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 |
|
lveclmod |
|- ( W e. LVec -> W e. LMod ) |
14 |
5 13
|
syl |
|- ( ph -> W e. LMod ) |
15 |
9
|
eldifad |
|- ( ph -> w e. V ) |
16 |
7
|
eldifad |
|- ( ph -> Y e. V ) |
17 |
1 2
|
lmodvacl |
|- ( ( W e. LMod /\ w e. V /\ Y e. V ) -> ( w .+ Y ) e. V ) |
18 |
14 15 16 17
|
syl3anc |
|- ( ph -> ( w .+ Y ) e. V ) |
19 |
6
|
eldifad |
|- ( ph -> X e. V ) |
20 |
1 4 5 15 19 16 12
|
lspindpi |
|- ( ph -> ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { Y } ) ) ) |
21 |
20
|
simprd |
|- ( ph -> ( N ` { w } ) =/= ( N ` { Y } ) ) |
22 |
21
|
necomd |
|- ( ph -> ( N ` { Y } ) =/= ( N ` { w } ) ) |
23 |
1 2 3 4 5 16 9 22
|
lspindp3 |
|- ( ph -> ( N ` { Y } ) =/= ( N ` { ( Y .+ w ) } ) ) |
24 |
1 2
|
lmodcom |
|- ( ( W e. LMod /\ w e. V /\ Y e. V ) -> ( w .+ Y ) = ( Y .+ w ) ) |
25 |
14 15 16 24
|
syl3anc |
|- ( ph -> ( w .+ Y ) = ( Y .+ w ) ) |
26 |
25
|
sneqd |
|- ( ph -> { ( w .+ Y ) } = { ( Y .+ w ) } ) |
27 |
26
|
fveq2d |
|- ( ph -> ( N ` { ( w .+ Y ) } ) = ( N ` { ( Y .+ w ) } ) ) |
28 |
23 27
|
neeqtrrd |
|- ( ph -> ( N ` { Y } ) =/= ( N ` { ( w .+ Y ) } ) ) |
29 |
10 28
|
eqnetrrd |
|- ( ph -> ( N ` { Z } ) =/= ( N ` { ( w .+ Y ) } ) ) |
30 |
1 3 4 5 6 16 15 11 12
|
lspindp1 |
|- ( ph -> ( ( N ` { w } ) =/= ( N ` { Y } ) /\ -. X e. ( N ` { w , Y } ) ) ) |
31 |
30
|
simprd |
|- ( ph -> -. X e. ( N ` { w , Y } ) ) |
32 |
|
eqid |
|- ( LSSum ` W ) = ( LSSum ` W ) |
33 |
8
|
eldifad |
|- ( ph -> Z e. V ) |
34 |
1 4 32 14 33 18
|
lsmpr |
|- ( ph -> ( N ` { Z , ( w .+ Y ) } ) = ( ( N ` { Z } ) ( LSSum ` W ) ( N ` { ( w .+ Y ) } ) ) ) |
35 |
1 2
|
lmodcom |
|- ( ( W e. LMod /\ Y e. V /\ w e. V ) -> ( Y .+ w ) = ( w .+ Y ) ) |
36 |
14 16 15 35
|
syl3anc |
|- ( ph -> ( Y .+ w ) = ( w .+ Y ) ) |
37 |
36
|
preq2d |
|- ( ph -> { Y , ( Y .+ w ) } = { Y , ( w .+ Y ) } ) |
38 |
37
|
fveq2d |
|- ( ph -> ( N ` { Y , ( Y .+ w ) } ) = ( N ` { Y , ( w .+ Y ) } ) ) |
39 |
1 2 4 14 16 15
|
lspprabs |
|- ( ph -> ( N ` { Y , ( Y .+ w ) } ) = ( N ` { Y , w } ) ) |
40 |
1 4 32 14 16 18
|
lsmpr |
|- ( ph -> ( N ` { Y , ( w .+ Y ) } ) = ( ( N ` { Y } ) ( LSSum ` W ) ( N ` { ( w .+ Y ) } ) ) ) |
41 |
38 39 40
|
3eqtr3rd |
|- ( ph -> ( ( N ` { Y } ) ( LSSum ` W ) ( N ` { ( w .+ Y ) } ) ) = ( N ` { Y , w } ) ) |
42 |
10
|
oveq1d |
|- ( ph -> ( ( N ` { Y } ) ( LSSum ` W ) ( N ` { ( w .+ Y ) } ) ) = ( ( N ` { Z } ) ( LSSum ` W ) ( N ` { ( w .+ Y ) } ) ) ) |
43 |
|
prcom |
|- { Y , w } = { w , Y } |
44 |
43
|
fveq2i |
|- ( N ` { Y , w } ) = ( N ` { w , Y } ) |
45 |
44
|
a1i |
|- ( ph -> ( N ` { Y , w } ) = ( N ` { w , Y } ) ) |
46 |
41 42 45
|
3eqtr3d |
|- ( ph -> ( ( N ` { Z } ) ( LSSum ` W ) ( N ` { ( w .+ Y ) } ) ) = ( N ` { w , Y } ) ) |
47 |
34 46
|
eqtrd |
|- ( ph -> ( N ` { Z , ( w .+ Y ) } ) = ( N ` { w , Y } ) ) |
48 |
31 47
|
neleqtrrd |
|- ( ph -> -. X e. ( N ` { Z , ( w .+ Y ) } ) ) |
49 |
1 3 4 5 8 18 19 29 48
|
lspindp1 |
|- ( ph -> ( ( N ` { X } ) =/= ( N ` { ( w .+ Y ) } ) /\ -. Z e. ( N ` { X , ( w .+ Y ) } ) ) ) |
50 |
49
|
simprd |
|- ( ph -> -. Z e. ( N ` { X , ( w .+ Y ) } ) ) |