ·𝑠 ‘ 𝐷 ) ( 𝐽 ‘ 𝑌 ) ) ) = ( 0g ‘ 𝐷 ) )
29 1 3 9 dvhlmod ⊢ ( 𝜑 → 𝑈 ∈ LMod )
30 21 29 lduallmod ⊢ ( 𝜑 → 𝐷 ∈ LMod )
31 eqid ⊢ ( LFnl ‘ 𝑈 ) = ( LFnl ‘ 𝑈 )
32 eqid ⊢ ( Base ‘ 𝐷 ) = ( Base ‘ 𝐷 )
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 31 20 21 33 34 18 9 10 lcfrlem10 ⊢ ( 𝜑 → ( 𝐽 ‘ 𝑋 ) ∈ ( LFnl ‘ 𝑈 ) )
36 31 21 32 29 35 ldualelvbase ⊢ ( 𝜑 → ( 𝐽 ‘ 𝑋 ) ∈ ( Base ‘ 𝐷 ) )
37 eqid ⊢ ( ·𝑠 ‘ 𝐷 ) = ( ·𝑠 ‘ 𝐷 )
38 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 )
39 1 2 3 4 5 14 15 17 6 31 20 21 33 34 18 9 11 lcfrlem10 ⊢ ( 𝜑 → ( 𝐽 ‘ 𝑌 ) ∈ ( LFnl ‘ 𝑈 ) )
40 31 15 17 21 37 29 38 39 ldualvscl ⊢ ( 𝜑 → ( ( ( 𝐹 ‘ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ) ( .r ‘ 𝑆 ) ( ( 𝐽 ‘ 𝑋 ) ‘ 𝐼 ) ) ( ·𝑠 ‘ 𝐷 ) ( 𝐽 ‘ 𝑌 ) ) ∈ ( LFnl ‘ 𝑈 ) )
41 31 21 32 29 40 ldualelvbase ⊢ ( 𝜑 → ( ( ( 𝐹 ‘ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ) ( .r ‘ 𝑆 ) ( ( 𝐽 ‘ 𝑋 ) ‘ 𝐼 ) ) ( ·𝑠 ‘ 𝐷 ) ( 𝐽 ‘ 𝑌 ) ) ∈ ( Base ‘ 𝐷 ) )
42 32 33 24 lmodsubeq0 ⊢ ( ( 𝐷 ∈ LMod ∧ ( 𝐽 ‘ 𝑋 ) ∈ ( Base ‘ 𝐷 ) ∧ ( ( ( 𝐹 ‘ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ) ( .r ‘ 𝑆 ) ( ( 𝐽 ‘ 𝑋 ) ‘ 𝐼 ) ) ( ·𝑠 ‘ 𝐷 ) ( 𝐽 ‘ 𝑌 ) ) ∈ ( Base ‘ 𝐷 ) ) → ( ( ( 𝐽 ‘ 𝑋 ) − ( ( ( 𝐹 ‘ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ) ( .r ‘ 𝑆 ) ( ( 𝐽 ‘ 𝑋 ) ‘ 𝐼 ) ) ( ·𝑠 ‘ 𝐷 ) ( 𝐽 ‘ 𝑌 ) ) ) = ( 0g ‘ 𝐷 ) ↔ ( 𝐽 ‘ 𝑋 ) = ( ( ( 𝐹 ‘ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ) ( .r ‘ 𝑆 ) ( ( 𝐽 ‘ 𝑋 ) ‘ 𝐼 ) ) ( ·𝑠 ‘ 𝐷 ) ( 𝐽 ‘ 𝑌 ) ) ) )
43 30 36 41 42 syl3anc ⊢ ( 𝜑 → ( ( ( 𝐽 ‘ 𝑋 ) − ( ( ( 𝐹 ‘ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ) ( .r ‘ 𝑆 ) ( ( 𝐽 ‘ 𝑋 ) ‘ 𝐼 ) ) ( ·𝑠 ‘ 𝐷 ) ( 𝐽 ‘ 𝑌 ) ) ) = ( 0g ‘ 𝐷 ) ↔ ( 𝐽 ‘ 𝑋 ) = ( ( ( 𝐹 ‘ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ) ( .r ‘ 𝑆 ) ( ( 𝐽 ‘ 𝑋 ) ‘ 𝐼 ) ) ( ·𝑠 ‘ 𝐷 ) ( 𝐽 ‘ 𝑌 ) ) ) )
44 28 43 mpbid ⊢ ( 𝜑 → ( 𝐽 ‘ 𝑋 ) = ( ( ( 𝐹 ‘ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ) ( .r ‘ 𝑆 ) ( ( 𝐽 ‘ 𝑋 ) ‘ 𝐼 ) ) ( ·𝑠 ‘ 𝐷 ) ( 𝐽 ‘ 𝑌 ) ) )
45 44 fveq2d ⊢ ( 𝜑 → ( 𝐿 ‘ ( 𝐽 ‘ 𝑋 ) ) = ( 𝐿 ‘ ( ( ( 𝐹 ‘ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ) ( .r ‘ 𝑆 ) ( ( 𝐽 ‘ 𝑋 ) ‘ 𝐼 ) ) ( ·𝑠 ‘ 𝐷 ) ( 𝐽 ‘ 𝑌 ) ) ) )
46 1 3 9 dvhlvec ⊢ ( 𝜑 → 𝑈 ∈ LVec )
47 15 lvecdrng ⊢ ( 𝑈 ∈ LVec → 𝑆 ∈ DivRing )
48 46 47 syl ⊢ ( 𝜑 → 𝑆 ∈ DivRing )
49 1 2 3 4 5 6 7 8 9 10 11 12 13 lcfrlem22 ⊢ ( 𝜑 → 𝐵 ∈ 𝐴 )
50 4 8 29 49 lsatssv ⊢ ( 𝜑 → 𝐵 ⊆ 𝑉 )
51 50 19 sseldd ⊢ ( 𝜑 → 𝐼 ∈ 𝑉 )
52 15 17 4 31 lflcl ⊢ ( ( 𝑈 ∈ LMod ∧ ( 𝐽 ‘ 𝑌 ) ∈ ( LFnl ‘ 𝑈 ) ∧ 𝐼 ∈ 𝑉 ) → ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ∈ 𝑅 )
53 29 39 51 52 syl3anc ⊢ ( 𝜑 → ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ∈ 𝑅 )
54 17 16 23 drnginvrn0 ⊢ ( ( 𝑆 ∈ DivRing ∧ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ∈ 𝑅 ∧ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ≠ 𝑄 ) → ( 𝐹 ‘ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ) ≠ 𝑄 )
55 48 53 22 54 syl3anc ⊢ ( 𝜑 → ( 𝐹 ‘ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ) ≠ 𝑄 )
56 eqid ⊢ ( .r ‘ 𝑆 ) = ( .r ‘ 𝑆 )
57 17 16 23 drnginvrcl ⊢ ( ( 𝑆 ∈ DivRing ∧ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ∈ 𝑅 ∧ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ≠ 𝑄 ) → ( 𝐹 ‘ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ) ∈ 𝑅 )
58 48 53 22 57 syl3anc ⊢ ( 𝜑 → ( 𝐹 ‘ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ) ∈ 𝑅 )
59 15 17 4 31 lflcl ⊢ ( ( 𝑈 ∈ LMod ∧ ( 𝐽 ‘ 𝑋 ) ∈ ( LFnl ‘ 𝑈 ) ∧ 𝐼 ∈ 𝑉 ) → ( ( 𝐽 ‘ 𝑋 ) ‘ 𝐼 ) ∈ 𝑅 )
60 29 35 51 59 syl3anc ⊢ ( 𝜑 → ( ( 𝐽 ‘ 𝑋 ) ‘ 𝐼 ) ∈ 𝑅 )
61 17 16 56 48 58 60 drngmulne0 ⊢ ( 𝜑 → ( ( ( 𝐹 ‘ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ) ( .r ‘ 𝑆 ) ( ( 𝐽 ‘ 𝑋 ) ‘ 𝐼 ) ) ≠ 𝑄 ↔ ( ( 𝐹 ‘ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ) ≠ 𝑄 ∧ ( ( 𝐽 ‘ 𝑋 ) ‘ 𝐼 ) ≠ 𝑄 ) ) )
62 55 26 61 mpbir2and ⊢ ( 𝜑 → ( ( 𝐹 ‘ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ) ( .r ‘ 𝑆 ) ( ( 𝐽 ‘ 𝑋 ) ‘ 𝐼 ) ) ≠ 𝑄 )
63 15 17 16 31 20 21 37 46 39 38 62 ldualkrsc ⊢ ( 𝜑 → ( 𝐿 ‘ ( ( ( 𝐹 ‘ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ) ( .r ‘ 𝑆 ) ( ( 𝐽 ‘ 𝑋 ) ‘ 𝐼 ) ) ( ·𝑠 ‘ 𝐷 ) ( 𝐽 ‘ 𝑌 ) ) ) = ( 𝐿 ‘ ( 𝐽 ‘ 𝑌 ) ) )
64 45 63 eqtrd ⊢ ( 𝜑 → ( 𝐿 ‘ ( 𝐽 ‘ 𝑋 ) ) = ( 𝐿 ‘ ( 𝐽 ‘ 𝑌 ) ) )
65 64 fveq2d ⊢ ( 𝜑 → ( ⊥ ‘ ( 𝐿 ‘ ( 𝐽 ‘ 𝑋 ) ) ) = ( ⊥ ‘ ( 𝐿 ‘ ( 𝐽 ‘ 𝑌 ) ) ) )
66 1 2 3 4 5 14 15 17 6 31 20 21 33 34 18 9 10 7 lcfrlem14 ⊢ ( 𝜑 → ( ⊥ ‘ ( 𝐿 ‘ ( 𝐽 ‘ 𝑋 ) ) ) = ( 𝑁 ‘ { 𝑋 } ) )
67 1 2 3 4 5 14 15 17 6 31 20 21 33 34 18 9 11 7 lcfrlem14 ⊢ ( 𝜑 → ( ⊥ ‘ ( 𝐿 ‘ ( 𝐽 ‘ 𝑌 ) ) ) = ( 𝑁 ‘ { 𝑌 } ) )
68 65 66 67 3eqtr3d ⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) = ( 𝑁 ‘ { 𝑌 } ) )