Theorem dochkr1OLDN 35433

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 33024. (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 2451	. . . 4 |- ( 0g ` U ) = ( 0g ` U )
2		eqid 2451	. . . 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 35064	. . . 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 35410	. . . . 5 |- ( ph -> ( ( ._|_ ` ( ._|_ ` ( L ` G ) ) ) =/= V <-> ( ._|_ ` ( L ` G ) ) e. (LSAtoms ` U ) ) )
14	7, 13	mpbid 210	. . . 4 |- ( ph -> ( ._|_ ` ( L ` G ) ) e. (LSAtoms ` U ) )
15	1, 2, 6, 14	lsateln0 32949	. . 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 725	. . . . . 6 |- ( ( ( ph /\ z e. ( ._|_ ` ( L ` G ) ) ) /\ z =/= ( 0g ` U ) ) -> ( K e. HL /\ W e. H ) )
19	12	ad2antrr 725	. . . . . 6 |- ( ( ( ph /\ z e. ( ._|_ ` ( L ` G ) ) ) /\ z =/= ( 0g ` U ) ) -> G e. F )
20		eldifsn 4101	. . . . . . . 8 |- ( z e. ( ( ._|_ ` ( L ` G ) ) \ { ( 0g ` U ) } ) <-> ( z e. ( ._|_ ` ( L ` G ) ) /\ z =/= ( 0g ` U ) ) )
21	20	biimpri 206	. . . . . . 7 |- ( ( z e. ( ._|_ ` ( L ` G ) ) /\ z =/= ( 0g ` U ) ) -> z e. ( ( ._|_ ` ( L ` G ) ) \ { ( 0g ` U ) } ) )
22	21	adantll 713	. . . . . 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 35431	. . . . 5 |- ( ( ( ph /\ z e. ( ._|_ ` ( L ` G ) ) ) /\ z =/= ( 0g ` U ) ) -> ( G ` z ) =/= .0. )
24	23	ex 434	. . . 4 |- ( ( ph /\ z e. ( ._|_ ` ( L ` G ) ) ) -> ( z =/= ( 0g ` U ) -> ( G ` z ) =/= .0. ) )
25	24	reximdva 2927	. . 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 33050	. . . . . . . 8 |- ( ph -> ( L ` G ) C_ V )
28		eqid 2451	. . . . . . . . 9 |- ( LSubSp ` U ) = ( LSubSp ` U )
29	3, 4, 9, 28, 8	dochlss 35308	. . . . . . . 8 |- ( ( ( K e. HL /\ W e. H ) /\ ( L ` G ) C_ V ) -> ( ._|_ ` ( L ` G ) ) e. ( LSubSp ` U ) )
30	5, 27, 29	syl2anc 661	. . . . . . 7 |- ( ph -> ( ._|_ ` ( L ` G ) ) e. ( LSubSp ` U ) )
31	6, 30	jca 532	. . . . . 6 |- ( ph -> ( U e. LMod /\ ( ._|_ ` ( L ` G ) ) e. ( LSubSp ` U ) ) )
32	31	3ad2ant1 1009	. . . . 5 |- ( ( ph /\ z e. ( ._|_ ` ( L ` G ) ) /\ ( G ` z ) =/= .0. ) -> ( U e. LMod /\ ( ._|_ ` ( L ` G ) ) e. ( LSubSp ` U ) ) )
33	3, 4, 5	dvhlvec 35063	. . . . . . . 8 |- ( ph -> U e. LVec )
34	33	3ad2ant1 1009	. . . . . . 7 |- ( ( ph /\ z e. ( ._|_ ` ( L ` G ) ) /\ ( G ` z ) =/= .0. ) -> U e. LVec )
35	16	lvecdrng 17301	. . . . . . 7 |- ( U e. LVec -> R e. DivRing )
36	34, 35	syl 16	. . . . . 6 |- ( ( ph /\ z e. ( ._|_ ` ( L ` G ) ) /\ ( G ` z ) =/= .0. ) -> R e. DivRing )
37	6	3ad2ant1 1009	. . . . . . 7 |- ( ( ph /\ z e. ( ._|_ ` ( L ` G ) ) /\ ( G ` z ) =/= .0. ) -> U e. LMod )
38	12	3ad2ant1 1009	. . . . . . 7 |- ( ( ph /\ z e. ( ._|_ ` ( L ` G ) ) /\ ( G ` z ) =/= .0. ) -> G e. F )
39	3, 4, 9, 8	dochssv 35309	. . . . . . . . . 10 |- ( ( ( K e. HL /\ W e. H ) /\ ( L ` G ) C_ V ) -> ( ._|_ ` ( L ` G ) ) C_ V )
40	5, 27, 39	syl2anc 661	. . . . . . . . 9 |- ( ph -> ( ._|_ ` ( L ` G ) ) C_ V )
41	40	sselda 3457	. . . . . . . 8 |- ( ( ph /\ z e. ( ._|_ ` ( L ` G ) ) ) -> z e. V )
42	41	3adant3 1008	. . . . . . 7 |- ( ( ph /\ z e. ( ._|_ ` ( L ` G ) ) /\ ( G ` z ) =/= .0. ) -> z e. V )
43		eqid 2451	. . . . . . . 8 |- ( Base ` R ) = ( Base ` R )
44	16, 43, 9, 10	lflcl 33018	. . . . . . 7 |- ( ( U e. LMod /\ G e. F /\ z e. V ) -> ( G ` z ) e. ( Base ` R ) )
45	37, 38, 42, 44	syl3anc 1219	. . . . . 6 |- ( ( ph /\ z e. ( ._|_ ` ( L ` G ) ) /\ ( G ` z ) =/= .0. ) -> ( G ` z ) e. ( Base ` R ) )
46		simp3 990	. . . . . 6 |- ( ( ph /\ z e. ( ._|_ ` ( L ` G ) ) /\ ( G ` z ) =/= .0. ) -> ( G ` z ) =/= .0. )
47		eqid 2451	. . . . . . 7 |- ( invr ` R ) = ( invr ` R )
48	43, 17, 47	drnginvrcl 16964	. . . . . 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 1219	. . . . 5 |- ( ( ph /\ z e. ( ._|_ ` ( L ` G ) ) /\ ( G ` z ) =/= .0. ) -> ( ( invr ` R ) ` ( G ` z ) ) e. ( Base ` R ) )
50		simp2 989	. . . . 5 |- ( ( ph /\ z e. ( ._|_ ` ( L ` G ) ) /\ ( G ` z ) =/= .0. ) -> z e. ( ._|_ ` ( L ` G ) ) )
51		eqid 2451	. . . . . 6 |- ( .s ` U ) = ( .s ` U )
52	16, 51, 43, 28	lssvscl 17151	. . . . 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 1217	. . . 4 |- ( ( ph /\ z e. ( ._|_ ` ( L ` G ) ) /\ ( G ` z ) =/= .0. ) -> ( ( ( invr ` R ) ` ( G ` z ) ) ( .s ` U ) z ) e. ( ._|_ ` ( L ` G ) ) )
54		eqid 2451	. . . . . . 7 |- ( .r ` R ) = ( .r ` R )
55	16, 43, 54, 9, 51, 10	lflmul 33022	. . . . . 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 1223	. . . . 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 16966	. . . . . 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 1219	. . . . 5 |- ( ( ph /\ z e. ( ._|_ ` ( L ` G ) ) /\ ( G ` z ) =/= .0. ) -> ( ( ( invr ` R ) ` ( G ` z ) ) ( .r ` R ) ( G ` z ) ) = .1. )
60	56, 59	eqtrd 2492	. . . 4 |- ( ( ph /\ z e. ( ._|_ ` ( L ` G ) ) /\ ( G ` z ) =/= .0. ) -> ( G ` ( ( ( invr ` R ) ` ( G ` z ) ) ( .s ` U ) z ) ) = .1. )
61		fveq2 5792	. . . . . 6 |- ( x = ( ( ( invr ` R ) ` ( G ` z ) ) ( .s ` U ) z ) -> ( G ` x ) = ( G ` ( ( ( invr ` R ) ` ( G ` z ) ) ( .s ` U ) z ) ) )
62	61	eqeq1d 2453	. . . . 5 |- ( x = ( ( ( invr ` R ) ` ( G ` z ) ) ( .s ` U ) z ) -> ( ( G ` x ) = .1. <-> ( G ` ( ( ( invr ` R ) ` ( G ` z ) ) ( .s ` U ) z ) ) = .1. ) )
63	62	rspcev 3172	. . . 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 661	. . 3 |- ( ( ph /\ z e. ( ._|_ ` ( L ` G ) ) /\ ( G ` z ) =/= .0. ) -> E. x e. ( ._|_ ` ( L ` G ) ) ( G ` x ) = .1. )
65	64	rexlimdv3a 2942	. 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 369   /\ w3a 965   = wceq 1370   e. wcel 1758   =/= wne 2644   E.wrex 2796   \ cdif 3426   C_ wss 3429   {csn 3978   ` cfv 5519  (class class class)co 6193   Basecbs 14285   .rcmulr 14350  Scalarcsca 14352   .scvsca 14353   0gc0g 14489   1rcur 16717   invrcinvr 16878   DivRingcdr 16947   LModclmod 17063   LSubSpclss 17128   LVecclvec 17298  LSAtomsclsa 32928  LFnlclfn 33011  LKerclk 33039   HLchlt 33304   LHypclh 33937   DVecHcdvh 35032   ocHcoch 35301

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 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463  ax-riotaBAD 32913

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-iin 4275  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-1st 6680  df-2nd 6681  df-tpos 6848  df-undef 6895  df-recs 6935  df-rdg 6969  df-1o 7023  df-oadd 7027  df-er 7204  df-map 7319  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-nn 10427  df-2 10484  df-3 10485  df-4 10486  df-5 10487  df-6 10488  df-n0 10684  df-z 10751  df-uz 10966  df-fz 11548  df-struct 14287  df-ndx 14288  df-slot 14289  df-base 14290  df-sets 14291  df-ress 14292  df-plusg 14362  df-mulr 14363  df-sca 14365  df-vsca 14366  df-0g 14491  df-poset 15227  df-plt 15239  df-lub 15255  df-glb 15256  df-join 15257  df-meet 15258  df-p0 15320  df-p1 15321  df-lat 15327  df-clat 15389  df-mnd 15526  df-submnd 15576  df-grp 15656  df-minusg 15657  df-sbg 15658  df-subg 15789  df-cntz 15946  df-lsm 16248  df-cmn 16392  df-abl 16393  df-mgp 16706  df-ur 16718  df-rng 16762  df-oppr 16830  df-dvdsr 16848  df-unit 16849  df-invr 16879  df-dvr 16890  df-drng 16949  df-lmod 17065  df-lss 17129  df-lsp 17168  df-lvec 17299  df-lsatoms 32930  df-lshyp 32931  df-lfl 33012  df-lkr 33040  df-oposet 33130  df-ol 33132  df-oml 33133  df-covers 33220  df-ats 33221  df-atl 33252  df-cvlat 33276  df-hlat 33305  df-llines 33451  df-lplanes 33452  df-lvols 33453  df-lines 33454  df-psubsp 33456  df-pmap 33457  df-padd 33749  df-lhyp 33941  df-laut 33942  df-ldil 34057  df-ltrn 34058  df-trl 34112  df-tgrp 34696  df-tendo 34708  df-edring 34710  df-dveca 34956  df-disoa 34983  df-dvech 35033  df-dib 35093  df-dic 35127  df-dih 35183  df-doch 35302  df-djh 35349

This theorem is referenced by: (None)