Metamath Proof Explorer


Theorem lshpset2N

Description: The set of all hyperplanes of a left module or left vector space equals the set of all kernels of nonzero functionals. (Contributed by NM, 17-Jul-2014) (New usage is discouraged.)

Ref Expression
Hypotheses lshpset2.v
|- V = ( Base ` W )
lshpset2.d
|- D = ( Scalar ` W )
lshpset2.z
|- .0. = ( 0g ` D )
lshpset2.h
|- H = ( LSHyp ` W )
lshpset2.f
|- F = ( LFnl ` W )
lshpset2.k
|- K = ( LKer ` W )
Assertion lshpset2N
|- ( W e. LVec -> H = { s | E. g e. F ( g =/= ( V X. { .0. } ) /\ s = ( K ` g ) ) } )

Proof

Step Hyp Ref Expression
1 lshpset2.v
 |-  V = ( Base ` W )
2 lshpset2.d
 |-  D = ( Scalar ` W )
3 lshpset2.z
 |-  .0. = ( 0g ` D )
4 lshpset2.h
 |-  H = ( LSHyp ` W )
5 lshpset2.f
 |-  F = ( LFnl ` W )
6 lshpset2.k
 |-  K = ( LKer ` W )
7 4 5 6 lshpkrex
 |-  ( ( W e. LVec /\ s e. H ) -> E. g e. F ( K ` g ) = s )
8 eleq1
 |-  ( ( K ` g ) = s -> ( ( K ` g ) e. H <-> s e. H ) )
9 8 biimparc
 |-  ( ( s e. H /\ ( K ` g ) = s ) -> ( K ` g ) e. H )
10 9 adantll
 |-  ( ( ( W e. LVec /\ s e. H ) /\ ( K ` g ) = s ) -> ( K ` g ) e. H )
11 10 adantlr
 |-  ( ( ( ( W e. LVec /\ s e. H ) /\ g e. F ) /\ ( K ` g ) = s ) -> ( K ` g ) e. H )
12 simplll
 |-  ( ( ( ( W e. LVec /\ s e. H ) /\ g e. F ) /\ ( K ` g ) = s ) -> W e. LVec )
13 simplr
 |-  ( ( ( ( W e. LVec /\ s e. H ) /\ g e. F ) /\ ( K ` g ) = s ) -> g e. F )
14 1 2 3 4 5 6 12 13 lkrshp3
 |-  ( ( ( ( W e. LVec /\ s e. H ) /\ g e. F ) /\ ( K ` g ) = s ) -> ( ( K ` g ) e. H <-> g =/= ( V X. { .0. } ) ) )
15 11 14 mpbid
 |-  ( ( ( ( W e. LVec /\ s e. H ) /\ g e. F ) /\ ( K ` g ) = s ) -> g =/= ( V X. { .0. } ) )
16 15 ex
 |-  ( ( ( W e. LVec /\ s e. H ) /\ g e. F ) -> ( ( K ` g ) = s -> g =/= ( V X. { .0. } ) ) )
17 eqimss2
 |-  ( ( K ` g ) = s -> s C_ ( K ` g ) )
18 eqimss
 |-  ( ( K ` g ) = s -> ( K ` g ) C_ s )
19 17 18 eqssd
 |-  ( ( K ` g ) = s -> s = ( K ` g ) )
20 19 a1i
 |-  ( ( ( W e. LVec /\ s e. H ) /\ g e. F ) -> ( ( K ` g ) = s -> s = ( K ` g ) ) )
21 16 20 jcad
 |-  ( ( ( W e. LVec /\ s e. H ) /\ g e. F ) -> ( ( K ` g ) = s -> ( g =/= ( V X. { .0. } ) /\ s = ( K ` g ) ) ) )
22 21 reximdva
 |-  ( ( W e. LVec /\ s e. H ) -> ( E. g e. F ( K ` g ) = s -> E. g e. F ( g =/= ( V X. { .0. } ) /\ s = ( K ` g ) ) ) )
23 7 22 mpd
 |-  ( ( W e. LVec /\ s e. H ) -> E. g e. F ( g =/= ( V X. { .0. } ) /\ s = ( K ` g ) ) )
24 23 ex
 |-  ( W e. LVec -> ( s e. H -> E. g e. F ( g =/= ( V X. { .0. } ) /\ s = ( K ` g ) ) ) )
25 1 2 3 4 5 6 lkrshp
 |-  ( ( W e. LVec /\ g e. F /\ g =/= ( V X. { .0. } ) ) -> ( K ` g ) e. H )
26 25 3adant3r
 |-  ( ( W e. LVec /\ g e. F /\ ( g =/= ( V X. { .0. } ) /\ s = ( K ` g ) ) ) -> ( K ` g ) e. H )
27 eqid
 |-  ( LSpan ` W ) = ( LSpan ` W )
28 eqid
 |-  ( LSubSp ` W ) = ( LSubSp ` W )
29 1 27 28 4 islshp
 |-  ( W e. LVec -> ( ( K ` g ) e. H <-> ( ( K ` g ) e. ( LSubSp ` W ) /\ ( K ` g ) =/= V /\ E. v e. V ( ( LSpan ` W ) ` ( ( K ` g ) u. { v } ) ) = V ) ) )
30 29 3ad2ant1
 |-  ( ( W e. LVec /\ g e. F /\ ( g =/= ( V X. { .0. } ) /\ s = ( K ` g ) ) ) -> ( ( K ` g ) e. H <-> ( ( K ` g ) e. ( LSubSp ` W ) /\ ( K ` g ) =/= V /\ E. v e. V ( ( LSpan ` W ) ` ( ( K ` g ) u. { v } ) ) = V ) ) )
31 26 30 mpbid
 |-  ( ( W e. LVec /\ g e. F /\ ( g =/= ( V X. { .0. } ) /\ s = ( K ` g ) ) ) -> ( ( K ` g ) e. ( LSubSp ` W ) /\ ( K ` g ) =/= V /\ E. v e. V ( ( LSpan ` W ) ` ( ( K ` g ) u. { v } ) ) = V ) )
32 eleq1
 |-  ( s = ( K ` g ) -> ( s e. ( LSubSp ` W ) <-> ( K ` g ) e. ( LSubSp ` W ) ) )
33 neeq1
 |-  ( s = ( K ` g ) -> ( s =/= V <-> ( K ` g ) =/= V ) )
34 uneq1
 |-  ( s = ( K ` g ) -> ( s u. { v } ) = ( ( K ` g ) u. { v } ) )
35 34 fveqeq2d
 |-  ( s = ( K ` g ) -> ( ( ( LSpan ` W ) ` ( s u. { v } ) ) = V <-> ( ( LSpan ` W ) ` ( ( K ` g ) u. { v } ) ) = V ) )
36 35 rexbidv
 |-  ( s = ( K ` g ) -> ( E. v e. V ( ( LSpan ` W ) ` ( s u. { v } ) ) = V <-> E. v e. V ( ( LSpan ` W ) ` ( ( K ` g ) u. { v } ) ) = V ) )
37 32 33 36 3anbi123d
 |-  ( s = ( K ` g ) -> ( ( s e. ( LSubSp ` W ) /\ s =/= V /\ E. v e. V ( ( LSpan ` W ) ` ( s u. { v } ) ) = V ) <-> ( ( K ` g ) e. ( LSubSp ` W ) /\ ( K ` g ) =/= V /\ E. v e. V ( ( LSpan ` W ) ` ( ( K ` g ) u. { v } ) ) = V ) ) )
38 37 adantl
 |-  ( ( g =/= ( V X. { .0. } ) /\ s = ( K ` g ) ) -> ( ( s e. ( LSubSp ` W ) /\ s =/= V /\ E. v e. V ( ( LSpan ` W ) ` ( s u. { v } ) ) = V ) <-> ( ( K ` g ) e. ( LSubSp ` W ) /\ ( K ` g ) =/= V /\ E. v e. V ( ( LSpan ` W ) ` ( ( K ` g ) u. { v } ) ) = V ) ) )
39 38 3ad2ant3
 |-  ( ( W e. LVec /\ g e. F /\ ( g =/= ( V X. { .0. } ) /\ s = ( K ` g ) ) ) -> ( ( s e. ( LSubSp ` W ) /\ s =/= V /\ E. v e. V ( ( LSpan ` W ) ` ( s u. { v } ) ) = V ) <-> ( ( K ` g ) e. ( LSubSp ` W ) /\ ( K ` g ) =/= V /\ E. v e. V ( ( LSpan ` W ) ` ( ( K ` g ) u. { v } ) ) = V ) ) )
40 31 39 mpbird
 |-  ( ( W e. LVec /\ g e. F /\ ( g =/= ( V X. { .0. } ) /\ s = ( K ` g ) ) ) -> ( s e. ( LSubSp ` W ) /\ s =/= V /\ E. v e. V ( ( LSpan ` W ) ` ( s u. { v } ) ) = V ) )
41 40 rexlimdv3a
 |-  ( W e. LVec -> ( E. g e. F ( g =/= ( V X. { .0. } ) /\ s = ( K ` g ) ) -> ( s e. ( LSubSp ` W ) /\ s =/= V /\ E. v e. V ( ( LSpan ` W ) ` ( s u. { v } ) ) = V ) ) )
42 1 27 28 4 islshp
 |-  ( W e. LVec -> ( s e. H <-> ( s e. ( LSubSp ` W ) /\ s =/= V /\ E. v e. V ( ( LSpan ` W ) ` ( s u. { v } ) ) = V ) ) )
43 41 42 sylibrd
 |-  ( W e. LVec -> ( E. g e. F ( g =/= ( V X. { .0. } ) /\ s = ( K ` g ) ) -> s e. H ) )
44 24 43 impbid
 |-  ( W e. LVec -> ( s e. H <-> E. g e. F ( g =/= ( V X. { .0. } ) /\ s = ( K ` g ) ) ) )
45 44 abbi2dv
 |-  ( W e. LVec -> H = { s | E. g e. F ( g =/= ( V X. { .0. } ) /\ s = ( K ` g ) ) } )