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 4
|
lspvadd |
|- ( ( W e. LMod /\ w e. V /\ Y e. V ) -> ( N ` { ( w .+ Y ) } ) C_ ( N ` { w , Y } ) ) |
18 |
14 15 16 17
|
syl3anc |
|- ( ph -> ( N ` { ( w .+ Y ) } ) C_ ( N ` { w , Y } ) ) |
19 |
1 3 4 5 6 16 15 11 12
|
lspindp1 |
|- ( ph -> ( ( N ` { w } ) =/= ( N ` { Y } ) /\ -. X e. ( N ` { w , Y } ) ) ) |
20 |
19
|
simprd |
|- ( ph -> -. X e. ( N ` { w , Y } ) ) |
21 |
18 20
|
ssneldd |
|- ( ph -> -. X e. ( N ` { ( w .+ Y ) } ) ) |
22 |
6
|
eldifad |
|- ( ph -> X e. V ) |
23 |
1 4
|
lspsnid |
|- ( ( W e. LMod /\ X e. V ) -> X e. ( N ` { X } ) ) |
24 |
14 22 23
|
syl2anc |
|- ( ph -> X e. ( N ` { X } ) ) |
25 |
|
eleq2 |
|- ( ( N ` { X } ) = ( N ` { ( w .+ Y ) } ) -> ( X e. ( N ` { X } ) <-> X e. ( N ` { ( w .+ Y ) } ) ) ) |
26 |
24 25
|
syl5ibcom |
|- ( ph -> ( ( N ` { X } ) = ( N ` { ( w .+ Y ) } ) -> X e. ( N ` { ( w .+ Y ) } ) ) ) |
27 |
26
|
necon3bd |
|- ( ph -> ( -. X e. ( N ` { ( w .+ Y ) } ) -> ( N ` { X } ) =/= ( N ` { ( w .+ Y ) } ) ) ) |
28 |
21 27
|
mpd |
|- ( ph -> ( N ` { X } ) =/= ( N ` { ( w .+ Y ) } ) ) |