Step |
Hyp |
Ref |
Expression |
1 |
|
lspsnneg.v |
|- V = ( Base ` W ) |
2 |
|
lspsnneg.m |
|- M = ( invg ` W ) |
3 |
|
lspsnneg.n |
|- N = ( LSpan ` W ) |
4 |
|
eqid |
|- ( Scalar ` W ) = ( Scalar ` W ) |
5 |
|
eqid |
|- ( .s ` W ) = ( .s ` W ) |
6 |
|
eqid |
|- ( 1r ` ( Scalar ` W ) ) = ( 1r ` ( Scalar ` W ) ) |
7 |
|
eqid |
|- ( invg ` ( Scalar ` W ) ) = ( invg ` ( Scalar ` W ) ) |
8 |
1 2 4 5 6 7
|
lmodvneg1 |
|- ( ( W e. LMod /\ X e. V ) -> ( ( ( invg ` ( Scalar ` W ) ) ` ( 1r ` ( Scalar ` W ) ) ) ( .s ` W ) X ) = ( M ` X ) ) |
9 |
8
|
sneqd |
|- ( ( W e. LMod /\ X e. V ) -> { ( ( ( invg ` ( Scalar ` W ) ) ` ( 1r ` ( Scalar ` W ) ) ) ( .s ` W ) X ) } = { ( M ` X ) } ) |
10 |
9
|
fveq2d |
|- ( ( W e. LMod /\ X e. V ) -> ( N ` { ( ( ( invg ` ( Scalar ` W ) ) ` ( 1r ` ( Scalar ` W ) ) ) ( .s ` W ) X ) } ) = ( N ` { ( M ` X ) } ) ) |
11 |
|
simpl |
|- ( ( W e. LMod /\ X e. V ) -> W e. LMod ) |
12 |
4
|
lmodfgrp |
|- ( W e. LMod -> ( Scalar ` W ) e. Grp ) |
13 |
|
eqid |
|- ( Base ` ( Scalar ` W ) ) = ( Base ` ( Scalar ` W ) ) |
14 |
4 13 6
|
lmod1cl |
|- ( W e. LMod -> ( 1r ` ( Scalar ` W ) ) e. ( Base ` ( Scalar ` W ) ) ) |
15 |
13 7
|
grpinvcl |
|- ( ( ( Scalar ` W ) e. Grp /\ ( 1r ` ( Scalar ` W ) ) e. ( Base ` ( Scalar ` W ) ) ) -> ( ( invg ` ( Scalar ` W ) ) ` ( 1r ` ( Scalar ` W ) ) ) e. ( Base ` ( Scalar ` W ) ) ) |
16 |
12 14 15
|
syl2anc |
|- ( W e. LMod -> ( ( invg ` ( Scalar ` W ) ) ` ( 1r ` ( Scalar ` W ) ) ) e. ( Base ` ( Scalar ` W ) ) ) |
17 |
16
|
adantr |
|- ( ( W e. LMod /\ X e. V ) -> ( ( invg ` ( Scalar ` W ) ) ` ( 1r ` ( Scalar ` W ) ) ) e. ( Base ` ( Scalar ` W ) ) ) |
18 |
|
simpr |
|- ( ( W e. LMod /\ X e. V ) -> X e. V ) |
19 |
4 13 1 5 3
|
lspsnvsi |
|- ( ( W e. LMod /\ ( ( invg ` ( Scalar ` W ) ) ` ( 1r ` ( Scalar ` W ) ) ) e. ( Base ` ( Scalar ` W ) ) /\ X e. V ) -> ( N ` { ( ( ( invg ` ( Scalar ` W ) ) ` ( 1r ` ( Scalar ` W ) ) ) ( .s ` W ) X ) } ) C_ ( N ` { X } ) ) |
20 |
11 17 18 19
|
syl3anc |
|- ( ( W e. LMod /\ X e. V ) -> ( N ` { ( ( ( invg ` ( Scalar ` W ) ) ` ( 1r ` ( Scalar ` W ) ) ) ( .s ` W ) X ) } ) C_ ( N ` { X } ) ) |
21 |
10 20
|
eqsstrrd |
|- ( ( W e. LMod /\ X e. V ) -> ( N ` { ( M ` X ) } ) C_ ( N ` { X } ) ) |
22 |
1 2
|
lmodvnegcl |
|- ( ( W e. LMod /\ X e. V ) -> ( M ` X ) e. V ) |
23 |
1 2 4 5 6 7
|
lmodvneg1 |
|- ( ( W e. LMod /\ ( M ` X ) e. V ) -> ( ( ( invg ` ( Scalar ` W ) ) ` ( 1r ` ( Scalar ` W ) ) ) ( .s ` W ) ( M ` X ) ) = ( M ` ( M ` X ) ) ) |
24 |
22 23
|
syldan |
|- ( ( W e. LMod /\ X e. V ) -> ( ( ( invg ` ( Scalar ` W ) ) ` ( 1r ` ( Scalar ` W ) ) ) ( .s ` W ) ( M ` X ) ) = ( M ` ( M ` X ) ) ) |
25 |
|
lmodgrp |
|- ( W e. LMod -> W e. Grp ) |
26 |
1 2
|
grpinvinv |
|- ( ( W e. Grp /\ X e. V ) -> ( M ` ( M ` X ) ) = X ) |
27 |
25 26
|
sylan |
|- ( ( W e. LMod /\ X e. V ) -> ( M ` ( M ` X ) ) = X ) |
28 |
24 27
|
eqtrd |
|- ( ( W e. LMod /\ X e. V ) -> ( ( ( invg ` ( Scalar ` W ) ) ` ( 1r ` ( Scalar ` W ) ) ) ( .s ` W ) ( M ` X ) ) = X ) |
29 |
28
|
sneqd |
|- ( ( W e. LMod /\ X e. V ) -> { ( ( ( invg ` ( Scalar ` W ) ) ` ( 1r ` ( Scalar ` W ) ) ) ( .s ` W ) ( M ` X ) ) } = { X } ) |
30 |
29
|
fveq2d |
|- ( ( W e. LMod /\ X e. V ) -> ( N ` { ( ( ( invg ` ( Scalar ` W ) ) ` ( 1r ` ( Scalar ` W ) ) ) ( .s ` W ) ( M ` X ) ) } ) = ( N ` { X } ) ) |
31 |
4 13 1 5 3
|
lspsnvsi |
|- ( ( W e. LMod /\ ( ( invg ` ( Scalar ` W ) ) ` ( 1r ` ( Scalar ` W ) ) ) e. ( Base ` ( Scalar ` W ) ) /\ ( M ` X ) e. V ) -> ( N ` { ( ( ( invg ` ( Scalar ` W ) ) ` ( 1r ` ( Scalar ` W ) ) ) ( .s ` W ) ( M ` X ) ) } ) C_ ( N ` { ( M ` X ) } ) ) |
32 |
11 17 22 31
|
syl3anc |
|- ( ( W e. LMod /\ X e. V ) -> ( N ` { ( ( ( invg ` ( Scalar ` W ) ) ` ( 1r ` ( Scalar ` W ) ) ) ( .s ` W ) ( M ` X ) ) } ) C_ ( N ` { ( M ` X ) } ) ) |
33 |
30 32
|
eqsstrrd |
|- ( ( W e. LMod /\ X e. V ) -> ( N ` { X } ) C_ ( N ` { ( M ` X ) } ) ) |
34 |
21 33
|
eqssd |
|- ( ( W e. LMod /\ X e. V ) -> ( N ` { ( M ` X ) } ) = ( N ` { X } ) ) |