Theorem dochkr1OLDN 34963

Description: A non-zero functional has a value of 1 at some argument belonging to the orthocomplement of its kernel (when its kernel is a closed hyperplane). Tighter version of lfl1 32552. (Contributed by NM, 2-Jan-2015.) (New usage is discouraged.)

Hypotheses

Ref	Expression
dochkr1OLD.h	|- H = ( LHyp ` K )
dochkr1OLD.o	|- ._|_ = ( ( ocH ` K ) ` W )
dochkr1OLD.u	|- U = ( ( DVecH ` K ) ` W )
dochkr1OLD.v	|- V = ( Base ` U )
dochkr1OLD.r	|- R = (Scalar ` U )
dochkr1OLD.z	|- .0. = ( 0g ` R )
dochkr1OLD.i	|- .1. = ( 1r ` R )
dochkr1OLD.f	|- F = (LFnl ` U )
dochkr1OLD.l	|- L = (LKer ` U )
dochkr1OLD.k	|- ( ph -> ( K e. HL /\ W e. H ) )
dochkr1OLD.g	|- ( ph -> G e. F )
dochkr1OLD.n	|- ( ph -> ( ._|_ ` ( ._|_ ` ( L ` G ) ) ) =/= V )

Assertion

Ref	Expression
dochkr1OLDN	|- ( ph -> E. x e. ( ._|_ ` ( L ` G ) ) ( G ` x ) = .1. )

Distinct variable groups:   x, .1.   x, G   x, L   x, R   x, U   x, ._|_

Allowed substitution hints:   ph( x)   F( x)   H( x)   K( x)   V( x)   W( x)   .0. ( x)

Proof of Theorem dochkr1OLDN

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2422	. . . 4 |- ( 0g ` U ) = ( 0g ` U )
2		eqid 2422	. . . 4 |- (LSAtoms ` U ) = (LSAtoms ` U )
3		dochkr1OLD.h	. . . . 5 |- H = ( LHyp ` K )
4		dochkr1OLD.u	. . . . 5 |- U = ( ( DVecH ` K ) ` W )
5		dochkr1OLD.k	. . . . 5 |- ( ph -> ( K e. HL /\ W e. H ) )
6	3, 4, 5	dvhlmod 34594	. . . 4 |- ( ph -> U e. LMod )
7		dochkr1OLD.n	. . . . 5 |- ( ph -> ( ._|_ ` ( ._|_ ` ( L ` G ) ) ) =/= V )
8		dochkr1OLD.o	. . . . . 6 |- ._|_ = ( ( ocH ` K ) ` W )
9		dochkr1OLD.v	. . . . . 6 |- V = ( Base ` U )
10		dochkr1OLD.f	. . . . . 6 |- F = (LFnl ` U )
11		dochkr1OLD.l	. . . . . 6 |- L = (LKer ` U )
12		dochkr1OLD.g	. . . . . 6 |- ( ph -> G e. F )
13	3, 8, 4, 9, 2, 10, 11, 5, 12	dochkrsat2 34940	. . . . 5 |- ( ph -> ( ( ._|_ ` ( ._|_ ` ( L ` G ) ) ) =/= V <-> ( ._|_ ` ( L ` G ) ) e. (LSAtoms ` U ) ) )
14	7, 13	mpbid 213	. . . 4 |- ( ph -> ( ._|_ ` ( L ` G ) ) e. (LSAtoms ` U ) )
15	1, 2, 6, 14	lsateln0 32477	. . 3 |- ( ph -> E. z e. ( ._|_ ` ( L ` G ) ) z =/= ( 0g ` U ) )
16		dochkr1OLD.r	. . . . . 6 |- R = (Scalar ` U )
17		dochkr1OLD.z	. . . . . 6 |- .0. = ( 0g ` R )
18	5	ad2antrr 730	. . . . . 6 |- ( ( ( ph /\ z e. ( ._|_ ` ( L ` G ) ) ) /\ z =/= ( 0g ` U ) ) -> ( K e. HL /\ W e. H ) )
19	12	ad2antrr 730	. . . . . 6 |- ( ( ( ph /\ z e. ( ._|_ ` ( L ` G ) ) ) /\ z =/= ( 0g ` U ) ) -> G e. F )
20		eldifsn 4122	. . . . . . . 8 |- ( z e. ( ( ._|_ ` ( L ` G ) ) \ { ( 0g ` U ) } ) <-> ( z e. ( ._|_ ` ( L ` G ) ) /\ z =/= ( 0g ` U ) ) )
21	20	biimpri 209	. . . . . . 7 |- ( ( z e. ( ._|_ ` ( L ` G ) ) /\ z =/= ( 0g ` U ) ) -> z e. ( ( ._|_ ` ( L ` G ) ) \ { ( 0g ` U ) } ) )
22	21	adantll 718	. . . . . 6 |- ( ( ( ph /\ z e. ( ._|_ ` ( L ` G ) ) ) /\ z =/= ( 0g ` U ) ) -> z e. ( ( ._|_ ` ( L ` G ) ) \ { ( 0g ` U ) } ) )
23	3, 8, 4, 9, 16, 17, 1, 10, 11, 18, 19, 22	dochfln0 34961	. . . . 5 |- ( ( ( ph /\ z e. ( ._|_ ` ( L ` G ) ) ) /\ z =/= ( 0g ` U ) ) -> ( G ` z ) =/= .0. )
24	23	ex 435	. . . 4 |- ( ( ph /\ z e. ( ._|_ ` ( L ` G ) ) ) -> ( z =/= ( 0g ` U ) -> ( G ` z ) =/= .0. ) )
25	24	reximdva 2900	. . 3 |- ( ph -> ( E. z e. ( ._|_ ` ( L ` G ) ) z =/= ( 0g ` U ) -> E. z e. ( ._|_ ` ( L ` G ) ) ( G ` z ) =/= .0. ) )
26	15, 25	mpd 15	. 2 |- ( ph -> E. z e. ( ._|_ ` ( L ` G ) ) ( G ` z ) =/= .0. )
27	9, 10, 11, 6, 12	lkrssv 32578	. . . . . . . 8 |- ( ph -> ( L ` G ) C_ V )
28		eqid 2422	. . . . . . . . 9 |- ( LSubSp ` U ) = ( LSubSp ` U )
29	3, 4, 9, 28, 8	dochlss 34838	. . . . . . . 8 |- ( ( ( K e. HL /\ W e. H ) /\ ( L ` G ) C_ V ) -> ( ._|_ ` ( L ` G ) ) e. ( LSubSp ` U ) )
30	5, 27, 29	syl2anc 665	. . . . . . 7 |- ( ph -> ( ._|_ ` ( L ` G ) ) e. ( LSubSp ` U ) )
31	6, 30	jca 534	. . . . . 6 |- ( ph -> ( U e. LMod /\ ( ._|_ ` ( L ` G ) ) e. ( LSubSp ` U ) ) )
32	31	3ad2ant1 1026	. . . . 5 |- ( ( ph /\ z e. ( ._|_ ` ( L ` G ) ) /\ ( G ` z ) =/= .0. ) -> ( U e. LMod /\ ( ._|_ ` ( L ` G ) ) e. ( LSubSp ` U ) ) )
33	3, 4, 5	dvhlvec 34593	. . . . . . . 8 |- ( ph -> U e. LVec )
34	33	3ad2ant1 1026	. . . . . . 7 |- ( ( ph /\ z e. ( ._|_ ` ( L ` G ) ) /\ ( G ` z ) =/= .0. ) -> U e. LVec )
35	16	lvecdrng 18313	. . . . . . 7 |- ( U e. LVec -> R e. DivRing )
36	34, 35	syl 17	. . . . . 6 |- ( ( ph /\ z e. ( ._|_ ` ( L ` G ) ) /\ ( G ` z ) =/= .0. ) -> R e. DivRing )
37	6	3ad2ant1 1026	. . . . . . 7 |- ( ( ph /\ z e. ( ._|_ ` ( L ` G ) ) /\ ( G ` z ) =/= .0. ) -> U e. LMod )
38	12	3ad2ant1 1026	. . . . . . 7 |- ( ( ph /\ z e. ( ._|_ ` ( L ` G ) ) /\ ( G ` z ) =/= .0. ) -> G e. F )
39	3, 4, 9, 8	dochssv 34839	. . . . . . . . . 10 |- ( ( ( K e. HL /\ W e. H ) /\ ( L ` G ) C_ V ) -> ( ._|_ ` ( L ` G ) ) C_ V )
40	5, 27, 39	syl2anc 665	. . . . . . . . 9 |- ( ph -> ( ._|_ ` ( L ` G ) ) C_ V )
41	40	sselda 3464	. . . . . . . 8 |- ( ( ph /\ z e. ( ._|_ ` ( L ` G ) ) ) -> z e. V )
42	41	3adant3 1025	. . . . . . 7 |- ( ( ph /\ z e. ( ._|_ ` ( L ` G ) ) /\ ( G ` z ) =/= .0. ) -> z e. V )
43		eqid 2422	. . . . . . . 8 |- ( Base ` R ) = ( Base ` R )
44	16, 43, 9, 10	lflcl 32546	. . . . . . 7 |- ( ( U e. LMod /\ G e. F /\ z e. V ) -> ( G ` z ) e. ( Base ` R ) )
45	37, 38, 42, 44	syl3anc 1264	. . . . . 6 |- ( ( ph /\ z e. ( ._|_ ` ( L ` G ) ) /\ ( G ` z ) =/= .0. ) -> ( G ` z ) e. ( Base ` R ) )
46		simp3 1007	. . . . . 6 |- ( ( ph /\ z e. ( ._|_ ` ( L ` G ) ) /\ ( G ` z ) =/= .0. ) -> ( G ` z ) =/= .0. )
47		eqid 2422	. . . . . . 7 |- ( invr ` R ) = ( invr ` R )
48	43, 17, 47	drnginvrcl 17977	. . . . . 6 |- ( ( R e. DivRing /\ ( G ` z ) e. ( Base ` R ) /\ ( G ` z ) =/= .0. ) -> ( ( invr ` R ) ` ( G ` z ) ) e. ( Base ` R ) )
49	36, 45, 46, 48	syl3anc 1264	. . . . 5 |- ( ( ph /\ z e. ( ._|_ ` ( L ` G ) ) /\ ( G ` z ) =/= .0. ) -> ( ( invr ` R ) ` ( G ` z ) ) e. ( Base ` R ) )
50		simp2 1006	. . . . 5 |- ( ( ph /\ z e. ( ._|_ ` ( L ` G ) ) /\ ( G ` z ) =/= .0. ) -> z e. ( ._|_ ` ( L ` G ) ) )
51		eqid 2422	. . . . . 6 |- ( .s ` U ) = ( .s ` U )
52	16, 51, 43, 28	lssvscl 18163	. . . . 5 |- ( ( ( U e. LMod /\ ( ._|_ ` ( L ` G ) ) e. ( LSubSp ` U ) ) /\ ( ( ( invr ` R ) ` ( G ` z ) ) e. ( Base ` R ) /\ z e. ( ._|_ ` ( L ` G ) ) ) ) -> ( ( ( invr ` R ) ` ( G ` z ) ) ( .s ` U ) z ) e. ( ._|_ ` ( L ` G ) ) )
53	32, 49, 50, 52	syl12anc 1262	. . . 4 |- ( ( ph /\ z e. ( ._|_ ` ( L ` G ) ) /\ ( G ` z ) =/= .0. ) -> ( ( ( invr ` R ) ` ( G ` z ) ) ( .s ` U ) z ) e. ( ._|_ ` ( L ` G ) ) )
54		eqid 2422	. . . . . . 7 |- ( .r ` R ) = ( .r ` R )
55	16, 43, 54, 9, 51, 10	lflmul 32550	. . . . . 6 |- ( ( U e. LMod /\ G e. F /\ ( ( ( invr ` R ) ` ( G ` z ) ) e. ( Base ` R ) /\ z e. V ) ) -> ( G ` ( ( ( invr ` R ) ` ( G ` z ) ) ( .s ` U ) z ) ) = ( ( ( invr ` R ) ` ( G ` z ) ) ( .r ` R ) ( G ` z ) ) )
56	37, 38, 49, 42, 55	syl112anc 1268	. . . . 5 |- ( ( ph /\ z e. ( ._|_ ` ( L ` G ) ) /\ ( G ` z ) =/= .0. ) -> ( G ` ( ( ( invr ` R ) ` ( G ` z ) ) ( .s ` U ) z ) ) = ( ( ( invr ` R ) ` ( G ` z ) ) ( .r ` R ) ( G ` z ) ) )
57		dochkr1OLD.i	. . . . . . 7 |- .1. = ( 1r ` R )
58	43, 17, 54, 57, 47	drnginvrl 17979	. . . . . 6 |- ( ( R e. DivRing /\ ( G ` z ) e. ( Base ` R ) /\ ( G ` z ) =/= .0. ) -> ( ( ( invr ` R ) ` ( G ` z ) ) ( .r ` R ) ( G ` z ) ) = .1. )
59	36, 45, 46, 58	syl3anc 1264	. . . . 5 |- ( ( ph /\ z e. ( ._|_ ` ( L ` G ) ) /\ ( G ` z ) =/= .0. ) -> ( ( ( invr ` R ) ` ( G ` z ) ) ( .r ` R ) ( G ` z ) ) = .1. )
60	56, 59	eqtrd 2463	. . . 4 |- ( ( ph /\ z e. ( ._|_ ` ( L ` G ) ) /\ ( G ` z ) =/= .0. ) -> ( G ` ( ( ( invr ` R ) ` ( G ` z ) ) ( .s ` U ) z ) ) = .1. )
61		fveq2 5877	. . . . . 6 |- ( x = ( ( ( invr ` R ) ` ( G ` z ) ) ( .s ` U ) z ) -> ( G ` x ) = ( G ` ( ( ( invr ` R ) ` ( G ` z ) ) ( .s ` U ) z ) ) )
62	61	eqeq1d 2424	. . . . 5 |- ( x = ( ( ( invr ` R ) ` ( G ` z ) ) ( .s ` U ) z ) -> ( ( G ` x ) = .1. <-> ( G ` ( ( ( invr ` R ) ` ( G ` z ) ) ( .s ` U ) z ) ) = .1. ) )
63	62	rspcev 3182	. . . 4 |- ( ( ( ( ( invr ` R ) ` ( G ` z ) ) ( .s ` U ) z ) e. ( ._|_ ` ( L ` G ) ) /\ ( G ` ( ( ( invr ` R ) ` ( G ` z ) ) ( .s ` U ) z ) ) = .1. ) -> E. x e. ( ._|_ ` ( L ` G ) ) ( G ` x ) = .1. )
64	53, 60, 63	syl2anc 665	. . 3 |- ( ( ph /\ z e. ( ._|_ ` ( L ` G ) ) /\ ( G ` z ) =/= .0. ) -> E. x e. ( ._|_ ` ( L ` G ) ) ( G ` x ) = .1. )
65	64	rexlimdv3a 2919	. 2 |- ( ph -> ( E. z e. ( ._|_ ` ( L ` G ) ) ( G ` z ) =/= .0. -> E. x e. ( ._|_ ` ( L ` G ) ) ( G ` x ) = .1. ) )
66	26, 65	mpd 15	1 |- ( ph -> E. x e. ( ._|_ ` ( L ` G ) ) ( G ` x ) = .1. )

Syntax hints:   -> wi 4   /\ wa 370   /\ w3a 982   = wceq 1437   e. wcel 1868   =/= wne 2618   E.wrex 2776   \ cdif 3433   C_ wss 3436   {csn 3996   ` cfv 5597  (class class class)co 6301   Basecbs 15106   .rcmulr 15176  Scalarcsca 15178   .scvsca 15179   0gc0g 15323   1rcur 17720   invrcinvr 17884   DivRingcdr 17960   LModclmod 18076   LSubSpclss 18140   LVecclvec 18310  LSAtomsclsa 32456  LFnlclfn 32539  LKerclk 32567   HLchlt 32832   LHypclh 33465   DVecHcdvh 34562   ocHcoch 34831

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-riotaBAD 32441

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-iin 4299  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4760  df-id 4764  df-po 4770  df-so 4771  df-fr 4808  df-we 4810  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-pred 5395  df-ord 5441  df-on 5442  df-lim 5443  df-suc 5444  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-riota 6263  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-om 6703  df-1st 6803  df-2nd 6804  df-tpos 6977  df-undef 7024  df-wrecs 7032  df-recs 7094  df-rdg 7132  df-1o 7186  df-oadd 7190  df-er 7367  df-map 7478  df-en 7574  df-dom 7575  df-sdom 7576  df-fin 7577  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11785  df-struct 15108  df-ndx 15109  df-slot 15110  df-base 15111  df-sets 15112  df-ress 15113  df-plusg 15188  df-mulr 15189  df-sca 15191  df-vsca 15192  df-0g 15325  df-preset 16158  df-poset 16176  df-plt 16189  df-lub 16205  df-glb 16206  df-join 16207  df-meet 16208  df-p0 16270  df-p1 16271  df-lat 16277  df-clat 16339  df-mgm 16473  df-sgrp 16512  df-mnd 16522  df-submnd 16568  df-grp 16658  df-minusg 16659  df-sbg 16660  df-subg 16799  df-cntz 16956  df-lsm 17273  df-cmn 17417  df-abl 17418  df-mgp 17709  df-ur 17721  df-ring 17767  df-oppr 17836  df-dvdsr 17854  df-unit 17855  df-invr 17885  df-dvr 17896  df-drng 17962  df-lmod 18078  df-lss 18141  df-lsp 18180  df-lvec 18311  df-lsatoms 32458  df-lshyp 32459  df-lfl 32540  df-lkr 32568  df-oposet 32658  df-ol 32660  df-oml 32661  df-covers 32748  df-ats 32749  df-atl 32780  df-cvlat 32804  df-hlat 32833  df-llines 32979  df-lplanes 32980  df-lvols 32981  df-lines 32982  df-psubsp 32984  df-pmap 32985  df-padd 33277  df-lhyp 33469  df-laut 33470  df-ldil 33585  df-ltrn 33586  df-trl 33641  df-tgrp 34226  df-tendo 34238  df-edring 34240  df-dveca 34486  df-disoa 34513  df-dvech 34563  df-dib 34623  df-dic 34657  df-dih 34713  df-doch 34832  df-djh 34879

This theorem is referenced by: (None)