Theorem lspdisj 17321

Description: The span of a vector not in a subspace is disjoint with the subspace. (Contributed by NM, 6-Apr-2015.)

Hypotheses

Ref	Expression
lspdisj.v	|- V = ( Base ` W )
lspdisj.o	|- .0. = ( 0g ` W )
lspdisj.n	|- N = ( LSpan ` W )
lspdisj.s	|- S = ( LSubSp ` W )
lspdisj.w	|- ( ph -> W e. LVec )
lspdisj.u	|- ( ph -> U e. S )
lspdisj.x	|- ( ph -> X e. V )
lspdisj.e	|- ( ph -> -. X e. U )

Assertion

Ref	Expression
lspdisj	|- ( ph -> ( ( N ` { X } ) i^i U ) = { .0. } )

Proof of Theorem lspdisj

Dummy variables v k are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lspdisj.w	. . . . . . . . . 10 |- ( ph -> W e. LVec )
2		lveclmod 17302	. . . . . . . . . 10 |- ( W e. LVec -> W e. LMod )
3	1, 2	syl 16	. . . . . . . . 9 |- ( ph -> W e. LMod )
4		lspdisj.x	. . . . . . . . 9 |- ( ph -> X e. V )
5		eqid 2451	. . . . . . . . . 10 |- (Scalar ` W ) = (Scalar ` W )
6		eqid 2451	. . . . . . . . . 10 |- ( Base ` (Scalar ` W ) ) = ( Base ` (Scalar ` W ) )
7		lspdisj.v	. . . . . . . . . 10 |- V = ( Base ` W )
8		eqid 2451	. . . . . . . . . 10 |- ( .s ` W ) = ( .s ` W )
9		lspdisj.n	. . . . . . . . . 10 |- N = ( LSpan ` W )
10	5, 6, 7, 8, 9	lspsnel 17199	. . . . . . . . 9 |- ( ( W e. LMod /\ X e. V ) -> ( v e. ( N ` { X } ) <-> E. k e. ( Base ` (Scalar ` W ) ) v = ( k ( .s ` W ) X ) ) )
11	3, 4, 10	syl2anc 661	. . . . . . . 8 |- ( ph -> ( v e. ( N ` { X } ) <-> E. k e. ( Base ` (Scalar ` W ) ) v = ( k ( .s ` W ) X ) ) )
12	11	biimpa 484	. . . . . . 7 |- ( ( ph /\ v e. ( N ` { X } ) ) -> E. k e. ( Base ` (Scalar ` W ) ) v = ( k ( .s ` W ) X ) )
13	12	adantrr 716	. . . . . 6 |- ( ( ph /\ ( v e. ( N ` { X } ) /\ v e. U ) ) -> E. k e. ( Base ` (Scalar ` W ) ) v = ( k ( .s ` W ) X ) )
14		simprr 756	. . . . . . . . . 10 |- ( ( ( ph /\ v e. U ) /\ ( k e. ( Base ` (Scalar ` W ) ) /\ v = ( k ( .s ` W ) X ) ) ) -> v = ( k ( .s ` W ) X ) )
15		lspdisj.e	. . . . . . . . . . . . 13 |- ( ph -> -. X e. U )
16	15	ad2antrr 725	. . . . . . . . . . . 12 |- ( ( ( ph /\ v e. U ) /\ ( k e. ( Base ` (Scalar ` W ) ) /\ v = ( k ( .s ` W ) X ) ) ) -> -. X e. U )
17		simplr 754	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ v e. U ) /\ ( k e. ( Base ` (Scalar ` W ) ) /\ v = ( k ( .s ` W ) X ) ) ) -> v e. U )
18	14, 17	eqeltrrd 2540	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ v e. U ) /\ ( k e. ( Base ` (Scalar ` W ) ) /\ v = ( k ( .s ` W ) X ) ) ) -> ( k ( .s ` W ) X ) e. U )
19		eqid 2451	. . . . . . . . . . . . . . . 16 |- ( 0g ` (Scalar ` W ) ) = ( 0g ` (Scalar ` W ) )
20		lspdisj.s	. . . . . . . . . . . . . . . 16 |- S = ( LSubSp ` W )
21	1	ad2antrr 725	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ v e. U ) /\ ( k e. ( Base ` (Scalar ` W ) ) /\ v = ( k ( .s ` W ) X ) ) ) -> W e. LVec )
22		lspdisj.u	. . . . . . . . . . . . . . . . 17 |- ( ph -> U e. S )
23	22	ad2antrr 725	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ v e. U ) /\ ( k e. ( Base ` (Scalar ` W ) ) /\ v = ( k ( .s ` W ) X ) ) ) -> U e. S )
24	4	ad2antrr 725	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ v e. U ) /\ ( k e. ( Base ` (Scalar ` W ) ) /\ v = ( k ( .s ` W ) X ) ) ) -> X e. V )
25		simprl 755	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ v e. U ) /\ ( k e. ( Base ` (Scalar ` W ) ) /\ v = ( k ( .s ` W ) X ) ) ) -> k e. ( Base ` (Scalar ` W ) ) )
26	7, 8, 5, 6, 19, 20, 21, 23, 24, 25	lssvs0or 17306	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ v e. U ) /\ ( k e. ( Base ` (Scalar ` W ) ) /\ v = ( k ( .s ` W ) X ) ) ) -> ( ( k ( .s ` W ) X ) e. U <-> ( k = ( 0g ` (Scalar ` W ) ) \/ X e. U ) ) )
27	18, 26	mpbid 210	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ v e. U ) /\ ( k e. ( Base ` (Scalar ` W ) ) /\ v = ( k ( .s ` W ) X ) ) ) -> ( k = ( 0g ` (Scalar ` W ) ) \/ X e. U ) )
28	27	orcomd 388	. . . . . . . . . . . . 13 |- ( ( ( ph /\ v e. U ) /\ ( k e. ( Base ` (Scalar ` W ) ) /\ v = ( k ( .s ` W ) X ) ) ) -> ( X e. U \/ k = ( 0g ` (Scalar ` W ) ) ) )
29	28	ord 377	. . . . . . . . . . . 12 |- ( ( ( ph /\ v e. U ) /\ ( k e. ( Base ` (Scalar ` W ) ) /\ v = ( k ( .s ` W ) X ) ) ) -> ( -. X e. U -> k = ( 0g ` (Scalar ` W ) ) ) )
30	16, 29	mpd 15	. . . . . . . . . . 11 |- ( ( ( ph /\ v e. U ) /\ ( k e. ( Base ` (Scalar ` W ) ) /\ v = ( k ( .s ` W ) X ) ) ) -> k = ( 0g ` (Scalar ` W ) ) )
31	30	oveq1d 6208	. . . . . . . . . 10 |- ( ( ( ph /\ v e. U ) /\ ( k e. ( Base ` (Scalar ` W ) ) /\ v = ( k ( .s ` W ) X ) ) ) -> ( k ( .s ` W ) X ) = ( ( 0g ` (Scalar ` W ) ) ( .s ` W ) X ) )
32	3	ad2antrr 725	. . . . . . . . . . 11 |- ( ( ( ph /\ v e. U ) /\ ( k e. ( Base ` (Scalar ` W ) ) /\ v = ( k ( .s ` W ) X ) ) ) -> W e. LMod )
33		lspdisj.o	. . . . . . . . . . . 12 |- .0. = ( 0g ` W )
34	7, 5, 8, 19, 33	lmod0vs 17096	. . . . . . . . . . 11 |- ( ( W e. LMod /\ X e. V ) -> ( ( 0g ` (Scalar ` W ) ) ( .s ` W ) X ) = .0. )
35	32, 24, 34	syl2anc 661	. . . . . . . . . 10 |- ( ( ( ph /\ v e. U ) /\ ( k e. ( Base ` (Scalar ` W ) ) /\ v = ( k ( .s ` W ) X ) ) ) -> ( ( 0g ` (Scalar ` W ) ) ( .s ` W ) X ) = .0. )
36	14, 31, 35	3eqtrd 2496	. . . . . . . . 9 |- ( ( ( ph /\ v e. U ) /\ ( k e. ( Base ` (Scalar ` W ) ) /\ v = ( k ( .s ` W ) X ) ) ) -> v = .0. )
37	36	exp32 605	. . . . . . . 8 |- ( ( ph /\ v e. U ) -> ( k e. ( Base ` (Scalar ` W ) ) -> ( v = ( k ( .s ` W ) X ) -> v = .0. ) ) )
38	37	adantrl 715	. . . . . . 7 |- ( ( ph /\ ( v e. ( N ` { X } ) /\ v e. U ) ) -> ( k e. ( Base ` (Scalar ` W ) ) -> ( v = ( k ( .s ` W ) X ) -> v = .0. ) ) )
39	38	rexlimdv 2939	. . . . . 6 |- ( ( ph /\ ( v e. ( N ` { X } ) /\ v e. U ) ) -> ( E. k e. ( Base ` (Scalar ` W ) ) v = ( k ( .s ` W ) X ) -> v = .0. ) )
40	13, 39	mpd 15	. . . . 5 |- ( ( ph /\ ( v e. ( N ` { X } ) /\ v e. U ) ) -> v = .0. )
41	40	ex 434	. . . 4 |- ( ph -> ( ( v e. ( N ` { X } ) /\ v e. U ) -> v = .0. ) )
42		elin 3640	. . . 4 |- ( v e. ( ( N ` { X } ) i^i U ) <-> ( v e. ( N ` { X } ) /\ v e. U ) )
43		elsn 3992	. . . 4 |- ( v e. { .0. } <-> v = .0. )
44	41, 42, 43	3imtr4g 270	. . 3 |- ( ph -> ( v e. ( ( N ` { X } ) i^i U ) -> v e. { .0. } ) )
45	44	ssrdv 3463	. 2 |- ( ph -> ( ( N ` { X } ) i^i U ) C_ { .0. } )
46	7, 20, 9	lspsncl 17173	. . . . 5 |- ( ( W e. LMod /\ X e. V ) -> ( N ` { X } ) e. S )
47	3, 4, 46	syl2anc 661	. . . 4 |- ( ph -> ( N ` { X } ) e. S )
48	33, 20	lss0ss 17145	. . . 4 |- ( ( W e. LMod /\ ( N ` { X } ) e. S ) -> { .0. } C_ ( N ` { X } ) )
49	3, 47, 48	syl2anc 661	. . 3 |- ( ph -> { .0. } C_ ( N ` { X } ) )
50	33, 20	lss0ss 17145	. . . 4 |- ( ( W e. LMod /\ U e. S ) -> { .0. } C_ U )
51	3, 22, 50	syl2anc 661	. . 3 |- ( ph -> { .0. } C_ U )
52	49, 51	ssind 3675	. 2 |- ( ph -> { .0. } C_ ( ( N ` { X } ) i^i U ) )
53	45, 52	eqssd 3474	1 |- ( ph -> ( ( N ` { X } ) i^i U ) = { .0. } )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   = wceq 1370   e. wcel 1758   E.wrex 2796   i^i cin 3428   C_ wss 3429   {csn 3978   ` cfv 5519  (class class class)co 6193   Basecbs 14285  Scalarcsca 14352   .scvsca 14353   0gc0g 14489   LModclmod 17063   LSubSpclss 17128   LSpanclspn 17167   LVecclvec 17298

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

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 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-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-recs 6935  df-rdg 6969  df-er 7204  df-en 7414  df-dom 7415  df-sdom 7416  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-ndx 14288  df-slot 14289  df-base 14290  df-sets 14291  df-ress 14292  df-plusg 14362  df-mulr 14363  df-0g 14491  df-mnd 15526  df-grp 15656  df-minusg 15657  df-sbg 15658  df-mgp 16706  df-ur 16718  df-rng 16762  df-oppr 16830  df-dvdsr 16848  df-unit 16849  df-invr 16879  df-drng 16949  df-lmod 17065  df-lss 17129  df-lsp 17168  df-lvec 17299

This theorem is referenced by:  lspdisjb  17322  lspdisj2  17323  lvecindp  17334