Step |
Hyp |
Ref |
Expression |
1 |
|
lspsnel5.v |
|- V = ( Base ` W ) |
2 |
|
lspsnel5.s |
|- S = ( LSubSp ` W ) |
3 |
|
lspsnel5.n |
|- N = ( LSpan ` W ) |
4 |
|
lspsnel5.w |
|- ( ph -> W e. LMod ) |
5 |
|
lspsnel5.a |
|- ( ph -> U e. S ) |
6 |
1 2
|
lssel |
|- ( ( U e. S /\ X e. U ) -> X e. V ) |
7 |
5 6
|
sylan |
|- ( ( ph /\ X e. U ) -> X e. V ) |
8 |
4
|
adantr |
|- ( ( ph /\ X e. U ) -> W e. LMod ) |
9 |
5
|
adantr |
|- ( ( ph /\ X e. U ) -> U e. S ) |
10 |
|
simpr |
|- ( ( ph /\ X e. U ) -> X e. U ) |
11 |
2 3
|
lspsnss |
|- ( ( W e. LMod /\ U e. S /\ X e. U ) -> ( N ` { X } ) C_ U ) |
12 |
8 9 10 11
|
syl3anc |
|- ( ( ph /\ X e. U ) -> ( N ` { X } ) C_ U ) |
13 |
7 12
|
jca |
|- ( ( ph /\ X e. U ) -> ( X e. V /\ ( N ` { X } ) C_ U ) ) |
14 |
1 3
|
lspsnid |
|- ( ( W e. LMod /\ X e. V ) -> X e. ( N ` { X } ) ) |
15 |
4 14
|
sylan |
|- ( ( ph /\ X e. V ) -> X e. ( N ` { X } ) ) |
16 |
|
ssel |
|- ( ( N ` { X } ) C_ U -> ( X e. ( N ` { X } ) -> X e. U ) ) |
17 |
15 16
|
syl5com |
|- ( ( ph /\ X e. V ) -> ( ( N ` { X } ) C_ U -> X e. U ) ) |
18 |
17
|
impr |
|- ( ( ph /\ ( X e. V /\ ( N ` { X } ) C_ U ) ) -> X e. U ) |
19 |
13 18
|
impbida |
|- ( ph -> ( X e. U <-> ( X e. V /\ ( N ` { X } ) C_ U ) ) ) |