Step |
Hyp |
Ref |
Expression |
1 |
|
mapdindp1.v |
|- V = ( Base ` W ) |
2 |
|
mapdindp1.p |
|- .+ = ( +g ` W ) |
3 |
|
mapdindp1.o |
|- .0. = ( 0g ` W ) |
4 |
|
mapdindp1.n |
|- N = ( LSpan ` W ) |
5 |
|
mapdindp1.w |
|- ( ph -> W e. LVec ) |
6 |
|
mapdindp1.x |
|- ( ph -> X e. ( V \ { .0. } ) ) |
7 |
|
mapdindp1.y |
|- ( ph -> Y e. ( V \ { .0. } ) ) |
8 |
|
mapdindp1.z |
|- ( ph -> Z e. ( V \ { .0. } ) ) |
9 |
|
mapdindp1.W |
|- ( ph -> w e. ( V \ { .0. } ) ) |
10 |
|
mapdindp1.e |
|- ( ph -> ( N ` { Y } ) = ( N ` { Z } ) ) |
11 |
|
mapdindp1.ne |
|- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) ) |
12 |
|
mapdindp1.f |
|- ( ph -> -. w e. ( N ` { X , Y } ) ) |
13 |
|
preq2 |
|- ( ( Y .+ Z ) = .0. -> { X , ( Y .+ Z ) } = { X , .0. } ) |
14 |
13
|
fveq2d |
|- ( ( Y .+ Z ) = .0. -> ( N ` { X , ( Y .+ Z ) } ) = ( N ` { X , .0. } ) ) |
15 |
|
lveclmod |
|- ( W e. LVec -> W e. LMod ) |
16 |
5 15
|
syl |
|- ( ph -> W e. LMod ) |
17 |
6
|
eldifad |
|- ( ph -> X e. V ) |
18 |
1 3 4 16 17
|
lsppr0 |
|- ( ph -> ( N ` { X , .0. } ) = ( N ` { X } ) ) |
19 |
14 18
|
sylan9eqr |
|- ( ( ph /\ ( Y .+ Z ) = .0. ) -> ( N ` { X , ( Y .+ Z ) } ) = ( N ` { X } ) ) |
20 |
7
|
eldifad |
|- ( ph -> Y e. V ) |
21 |
|
prssi |
|- ( ( X e. V /\ Y e. V ) -> { X , Y } C_ V ) |
22 |
17 20 21
|
syl2anc |
|- ( ph -> { X , Y } C_ V ) |
23 |
|
snsspr1 |
|- { X } C_ { X , Y } |
24 |
23
|
a1i |
|- ( ph -> { X } C_ { X , Y } ) |
25 |
1 4
|
lspss |
|- ( ( W e. LMod /\ { X , Y } C_ V /\ { X } C_ { X , Y } ) -> ( N ` { X } ) C_ ( N ` { X , Y } ) ) |
26 |
16 22 24 25
|
syl3anc |
|- ( ph -> ( N ` { X } ) C_ ( N ` { X , Y } ) ) |
27 |
26
|
adantr |
|- ( ( ph /\ ( Y .+ Z ) = .0. ) -> ( N ` { X } ) C_ ( N ` { X , Y } ) ) |
28 |
19 27
|
eqsstrd |
|- ( ( ph /\ ( Y .+ Z ) = .0. ) -> ( N ` { X , ( Y .+ Z ) } ) C_ ( N ` { X , Y } ) ) |
29 |
12
|
adantr |
|- ( ( ph /\ ( Y .+ Z ) = .0. ) -> -. w e. ( N ` { X , Y } ) ) |
30 |
28 29
|
ssneldd |
|- ( ( ph /\ ( Y .+ Z ) = .0. ) -> -. w e. ( N ` { X , ( Y .+ Z ) } ) ) |
31 |
12
|
adantr |
|- ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> -. w e. ( N ` { X , Y } ) ) |
32 |
5
|
adantr |
|- ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> W e. LVec ) |
33 |
6
|
adantr |
|- ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> X e. ( V \ { .0. } ) ) |
34 |
7
|
adantr |
|- ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> Y e. ( V \ { .0. } ) ) |
35 |
8
|
adantr |
|- ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> Z e. ( V \ { .0. } ) ) |
36 |
9
|
adantr |
|- ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> w e. ( V \ { .0. } ) ) |
37 |
10
|
adantr |
|- ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> ( N ` { Y } ) = ( N ` { Z } ) ) |
38 |
11
|
adantr |
|- ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> ( N ` { X } ) =/= ( N ` { Y } ) ) |
39 |
|
simpr |
|- ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> ( Y .+ Z ) =/= .0. ) |
40 |
1 2 3 4 32 33 34 35 36 37 38 31 39
|
mapdindp0 |
|- ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> ( N ` { ( Y .+ Z ) } ) = ( N ` { Y } ) ) |
41 |
40
|
oveq2d |
|- ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> ( ( N ` { X } ) ( LSSum ` W ) ( N ` { ( Y .+ Z ) } ) ) = ( ( N ` { X } ) ( LSSum ` W ) ( N ` { Y } ) ) ) |
42 |
|
eqid |
|- ( LSSum ` W ) = ( LSSum ` W ) |
43 |
8
|
eldifad |
|- ( ph -> Z e. V ) |
44 |
1 2
|
lmodvacl |
|- ( ( W e. LMod /\ Y e. V /\ Z e. V ) -> ( Y .+ Z ) e. V ) |
45 |
16 20 43 44
|
syl3anc |
|- ( ph -> ( Y .+ Z ) e. V ) |
46 |
1 4 42 16 17 45
|
lsmpr |
|- ( ph -> ( N ` { X , ( Y .+ Z ) } ) = ( ( N ` { X } ) ( LSSum ` W ) ( N ` { ( Y .+ Z ) } ) ) ) |
47 |
46
|
adantr |
|- ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> ( N ` { X , ( Y .+ Z ) } ) = ( ( N ` { X } ) ( LSSum ` W ) ( N ` { ( Y .+ Z ) } ) ) ) |
48 |
1 4 42 16 17 20
|
lsmpr |
|- ( ph -> ( N ` { X , Y } ) = ( ( N ` { X } ) ( LSSum ` W ) ( N ` { Y } ) ) ) |
49 |
48
|
adantr |
|- ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> ( N ` { X , Y } ) = ( ( N ` { X } ) ( LSSum ` W ) ( N ` { Y } ) ) ) |
50 |
41 47 49
|
3eqtr4d |
|- ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> ( N ` { X , ( Y .+ Z ) } ) = ( N ` { X , Y } ) ) |
51 |
31 50
|
neleqtrrd |
|- ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> -. w e. ( N ` { X , ( Y .+ Z ) } ) ) |
52 |
30 51
|
pm2.61dane |
|- ( ph -> -. w e. ( N ` { X , ( Y .+ Z ) } ) ) |