Theorem lshpset2N 33103

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.)

Hypotheses

Ref	Expression
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

Ref	Expression
lshpset2N	|- ( W e. LVec -> H = { s | E. g e. F ( g =/= ( V X. { .0. } ) /\ s = ( K ` g ) ) } )

Distinct variable groups:   g, F   g, s, H   g, K   g, V   g, W, s

Allowed substitution hints:   D( g, s)   F( s)   K( s)   V( s)   .0. ( g, s)

Proof of Theorem lshpset2N

Dummy variable v is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lshpset2.h	. . . . . 6 |- H = (LSHyp ` W )
2		lshpset2.f	. . . . . 6 |- F = (LFnl ` W )
3		lshpset2.k	. . . . . 6 |- K = (LKer ` W )
4	1, 2, 3	lshpkrex 33102	. . . . 5 |- ( ( W e. LVec /\ s e. H ) -> E. g e. F ( K ` g ) = s )
5		eleq1 2526	. . . . . . . . . . . 12 |- ( ( K ` g ) = s -> ( ( K ` g ) e. H <-> s e. H ) )
6	5	biimparc 487	. . . . . . . . . . 11 |- ( ( s e. H /\ ( K ` g ) = s ) -> ( K ` g ) e. H )
7	6	adantll 713	. . . . . . . . . 10 |- ( ( ( W e. LVec /\ s e. H ) /\ ( K ` g ) = s ) -> ( K ` g ) e. H )
8	7	adantlr 714	. . . . . . . . 9 |- ( ( ( ( W e. LVec /\ s e. H ) /\ g e. F ) /\ ( K ` g ) = s ) -> ( K ` g ) e. H )
9		lshpset2.v	. . . . . . . . . 10 |- V = ( Base ` W )
10		lshpset2.d	. . . . . . . . . 10 |- D = (Scalar ` W )
11		lshpset2.z	. . . . . . . . . 10 |- .0. = ( 0g ` D )
12		simplll 757	. . . . . . . . . 10 |- ( ( ( ( W e. LVec /\ s e. H ) /\ g e. F ) /\ ( K ` g ) = s ) -> W e. LVec )
13		simplr 754	. . . . . . . . . 10 |- ( ( ( ( W e. LVec /\ s e. H ) /\ g e. F ) /\ ( K ` g ) = s ) -> g e. F )
14	9, 10, 11, 1, 2, 3, 12, 13	lkrshp3 33090	. . . . . . . . 9 |- ( ( ( ( W e. LVec /\ s e. H ) /\ g e. F ) /\ ( K ` g ) = s ) -> ( ( K ` g ) e. H <-> g =/= ( V X. { .0. } ) ) )
15	8, 14	mpbid 210	. . . . . . . 8 |- ( ( ( ( W e. LVec /\ s e. H ) /\ g e. F ) /\ ( K ` g ) = s ) -> g =/= ( V X. { .0. } ) )
16	15	ex 434	. . . . . . 7 |- ( ( ( W e. LVec /\ s e. H ) /\ g e. F ) -> ( ( K ` g ) = s -> g =/= ( V X. { .0. } ) ) )
17		eqimss2 3518	. . . . . . . . 9 |- ( ( K ` g ) = s -> s C_ ( K ` g ) )
18		eqimss 3517	. . . . . . . . 9 |- ( ( K ` g ) = s -> ( K ` g ) C_ s )
19	17, 18	eqssd 3482	. . . . . . . 8 |- ( ( K ` g ) = s -> s = ( K ` g ) )
20	19	a1i 11	. . . . . . 7 |- ( ( ( W e. LVec /\ s e. H ) /\ g e. F ) -> ( ( K ` g ) = s -> s = ( K ` g ) ) )
21	16, 20	jcad 533	. . . . . 6 |- ( ( ( W e. LVec /\ s e. H ) /\ g e. F ) -> ( ( K ` g ) = s -> ( g =/= ( V X. { .0. } ) /\ s = ( K ` g ) ) ) )
22	21	reximdva 2934	. . . . 5 |- ( ( 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	4, 22	mpd 15	. . . 4 |- ( ( W e. LVec /\ s e. H ) -> E. g e. F ( g =/= ( V X. { .0. } ) /\ s = ( K ` g ) ) )
24	23	ex 434	. . 3 |- ( W e. LVec -> ( s e. H -> E. g e. F ( g =/= ( V X. { .0. } ) /\ s = ( K ` g ) ) ) )
25	9, 10, 11, 1, 2, 3	lkrshp 33089	. . . . . . . 8 |- ( ( W e. LVec /\ g e. F /\ g =/= ( V X. { .0. } ) ) -> ( K ` g ) e. H )
26	25	3adant3r 1216	. . . . . . 7 |- ( ( W e. LVec /\ g e. F /\ ( g =/= ( V X. { .0. } ) /\ s = ( K ` g ) ) ) -> ( K ` g ) e. H )
27		eqid 2454	. . . . . . . . 9 |- ( LSpan ` W ) = ( LSpan ` W )
28		eqid 2454	. . . . . . . . 9 |- ( LSubSp ` W ) = ( LSubSp ` W )
29	9, 27, 28, 1	islshp 32963	. . . . . . . 8 |- ( 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 1009	. . . . . . 7 |- ( ( 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 210	. . . . . 6 |- ( ( 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 2526	. . . . . . . . 9 |- ( s = ( K ` g ) -> ( s e. ( LSubSp ` W ) <-> ( K ` g ) e. ( LSubSp ` W ) ) )
33		neeq1 2733	. . . . . . . . 9 |- ( s = ( K ` g ) -> ( s =/= V <-> ( K ` g ) =/= V ) )
34		uneq1 3612	. . . . . . . . . . . 12 |- ( s = ( K ` g ) -> ( s u. { v } ) = ( ( K ` g ) u. { v } ) )
35	34	fveq2d 5804	. . . . . . . . . . 11 |- ( s = ( K ` g ) -> ( ( LSpan ` W ) ` ( s u. { v } ) ) = ( ( LSpan ` W ) ` ( ( K ` g ) u. { v } ) ) )
36	35	eqeq1d 2456	. . . . . . . . . 10 |- ( s = ( K ` g ) -> ( ( ( LSpan ` W ) ` ( s u. { v } ) ) = V <-> ( ( LSpan ` W ) ` ( ( K ` g ) u. { v } ) ) = V ) )
37	36	rexbidv 2868	. . . . . . . . 9 |- ( s = ( K ` g ) -> ( E. v e. V ( ( LSpan ` W ) ` ( s u. { v } ) ) = V <-> E. v e. V ( ( LSpan ` W ) ` ( ( K ` g ) u. { v } ) ) = V ) )
38	32, 33, 37	3anbi123d 1290	. . . . . . . 8 |- ( 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	adantl 466	. . . . . . 7 |- ( ( 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	39	3ad2ant3 1011	. . . . . 6 |- ( ( 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 ) ) )
41	31, 40	mpbird 232	. . . . 5 |- ( ( 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 ) )
42	41	rexlimdv3a 2949	. . . 4 |- ( 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 ) ) )
43	9, 27, 28, 1	islshp 32963	. . . 4 |- ( W e. LVec -> ( s e. H <-> ( s e. ( LSubSp ` W ) /\ s =/= V /\ E. v e. V ( ( LSpan ` W ) ` ( s u. { v } ) ) = V ) ) )
44	42, 43	sylibrd 234	. . 3 |- ( W e. LVec -> ( E. g e. F ( g =/= ( V X. { .0. } ) /\ s = ( K ` g ) ) -> s e. H ) )
45	24, 44	impbid 191	. 2 |- ( W e. LVec -> ( s e. H <-> E. g e. F ( g =/= ( V X. { .0. } ) /\ s = ( K ` g ) ) ) )
46	45	abbi2dv 2591	1 |- ( W e. LVec -> H = { s | E. g e. F ( g =/= ( V X. { .0. } ) /\ s = ( K ` g ) ) } )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 965   = wceq 1370   e. wcel 1758   {cab 2439   =/= wne 2648   E.wrex 2800   u. cun 3435   {csn 3986   X. cxp 4947   ` cfv 5527   Basecbs 14293  Scalarcsca 14361   0gc0g 14498   LSubSpclss 17137   LSpanclspn 17176   LVecclvec 17307  LSHypclsh 32959  LFnlclfn 33041  LKerclk 33069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-1st 6688  df-2nd 6689  df-tpos 6856  df-recs 6943  df-rdg 6977  df-er 7212  df-map 7327  df-en 7422  df-dom 7423  df-sdom 7424  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-nn 10435  df-2 10492  df-3 10493  df-ndx 14296  df-slot 14297  df-base 14298  df-sets 14299  df-ress 14300  df-plusg 14371  df-mulr 14372  df-0g 14500  df-mnd 15535  df-submnd 15585  df-grp 15665  df-minusg 15666  df-sbg 15667  df-subg 15798  df-cntz 15955  df-lsm 16257  df-cmn 16401  df-abl 16402  df-mgp 16715  df-ur 16727  df-rng 16771  df-oppr 16839  df-dvdsr 16857  df-unit 16858  df-invr 16888  df-drng 16958  df-lmod 17074  df-lss 17138  df-lsp 17177  df-lvec 17308  df-lshyp 32961  df-lfl 33042  df-lkr 33070

This theorem is referenced by:  islshpkrN  33104