Metamath Proof Explorer


Theorem lclkrlem2e

Description: Lemma for lclkr . The kernel of the sum is closed when the kernels of the summands are equal and closed. (Contributed by NM, 17-Jan-2015)

Ref Expression
Hypotheses lclkrlem2e.h H = LHyp K
lclkrlem2e.o ˙ = ocH K W
lclkrlem2e.u U = DVecH K W
lclkrlem2e.v V = Base U
lclkrlem2e.z 0 ˙ = 0 U
lclkrlem2e.f F = LFnl U
lclkrlem2e.l L = LKer U
lclkrlem2e.d D = LDual U
lclkrlem2e.p + ˙ = + D
lclkrlem2e.k φ K HL W H
lclkrlem2e.x φ X V 0 ˙
lclkrlem2e.e φ E F
lclkrlem2e.g φ G F
lclkrlem2e.le φ L E = ˙ X
lclkrlem2e.ne φ L E = L G
Assertion lclkrlem2e φ ˙ ˙ L E + ˙ G = L E + ˙ G

Proof

Step Hyp Ref Expression
1 lclkrlem2e.h H = LHyp K
2 lclkrlem2e.o ˙ = ocH K W
3 lclkrlem2e.u U = DVecH K W
4 lclkrlem2e.v V = Base U
5 lclkrlem2e.z 0 ˙ = 0 U
6 lclkrlem2e.f F = LFnl U
7 lclkrlem2e.l L = LKer U
8 lclkrlem2e.d D = LDual U
9 lclkrlem2e.p + ˙ = + D
10 lclkrlem2e.k φ K HL W H
11 lclkrlem2e.x φ X V 0 ˙
12 lclkrlem2e.e φ E F
13 lclkrlem2e.g φ G F
14 lclkrlem2e.le φ L E = ˙ X
15 lclkrlem2e.ne φ L E = L G
16 10 adantr φ L E + ˙ G LSHyp U K HL W H
17 11 eldifad φ X V
18 17 snssd φ X V
19 18 adantr φ L E + ˙ G LSHyp U X V
20 eqid DIsoH K W = DIsoH K W
21 1 20 3 4 2 dochcl K HL W H X V ˙ X ran DIsoH K W
22 16 19 21 syl2anc φ L E + ˙ G LSHyp U ˙ X ran DIsoH K W
23 1 20 2 dochoc K HL W H ˙ X ran DIsoH K W ˙ ˙ ˙ X = ˙ X
24 16 22 23 syl2anc φ L E + ˙ G LSHyp U ˙ ˙ ˙ X = ˙ X
25 14 adantr φ L E + ˙ G LSHyp U L E = ˙ X
26 inidm L E L E = L E
27 15 ineq2d φ L E L E = L E L G
28 26 27 eqtr3id φ L E = L E L G
29 1 3 10 dvhlmod φ U LMod
30 6 7 8 9 29 12 13 lkrin φ L E L G L E + ˙ G
31 28 30 eqsstrd φ L E L E + ˙ G
32 31 adantr φ L E + ˙ G LSHyp U L E L E + ˙ G
33 eqid LSHyp U = LSHyp U
34 1 3 10 dvhlvec φ U LVec
35 34 adantr φ L E + ˙ G LSHyp U U LVec
36 eqid LSpan U = LSpan U
37 1 3 2 4 36 10 18 dochocsp φ ˙ LSpan U X = ˙ X
38 37 adantr φ L E + ˙ G LSHyp U ˙ LSpan U X = ˙ X
39 25 38 eqtr4d φ L E + ˙ G LSHyp U L E = ˙ LSpan U X
40 eqid LSAtoms U = LSAtoms U
41 4 36 5 40 29 11 lsatlspsn φ LSpan U X LSAtoms U
42 41 adantr φ L E + ˙ G LSHyp U LSpan U X LSAtoms U
43 1 3 2 40 33 16 42 dochsatshp φ L E + ˙ G LSHyp U ˙ LSpan U X LSHyp U
44 39 43 eqeltrd φ L E + ˙ G LSHyp U L E LSHyp U
45 simpr φ L E + ˙ G LSHyp U L E + ˙ G LSHyp U
46 33 35 44 45 lshpcmp φ L E + ˙ G LSHyp U L E L E + ˙ G L E = L E + ˙ G
47 32 46 mpbid φ L E + ˙ G LSHyp U L E = L E + ˙ G
48 25 47 eqtr3d φ L E + ˙ G LSHyp U ˙ X = L E + ˙ G
49 48 fveq2d φ L E + ˙ G LSHyp U ˙ ˙ X = ˙ L E + ˙ G
50 49 fveq2d φ L E + ˙ G LSHyp U ˙ ˙ ˙ X = ˙ ˙ L E + ˙ G
51 24 50 48 3eqtr3d φ L E + ˙ G LSHyp U ˙ ˙ L E + ˙ G = L E + ˙ G
52 51 ex φ L E + ˙ G LSHyp U ˙ ˙ L E + ˙ G = L E + ˙ G
53 1 3 2 4 10 dochoc1 φ ˙ ˙ V = V
54 2fveq3 L E + ˙ G = V ˙ ˙ L E + ˙ G = ˙ ˙ V
55 id L E + ˙ G = V L E + ˙ G = V
56 54 55 eqeq12d L E + ˙ G = V ˙ ˙ L E + ˙ G = L E + ˙ G ˙ ˙ V = V
57 53 56 syl5ibrcom φ L E + ˙ G = V ˙ ˙ L E + ˙ G = L E + ˙ G
58 6 8 9 29 12 13 ldualvaddcl φ E + ˙ G F
59 4 33 6 7 34 58 lkrshpor φ L E + ˙ G LSHyp U L E + ˙ G = V
60 52 57 59 mpjaod φ ˙ ˙ L E + ˙ G = L E + ˙ G