Step |
Hyp |
Ref |
Expression |
1 |
|
lsslsp.x |
|- X = ( W |`s U ) |
2 |
|
lsslsp.m |
|- M = ( LSpan ` W ) |
3 |
|
lsslsp.n |
|- N = ( LSpan ` X ) |
4 |
|
lsslsp.l |
|- L = ( LSubSp ` W ) |
5 |
|
simp1 |
|- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> W e. LMod ) |
6 |
1 4
|
lsslmod |
|- ( ( W e. LMod /\ U e. L ) -> X e. LMod ) |
7 |
6
|
3adant3 |
|- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> X e. LMod ) |
8 |
|
simp3 |
|- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> G C_ U ) |
9 |
|
eqid |
|- ( Base ` W ) = ( Base ` W ) |
10 |
9 4
|
lssss |
|- ( U e. L -> U C_ ( Base ` W ) ) |
11 |
10
|
3ad2ant2 |
|- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> U C_ ( Base ` W ) ) |
12 |
1 9
|
ressbas2 |
|- ( U C_ ( Base ` W ) -> U = ( Base ` X ) ) |
13 |
11 12
|
syl |
|- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> U = ( Base ` X ) ) |
14 |
8 13
|
sseqtrd |
|- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> G C_ ( Base ` X ) ) |
15 |
|
eqid |
|- ( Base ` X ) = ( Base ` X ) |
16 |
|
eqid |
|- ( LSubSp ` X ) = ( LSubSp ` X ) |
17 |
15 16 3
|
lspcl |
|- ( ( X e. LMod /\ G C_ ( Base ` X ) ) -> ( N ` G ) e. ( LSubSp ` X ) ) |
18 |
7 14 17
|
syl2anc |
|- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> ( N ` G ) e. ( LSubSp ` X ) ) |
19 |
1 4 16
|
lsslss |
|- ( ( W e. LMod /\ U e. L ) -> ( ( N ` G ) e. ( LSubSp ` X ) <-> ( ( N ` G ) e. L /\ ( N ` G ) C_ U ) ) ) |
20 |
19
|
3adant3 |
|- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> ( ( N ` G ) e. ( LSubSp ` X ) <-> ( ( N ` G ) e. L /\ ( N ` G ) C_ U ) ) ) |
21 |
18 20
|
mpbid |
|- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> ( ( N ` G ) e. L /\ ( N ` G ) C_ U ) ) |
22 |
21
|
simpld |
|- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> ( N ` G ) e. L ) |
23 |
15 3
|
lspssid |
|- ( ( X e. LMod /\ G C_ ( Base ` X ) ) -> G C_ ( N ` G ) ) |
24 |
7 14 23
|
syl2anc |
|- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> G C_ ( N ` G ) ) |
25 |
4 2
|
lspssp |
|- ( ( W e. LMod /\ ( N ` G ) e. L /\ G C_ ( N ` G ) ) -> ( M ` G ) C_ ( N ` G ) ) |
26 |
5 22 24 25
|
syl3anc |
|- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> ( M ` G ) C_ ( N ` G ) ) |
27 |
8 11
|
sstrd |
|- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> G C_ ( Base ` W ) ) |
28 |
9 4 2
|
lspcl |
|- ( ( W e. LMod /\ G C_ ( Base ` W ) ) -> ( M ` G ) e. L ) |
29 |
5 27 28
|
syl2anc |
|- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> ( M ` G ) e. L ) |
30 |
4 2
|
lspssp |
|- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> ( M ` G ) C_ U ) |
31 |
1 4 16
|
lsslss |
|- ( ( W e. LMod /\ U e. L ) -> ( ( M ` G ) e. ( LSubSp ` X ) <-> ( ( M ` G ) e. L /\ ( M ` G ) C_ U ) ) ) |
32 |
31
|
3adant3 |
|- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> ( ( M ` G ) e. ( LSubSp ` X ) <-> ( ( M ` G ) e. L /\ ( M ` G ) C_ U ) ) ) |
33 |
29 30 32
|
mpbir2and |
|- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> ( M ` G ) e. ( LSubSp ` X ) ) |
34 |
9 2
|
lspssid |
|- ( ( W e. LMod /\ G C_ ( Base ` W ) ) -> G C_ ( M ` G ) ) |
35 |
5 27 34
|
syl2anc |
|- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> G C_ ( M ` G ) ) |
36 |
16 3
|
lspssp |
|- ( ( X e. LMod /\ ( M ` G ) e. ( LSubSp ` X ) /\ G C_ ( M ` G ) ) -> ( N ` G ) C_ ( M ` G ) ) |
37 |
7 33 35 36
|
syl3anc |
|- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> ( N ` G ) C_ ( M ` G ) ) |
38 |
26 37
|
eqssd |
|- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> ( M ` G ) = ( N ` G ) ) |