Theorem lshpset2N 35241

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 35240	. . . . 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 485	. . . . . . . . . . 11 |- ( ( s e. H /\ ( K ` g ) = s ) -> ( K ` g ) e. H )
7	6	adantll 711	. . . . . . . . . 10 |- ( ( ( W e. LVec /\ s e. H ) /\ ( K ` g ) = s ) -> ( K ` g ) e. H )
8	7	adantlr 712	. . . . . . . . 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 753	. . . . . . . . . 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 35228	. . . . . . . . 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 432	. . . . . . 7 |- ( ( ( W e. LVec /\ s e. H ) /\ g e. F ) -> ( ( K ` g ) = s -> g =/= ( V X. { .0. } ) ) )
17		eqimss2 3542	. . . . . . . . 9 |- ( ( K ` g ) = s -> s C_ ( K ` g ) )
18		eqimss 3541	. . . . . . . . 9 |- ( ( K ` g ) = s -> ( K ` g ) C_ s )
19	17, 18	eqssd 3506	. . . . . . . 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 531	. . . . . 6 |- ( ( ( W e. LVec /\ s e. H ) /\ g e. F ) -> ( ( K ` g ) = s -> ( g =/= ( V X. { .0. } ) /\ s = ( K ` g ) ) ) )
22	21	reximdva 2929	. . . . 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 432	. . 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 35227	. . . . . . . 8 |- ( ( W e. LVec /\ g e. F /\ g =/= ( V X. { .0. } ) ) -> ( K ` g ) e. H )
26	25	3adant3r 1223	. . . . . . 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 35101	. . . . . . . 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 1015	. . . . . . 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 2735	. . . . . . . . 9 |- ( s = ( K ` g ) -> ( s =/= V <-> ( K ` g ) =/= V ) )
34		uneq1 3637	. . . . . . . . . . . 12 |- ( s = ( K ` g ) -> ( s u. { v } ) = ( ( K ` g ) u. { v } ) )
35	34	fveq2d 5852	. . . . . . . . . . 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 2965	. . . . . . . . 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 1297	. . . . . . . 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 464	. . . . . . 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 1017	. . . . . 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 2948	. . . 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 35101	. . . 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 367   /\ w3a 971   = wceq 1398   e. wcel 1823   {cab 2439   =/= wne 2649   E.wrex 2805   u. cun 3459   {csn 4016   X. cxp 4986   ` cfv 5570   Basecbs 14716  Scalarcsca 14787   0gc0g 14929   LSubSpclss 17773   LSpanclspn 17812   LVecclvec 17943  LSHypclsh 35097  LFnlclfn 35179  LKerclk 35207

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-tpos 6947  df-recs 7034  df-rdg 7068  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ress 14723  df-plusg 14797  df-mulr 14798  df-0g 14931  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-submnd 16166  df-grp 16256  df-minusg 16257  df-sbg 16258  df-subg 16397  df-cntz 16554  df-lsm 16855  df-cmn 16999  df-abl 17000  df-mgp 17337  df-ur 17349  df-ring 17395  df-oppr 17467  df-dvdsr 17485  df-unit 17486  df-invr 17516  df-drng 17593  df-lmod 17709  df-lss 17774  df-lsp 17813  df-lvec 17944  df-lshyp 35099  df-lfl 35180  df-lkr 35208

This theorem is referenced by:  islshpkrN  35242