Metamath Proof Explorer

Theorem lcfrlem33

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 ) ) )
lcfrlem33.xi ⊢ ( 𝜑 → ( ( 𝐽 ‘ 𝑋 ) ‘ 𝐼 ) = 𝑄 )
Assertion lcfrlem33 ( 𝜑 → 𝐶 ≠ ( 0g ‘ 𝐷 ) )

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 lcfrlem33.xi ⊢ ( 𝜑 → ( ( 𝐽 ‘ 𝑋 ) ‘ 𝐼 ) = 𝑄 )
27 26 oveq2d ⊢ ( 𝜑 → ( ( 𝐹 ‘ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ) ( .r ‘ 𝑆 ) ( ( 𝐽 ‘ 𝑋 ) ‘ 𝐼 ) ) = ( ( 𝐹 ‘ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ) ( .r ‘ 𝑆 ) 𝑄 ) )
28 1 3 9 dvhlmod ⊢ ( 𝜑 → 𝑈 ∈ LMod )
29 15 lmodring ⊢ ( 𝑈 ∈ LMod → 𝑆 ∈ Ring )
30 28 29 syl ⊢ ( 𝜑 → 𝑆 ∈ Ring )
31 1 3 9 dvhlvec ⊢ ( 𝜑 → 𝑈 ∈ LVec )
32 15 lvecdrng ⊢ ( 𝑈 ∈ LVec → 𝑆 ∈ DivRing )
33 31 32 syl ⊢ ( 𝜑 → 𝑆 ∈ DivRing )
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 1 2 3 4 5 6 7 8 9 10 11 12 13 lcfrlem22 ⊢ ( 𝜑 → 𝐵 ∈ 𝐴 )
39 4 8 28 38 lsatssv ⊢ ( 𝜑 → 𝐵 ⊆ 𝑉 )
40 39 19 sseldd ⊢ ( 𝜑 → 𝐼 ∈ 𝑉 )
41 15 17 4 34 lflcl ⊢ ( ( 𝑈 ∈ LMod ∧ ( 𝐽 ‘ 𝑌 ) ∈ ( LFnl ‘ 𝑈 ) ∧ 𝐼 ∈ 𝑉 ) → ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ∈ 𝑅 )
42 28 37 40 41 syl3anc ⊢ ( 𝜑 → ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ∈ 𝑅 )
43 17 16 23 drnginvrcl ⊢ ( ( 𝑆 ∈ DivRing ∧ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ∈ 𝑅 ∧ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ≠ 𝑄 ) → ( 𝐹 ‘ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ) ∈ 𝑅 )
44 33 42 22 43 syl3anc ⊢ ( 𝜑 → ( 𝐹 ‘ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ) ∈ 𝑅 )
45 eqid ⊢ ( .r ‘ 𝑆 ) = ( .r ‘ 𝑆 )
46 17 45 16 ringrz ⊢ ( ( 𝑆 ∈ Ring ∧ ( 𝐹 ‘ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ) ∈ 𝑅 ) → ( ( 𝐹 ‘ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ) ( .r ‘ 𝑆 ) 𝑄 ) = 𝑄 )
47 30 44 46 syl2anc ⊢ ( 𝜑 → ( ( 𝐹 ‘ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ) ( .r ‘ 𝑆 ) 𝑄 ) = 𝑄 )
48 27 47 eqtrd ⊢ ( 𝜑 → ( ( 𝐹 ‘ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ) ( .r ‘ 𝑆 ) ( ( 𝐽 ‘ 𝑋 ) ‘ 𝐼 ) ) = 𝑄 )
49 48 oveq1d ⊢ ( 𝜑 → ( ( ( 𝐹 ‘ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ) ( .r ‘ 𝑆 ) ( ( 𝐽 ‘ 𝑋 ) ‘ 𝐼 ) ) ( ·𝑠 ‘ 𝐷 ) ( 𝐽 ‘ 𝑌 ) ) = ( 𝑄 ( ·𝑠 ‘ 𝐷 ) ( 𝐽 ‘ 𝑌 ) ) )
50 eqid ⊢ ( ·𝑠 ‘ 𝐷 ) = ( ·𝑠 ‘ 𝐷 )
51 34 15 16 21 50 35 28 37 ldual0vs ⊢ ( 𝜑 → ( 𝑄 ( ·𝑠 ‘ 𝐷 ) ( 𝐽 ‘ 𝑌 ) ) = ( 0g ‘ 𝐷 ) )
52 49 51 eqtrd ⊢ ( 𝜑 → ( ( ( 𝐹 ‘ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ) ( .r ‘ 𝑆 ) ( ( 𝐽 ‘ 𝑋 ) ‘ 𝐼 ) ) ( ·𝑠 ‘ 𝐷 ) ( 𝐽 ‘ 𝑌 ) ) = ( 0g ‘ 𝐷 ) )
53 52 oveq2d ⊢ ( 𝜑 → ( ( 𝐽 ‘ 𝑋 ) − ( ( ( 𝐹 ‘ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ) ( .r ‘ 𝑆 ) ( ( 𝐽 ‘ 𝑋 ) ‘ 𝐼 ) ) ( ·𝑠 ‘ 𝐷 ) ( 𝐽 ‘ 𝑌 ) ) ) = ( ( 𝐽 ‘ 𝑋 ) − ( 0g ‘ 𝐷 ) ) )
54 21 28 ldualgrp ⊢ ( 𝜑 → 𝐷 ∈ Grp )
55 eqid ⊢ ( Base ‘ 𝐷 ) = ( Base ‘ 𝐷 )
56 1 2 3 4 5 14 15 17 6 34 20 21 35 36 18 9 10 lcfrlem10 ⊢ ( 𝜑 → ( 𝐽 ‘ 𝑋 ) ∈ ( LFnl ‘ 𝑈 ) )
57 34 21 55 28 56 ldualelvbase ⊢ ( 𝜑 → ( 𝐽 ‘ 𝑋 ) ∈ ( Base ‘ 𝐷 ) )
58 55 35 24 grpsubid1 ⊢ ( ( 𝐷 ∈ Grp ∧ ( 𝐽 ‘ 𝑋 ) ∈ ( Base ‘ 𝐷 ) ) → ( ( 𝐽 ‘ 𝑋 ) − ( 0g ‘ 𝐷 ) ) = ( 𝐽 ‘ 𝑋 ) )
59 54 57 58 syl2anc ⊢ ( 𝜑 → ( ( 𝐽 ‘ 𝑋 ) − ( 0g ‘ 𝐷 ) ) = ( 𝐽 ‘ 𝑋 ) )
60 53 59 eqtrd ⊢ ( 𝜑 → ( ( 𝐽 ‘ 𝑋 ) − ( ( ( 𝐹 ‘ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ) ( .r ‘ 𝑆 ) ( ( 𝐽 ‘ 𝑋 ) ‘ 𝐼 ) ) ( ·𝑠 ‘ 𝐷 ) ( 𝐽 ‘ 𝑌 ) ) ) = ( 𝐽 ‘ 𝑋 ) )
61 25 60 syl5eq ⊢ ( 𝜑 → 𝐶 = ( 𝐽 ‘ 𝑋 ) )
62 1 2 3 4 5 14 15 17 6 34 20 21 35 36 18 9 10 lcfrlem13 ⊢ ( 𝜑 → ( 𝐽 ‘ 𝑋 ) ∈ ( { 𝑓 ∈ ( LFnl ‘ 𝑈 ) ∣ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) } ∖ { ( 0g ‘ 𝐷 ) } ) )
63 eldifsni ⊢ ( ( 𝐽 ‘ 𝑋 ) ∈ ( { 𝑓 ∈ ( LFnl ‘ 𝑈 ) ∣ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) } ∖ { ( 0g ‘ 𝐷 ) } ) → ( 𝐽 ‘ 𝑋 ) ≠ ( 0g ‘ 𝐷 ) )
64 62 63 syl ⊢ ( 𝜑 → ( 𝐽 ‘ 𝑋 ) ≠ ( 0g ‘ 𝐷 ) )
65 61 64 eqnetrd ⊢ ( 𝜑 → 𝐶 ≠ ( 0g ‘ 𝐷 ) )