Metamath Proof Explorer


Theorem lkrlsp

Description: The subspace sum of a kernel and the span of a vector not in the kernel (by ellkr ) is the whole vector space. (Contributed by NM, 19-Apr-2014)

Ref Expression
Hypotheses lkrlsp.d D = Scalar W
lkrlsp.o 0 ˙ = 0 D
lkrlsp.v V = Base W
lkrlsp.n N = LSpan W
lkrlsp.p ˙ = LSSum W
lkrlsp.f F = LFnl W
lkrlsp.k K = LKer W
Assertion lkrlsp W LVec X V G F G X 0 ˙ K G ˙ N X = V

Proof

Step Hyp Ref Expression
1 lkrlsp.d D = Scalar W
2 lkrlsp.o 0 ˙ = 0 D
3 lkrlsp.v V = Base W
4 lkrlsp.n N = LSpan W
5 lkrlsp.p ˙ = LSSum W
6 lkrlsp.f F = LFnl W
7 lkrlsp.k K = LKer W
8 lveclmod W LVec W LMod
9 8 3ad2ant1 W LVec X V G F G X 0 ˙ W LMod
10 simp2r W LVec X V G F G X 0 ˙ G F
11 eqid LSubSp W = LSubSp W
12 6 7 11 lkrlss W LMod G F K G LSubSp W
13 9 10 12 syl2anc W LVec X V G F G X 0 ˙ K G LSubSp W
14 simp2l W LVec X V G F G X 0 ˙ X V
15 3 11 4 lspsncl W LMod X V N X LSubSp W
16 9 14 15 syl2anc W LVec X V G F G X 0 ˙ N X LSubSp W
17 11 5 lsmcl W LMod K G LSubSp W N X LSubSp W K G ˙ N X LSubSp W
18 9 13 16 17 syl3anc W LVec X V G F G X 0 ˙ K G ˙ N X LSubSp W
19 3 11 lssss K G ˙ N X LSubSp W K G ˙ N X V
20 18 19 syl W LVec X V G F G X 0 ˙ K G ˙ N X V
21 simpl1 W LVec X V G F G X 0 ˙ u V W LVec
22 21 8 syl W LVec X V G F G X 0 ˙ u V W LMod
23 simpr W LVec X V G F G X 0 ˙ u V u V
24 1 lmodring W LMod D Ring
25 22 24 syl W LVec X V G F G X 0 ˙ u V D Ring
26 simpl2r W LVec X V G F G X 0 ˙ u V G F
27 eqid Base D = Base D
28 1 27 3 6 lflcl W LVec G F u V G u Base D
29 21 26 23 28 syl3anc W LVec X V G F G X 0 ˙ u V G u Base D
30 1 lvecdrng W LVec D DivRing
31 21 30 syl W LVec X V G F G X 0 ˙ u V D DivRing
32 simpl2l W LVec X V G F G X 0 ˙ u V X V
33 1 27 3 6 lflcl W LVec G F X V G X Base D
34 21 26 32 33 syl3anc W LVec X V G F G X 0 ˙ u V G X Base D
35 simpl3 W LVec X V G F G X 0 ˙ u V G X 0 ˙
36 eqid inv r D = inv r D
37 27 2 36 drnginvrcl D DivRing G X Base D G X 0 ˙ inv r D G X Base D
38 31 34 35 37 syl3anc W LVec X V G F G X 0 ˙ u V inv r D G X Base D
39 eqid D = D
40 27 39 ringcl D Ring G u Base D inv r D G X Base D G u D inv r D G X Base D
41 25 29 38 40 syl3anc W LVec X V G F G X 0 ˙ u V G u D inv r D G X Base D
42 eqid W = W
43 3 1 42 27 lmodvscl W LMod G u D inv r D G X Base D X V G u D inv r D G X W X V
44 22 41 32 43 syl3anc W LVec X V G F G X 0 ˙ u V G u D inv r D G X W X V
45 eqid + W = + W
46 eqid - W = - W
47 3 45 46 lmodvnpcan W LMod u V G u D inv r D G X W X V u - W G u D inv r D G X W X + W G u D inv r D G X W X = u
48 22 23 44 47 syl3anc W LVec X V G F G X 0 ˙ u V u - W G u D inv r D G X W X + W G u D inv r D G X W X = u
49 11 lsssssubg W LMod LSubSp W SubGrp W
50 22 49 syl W LVec X V G F G X 0 ˙ u V LSubSp W SubGrp W
51 13 adantr W LVec X V G F G X 0 ˙ u V K G LSubSp W
52 50 51 sseldd W LVec X V G F G X 0 ˙ u V K G SubGrp W
53 16 adantr W LVec X V G F G X 0 ˙ u V N X LSubSp W
54 50 53 sseldd W LVec X V G F G X 0 ˙ u V N X SubGrp W
55 3 46 lmodvsubcl W LMod u V G u D inv r D G X W X V u - W G u D inv r D G X W X V
56 22 23 44 55 syl3anc W LVec X V G F G X 0 ˙ u V u - W G u D inv r D G X W X V
57 eqid - D = - D
58 1 57 3 46 6 lflsub W LMod G F u V G u D inv r D G X W X V G u - W G u D inv r D G X W X = G u - D G G u D inv r D G X W X
59 22 26 23 44 58 syl112anc W LVec X V G F G X 0 ˙ u V G u - W G u D inv r D G X W X = G u - D G G u D inv r D G X W X
60 1 27 39 3 42 6 lflmul W LMod G F G u D inv r D G X Base D X V G G u D inv r D G X W X = G u D inv r D G X D G X
61 22 26 41 32 60 syl112anc W LVec X V G F G X 0 ˙ u V G G u D inv r D G X W X = G u D inv r D G X D G X
62 27 39 ringass D Ring G u Base D inv r D G X Base D G X Base D G u D inv r D G X D G X = G u D inv r D G X D G X
63 25 29 38 34 62 syl13anc W LVec X V G F G X 0 ˙ u V G u D inv r D G X D G X = G u D inv r D G X D G X
64 eqid 1 D = 1 D
65 27 2 39 64 36 drnginvrl D DivRing G X Base D G X 0 ˙ inv r D G X D G X = 1 D
66 31 34 35 65 syl3anc W LVec X V G F G X 0 ˙ u V inv r D G X D G X = 1 D
67 66 oveq2d W LVec X V G F G X 0 ˙ u V G u D inv r D G X D G X = G u D 1 D
68 27 39 64 ringridm D Ring G u Base D G u D 1 D = G u
69 25 29 68 syl2anc W LVec X V G F G X 0 ˙ u V G u D 1 D = G u
70 67 69 eqtrd W LVec X V G F G X 0 ˙ u V G u D inv r D G X D G X = G u
71 61 63 70 3eqtrd W LVec X V G F G X 0 ˙ u V G G u D inv r D G X W X = G u
72 71 oveq2d W LVec X V G F G X 0 ˙ u V G u - D G G u D inv r D G X W X = G u - D G u
73 1 lmodfgrp W LMod D Grp
74 22 73 syl W LVec X V G F G X 0 ˙ u V D Grp
75 27 2 57 grpsubid D Grp G u Base D G u - D G u = 0 ˙
76 74 29 75 syl2anc W LVec X V G F G X 0 ˙ u V G u - D G u = 0 ˙
77 59 72 76 3eqtrd W LVec X V G F G X 0 ˙ u V G u - W G u D inv r D G X W X = 0 ˙
78 3 1 2 6 7 ellkr W LVec G F u - W G u D inv r D G X W X K G u - W G u D inv r D G X W X V G u - W G u D inv r D G X W X = 0 ˙
79 21 26 78 syl2anc W LVec X V G F G X 0 ˙ u V u - W G u D inv r D G X W X K G u - W G u D inv r D G X W X V G u - W G u D inv r D G X W X = 0 ˙
80 56 77 79 mpbir2and W LVec X V G F G X 0 ˙ u V u - W G u D inv r D G X W X K G
81 3 42 1 27 4 22 41 32 lspsneli W LVec X V G F G X 0 ˙ u V G u D inv r D G X W X N X
82 45 5 lsmelvali K G SubGrp W N X SubGrp W u - W G u D inv r D G X W X K G G u D inv r D G X W X N X u - W G u D inv r D G X W X + W G u D inv r D G X W X K G ˙ N X
83 52 54 80 81 82 syl22anc W LVec X V G F G X 0 ˙ u V u - W G u D inv r D G X W X + W G u D inv r D G X W X K G ˙ N X
84 48 83 eqeltrrd W LVec X V G F G X 0 ˙ u V u K G ˙ N X
85 20 84 eqelssd W LVec X V G F G X 0 ˙ K G ˙ N X = V