Step |
Hyp |
Ref |
Expression |
1 |
|
lkrss2.s |
|- S = ( Scalar ` W ) |
2 |
|
lkrss2.r |
|- R = ( Base ` S ) |
3 |
|
lkrss2.f |
|- F = ( LFnl ` W ) |
4 |
|
lkrss2.k |
|- K = ( LKer ` W ) |
5 |
|
lkrss2.d |
|- D = ( LDual ` W ) |
6 |
|
lkrss2.t |
|- .x. = ( .s ` D ) |
7 |
|
lkrss2.w |
|- ( ph -> W e. LVec ) |
8 |
|
lkrss2.g |
|- ( ph -> G e. F ) |
9 |
|
lkrss2.h |
|- ( ph -> H e. F ) |
10 |
|
sspss |
|- ( ( K ` G ) C_ ( K ` H ) <-> ( ( K ` G ) C. ( K ` H ) \/ ( K ` G ) = ( K ` H ) ) ) |
11 |
|
eqid |
|- ( 0g ` D ) = ( 0g ` D ) |
12 |
3 4 5 11 7 8 9
|
lkrpssN |
|- ( ph -> ( ( K ` G ) C. ( K ` H ) <-> ( G =/= ( 0g ` D ) /\ H = ( 0g ` D ) ) ) ) |
13 |
|
lveclmod |
|- ( W e. LVec -> W e. LMod ) |
14 |
7 13
|
syl |
|- ( ph -> W e. LMod ) |
15 |
|
eqid |
|- ( 0g ` S ) = ( 0g ` S ) |
16 |
1 2 15
|
lmod0cl |
|- ( W e. LMod -> ( 0g ` S ) e. R ) |
17 |
14 16
|
syl |
|- ( ph -> ( 0g ` S ) e. R ) |
18 |
17
|
adantr |
|- ( ( ph /\ H = ( 0g ` D ) ) -> ( 0g ` S ) e. R ) |
19 |
|
simpr |
|- ( ( ph /\ H = ( 0g ` D ) ) -> H = ( 0g ` D ) ) |
20 |
3 1 15 5 6 11 14 8
|
ldual0vs |
|- ( ph -> ( ( 0g ` S ) .x. G ) = ( 0g ` D ) ) |
21 |
20
|
adantr |
|- ( ( ph /\ H = ( 0g ` D ) ) -> ( ( 0g ` S ) .x. G ) = ( 0g ` D ) ) |
22 |
19 21
|
eqtr4d |
|- ( ( ph /\ H = ( 0g ` D ) ) -> H = ( ( 0g ` S ) .x. G ) ) |
23 |
|
oveq1 |
|- ( r = ( 0g ` S ) -> ( r .x. G ) = ( ( 0g ` S ) .x. G ) ) |
24 |
23
|
rspceeqv |
|- ( ( ( 0g ` S ) e. R /\ H = ( ( 0g ` S ) .x. G ) ) -> E. r e. R H = ( r .x. G ) ) |
25 |
18 22 24
|
syl2anc |
|- ( ( ph /\ H = ( 0g ` D ) ) -> E. r e. R H = ( r .x. G ) ) |
26 |
25
|
ex |
|- ( ph -> ( H = ( 0g ` D ) -> E. r e. R H = ( r .x. G ) ) ) |
27 |
26
|
adantld |
|- ( ph -> ( ( G =/= ( 0g ` D ) /\ H = ( 0g ` D ) ) -> E. r e. R H = ( r .x. G ) ) ) |
28 |
12 27
|
sylbid |
|- ( ph -> ( ( K ` G ) C. ( K ` H ) -> E. r e. R H = ( r .x. G ) ) ) |
29 |
28
|
imp |
|- ( ( ph /\ ( K ` G ) C. ( K ` H ) ) -> E. r e. R H = ( r .x. G ) ) |
30 |
7
|
adantr |
|- ( ( ph /\ ( K ` G ) = ( K ` H ) ) -> W e. LVec ) |
31 |
8
|
adantr |
|- ( ( ph /\ ( K ` G ) = ( K ` H ) ) -> G e. F ) |
32 |
9
|
adantr |
|- ( ( ph /\ ( K ` G ) = ( K ` H ) ) -> H e. F ) |
33 |
|
simpr |
|- ( ( ph /\ ( K ` G ) = ( K ` H ) ) -> ( K ` G ) = ( K ` H ) ) |
34 |
1 2 3 4 5 6 30 31 32 33
|
eqlkr4 |
|- ( ( ph /\ ( K ` G ) = ( K ` H ) ) -> E. r e. R H = ( r .x. G ) ) |
35 |
29 34
|
jaodan |
|- ( ( ph /\ ( ( K ` G ) C. ( K ` H ) \/ ( K ` G ) = ( K ` H ) ) ) -> E. r e. R H = ( r .x. G ) ) |
36 |
10 35
|
sylan2b |
|- ( ( ph /\ ( K ` G ) C_ ( K ` H ) ) -> E. r e. R H = ( r .x. G ) ) |
37 |
7
|
adantr |
|- ( ( ph /\ r e. R ) -> W e. LVec ) |
38 |
8
|
adantr |
|- ( ( ph /\ r e. R ) -> G e. F ) |
39 |
|
simpr |
|- ( ( ph /\ r e. R ) -> r e. R ) |
40 |
1 2 3 4 5 6 37 38 39
|
lkrss |
|- ( ( ph /\ r e. R ) -> ( K ` G ) C_ ( K ` ( r .x. G ) ) ) |
41 |
40
|
ex |
|- ( ph -> ( r e. R -> ( K ` G ) C_ ( K ` ( r .x. G ) ) ) ) |
42 |
|
fveq2 |
|- ( H = ( r .x. G ) -> ( K ` H ) = ( K ` ( r .x. G ) ) ) |
43 |
42
|
sseq2d |
|- ( H = ( r .x. G ) -> ( ( K ` G ) C_ ( K ` H ) <-> ( K ` G ) C_ ( K ` ( r .x. G ) ) ) ) |
44 |
43
|
biimprcd |
|- ( ( K ` G ) C_ ( K ` ( r .x. G ) ) -> ( H = ( r .x. G ) -> ( K ` G ) C_ ( K ` H ) ) ) |
45 |
41 44
|
syl6 |
|- ( ph -> ( r e. R -> ( H = ( r .x. G ) -> ( K ` G ) C_ ( K ` H ) ) ) ) |
46 |
45
|
rexlimdv |
|- ( ph -> ( E. r e. R H = ( r .x. G ) -> ( K ` G ) C_ ( K ` H ) ) ) |
47 |
46
|
imp |
|- ( ( ph /\ E. r e. R H = ( r .x. G ) ) -> ( K ` G ) C_ ( K ` H ) ) |
48 |
36 47
|
impbida |
|- ( ph -> ( ( K ` G ) C_ ( K ` H ) <-> E. r e. R H = ( r .x. G ) ) ) |