Step |
Hyp |
Ref |
Expression |
1 |
|
lspun0.v |
|- V = ( Base ` W ) |
2 |
|
lspun0.o |
|- .0. = ( 0g ` W ) |
3 |
|
lspun0.n |
|- N = ( LSpan ` W ) |
4 |
|
lspun0.w |
|- ( ph -> W e. LMod ) |
5 |
|
lspun0.x |
|- ( ph -> X C_ V ) |
6 |
1 2
|
lmod0vcl |
|- ( W e. LMod -> .0. e. V ) |
7 |
4 6
|
syl |
|- ( ph -> .0. e. V ) |
8 |
7
|
snssd |
|- ( ph -> { .0. } C_ V ) |
9 |
1 3
|
lspun |
|- ( ( W e. LMod /\ X C_ V /\ { .0. } C_ V ) -> ( N ` ( X u. { .0. } ) ) = ( N ` ( ( N ` X ) u. ( N ` { .0. } ) ) ) ) |
10 |
4 5 8 9
|
syl3anc |
|- ( ph -> ( N ` ( X u. { .0. } ) ) = ( N ` ( ( N ` X ) u. ( N ` { .0. } ) ) ) ) |
11 |
2 3
|
lspsn0 |
|- ( W e. LMod -> ( N ` { .0. } ) = { .0. } ) |
12 |
4 11
|
syl |
|- ( ph -> ( N ` { .0. } ) = { .0. } ) |
13 |
12
|
uneq2d |
|- ( ph -> ( ( N ` X ) u. ( N ` { .0. } ) ) = ( ( N ` X ) u. { .0. } ) ) |
14 |
|
eqid |
|- ( LSubSp ` W ) = ( LSubSp ` W ) |
15 |
1 14 3
|
lspcl |
|- ( ( W e. LMod /\ X C_ V ) -> ( N ` X ) e. ( LSubSp ` W ) ) |
16 |
4 5 15
|
syl2anc |
|- ( ph -> ( N ` X ) e. ( LSubSp ` W ) ) |
17 |
2 14
|
lss0ss |
|- ( ( W e. LMod /\ ( N ` X ) e. ( LSubSp ` W ) ) -> { .0. } C_ ( N ` X ) ) |
18 |
4 16 17
|
syl2anc |
|- ( ph -> { .0. } C_ ( N ` X ) ) |
19 |
|
ssequn2 |
|- ( { .0. } C_ ( N ` X ) <-> ( ( N ` X ) u. { .0. } ) = ( N ` X ) ) |
20 |
18 19
|
sylib |
|- ( ph -> ( ( N ` X ) u. { .0. } ) = ( N ` X ) ) |
21 |
13 20
|
eqtrd |
|- ( ph -> ( ( N ` X ) u. ( N ` { .0. } ) ) = ( N ` X ) ) |
22 |
21
|
fveq2d |
|- ( ph -> ( N ` ( ( N ` X ) u. ( N ` { .0. } ) ) ) = ( N ` ( N ` X ) ) ) |
23 |
1 3
|
lspidm |
|- ( ( W e. LMod /\ X C_ V ) -> ( N ` ( N ` X ) ) = ( N ` X ) ) |
24 |
4 5 23
|
syl2anc |
|- ( ph -> ( N ` ( N ` X ) ) = ( N ` X ) ) |
25 |
22 24
|
eqtrd |
|- ( ph -> ( N ` ( ( N ` X ) u. ( N ` { .0. } ) ) ) = ( N ` X ) ) |
26 |
10 25
|
eqtrd |
|- ( ph -> ( N ` ( X u. { .0. } ) ) = ( N ` X ) ) |