Theorem lspdisj 18426

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 18407	. . . . . . . . . 10 |- ( W e. LVec -> W e. LMod )
3	1, 2	syl 17	. . . . . . . . 9 |- ( ph -> W e. LMod )
4		lspdisj.x	. . . . . . . . 9 |- ( ph -> X e. V )
5		eqid 2471	. . . . . . . . . 10 |- (Scalar ` W ) = (Scalar ` W )
6		eqid 2471	. . . . . . . . . 10 |- ( Base ` (Scalar ` W ) ) = ( Base ` (Scalar ` W ) )
7		lspdisj.v	. . . . . . . . . 10 |- V = ( Base ` W )
8		eqid 2471	. . . . . . . . . 10 |- ( .s ` W ) = ( .s ` W )
9		lspdisj.n	. . . . . . . . . 10 |- N = ( LSpan ` W )
10	5, 6, 7, 8, 9	lspsnel 18304	. . . . . . . . 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 673	. . . . . . . 8 |- ( ph -> ( v e. ( N ` { X } ) <-> E. k e. ( Base ` (Scalar ` W ) ) v = ( k ( .s ` W ) X ) ) )
12	11	biimpa 492	. . . . . . 7 |- ( ( ph /\ v e. ( N ` { X } ) ) -> E. k e. ( Base ` (Scalar ` W ) ) v = ( k ( .s ` W ) X ) )
13	12	adantrr 731	. . . . . 6 |- ( ( ph /\ ( v e. ( N ` { X } ) /\ v e. U ) ) -> E. k e. ( Base ` (Scalar ` W ) ) v = ( k ( .s ` W ) X ) )
14		simprr 774	. . . . . . . . . 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 740	. . . . . . . . . . . 12 |- ( ( ( ph /\ v e. U ) /\ ( k e. ( Base ` (Scalar ` W ) ) /\ v = ( k ( .s ` W ) X ) ) ) -> -. X e. U )
17		simplr 770	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ v e. U ) /\ ( k e. ( Base ` (Scalar ` W ) ) /\ v = ( k ( .s ` W ) X ) ) ) -> v e. U )
18	14, 17	eqeltrrd 2550	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ v e. U ) /\ ( k e. ( Base ` (Scalar ` W ) ) /\ v = ( k ( .s ` W ) X ) ) ) -> ( k ( .s ` W ) X ) e. U )
19		eqid 2471	. . . . . . . . . . . . . . . 16 |- ( 0g ` (Scalar ` W ) ) = ( 0g ` (Scalar ` W ) )
20		lspdisj.s	. . . . . . . . . . . . . . . 16 |- S = ( LSubSp ` W )
21	1	ad2antrr 740	. . . . . . . . . . . . . . . 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 740	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ v e. U ) /\ ( k e. ( Base ` (Scalar ` W ) ) /\ v = ( k ( .s ` W ) X ) ) ) -> U e. S )
24	4	ad2antrr 740	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ v e. U ) /\ ( k e. ( Base ` (Scalar ` W ) ) /\ v = ( k ( .s ` W ) X ) ) ) -> X e. V )
25		simprl 772	. . . . . . . . . . . . . . . 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 18411	. . . . . . . . . . . . . . 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 215	. . . . . . . . . . . . . 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 395	. . . . . . . . . . . . 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 384	. . . . . . . . . . . 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 6323	. . . . . . . . . 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 740	. . . . . . . . . . 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 18202	. . . . . . . . . . 11 |- ( ( W e. LMod /\ X e. V ) -> ( ( 0g ` (Scalar ` W ) ) ( .s ` W ) X ) = .0. )
35	32, 24, 34	syl2anc 673	. . . . . . . . . 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 2509	. . . . . . . . 9 |- ( ( ( ph /\ v e. U ) /\ ( k e. ( Base ` (Scalar ` W ) ) /\ v = ( k ( .s ` W ) X ) ) ) -> v = .0. )
37	36	exp32 616	. . . . . . . 8 |- ( ( ph /\ v e. U ) -> ( k e. ( Base ` (Scalar ` W ) ) -> ( v = ( k ( .s ` W ) X ) -> v = .0. ) ) )
38	37	adantrl 730	. . . . . . 7 |- ( ( ph /\ ( v e. ( N ` { X } ) /\ v e. U ) ) -> ( k e. ( Base ` (Scalar ` W ) ) -> ( v = ( k ( .s ` W ) X ) -> v = .0. ) ) )
39	38	rexlimdv 2870	. . . . . 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 441	. . . 4 |- ( ph -> ( ( v e. ( N ` { X } ) /\ v e. U ) -> v = .0. ) )
42		elin 3608	. . . 4 |- ( v e. ( ( N ` { X } ) i^i U ) <-> ( v e. ( N ` { X } ) /\ v e. U ) )
43		elsn 3973	. . . 4 |- ( v e. { .0. } <-> v = .0. )
44	41, 42, 43	3imtr4g 278	. . 3 |- ( ph -> ( v e. ( ( N ` { X } ) i^i U ) -> v e. { .0. } ) )
45	44	ssrdv 3424	. 2 |- ( ph -> ( ( N ` { X } ) i^i U ) C_ { .0. } )
46	7, 20, 9	lspsncl 18278	. . . . 5 |- ( ( W e. LMod /\ X e. V ) -> ( N ` { X } ) e. S )
47	3, 4, 46	syl2anc 673	. . . 4 |- ( ph -> ( N ` { X } ) e. S )
48	33, 20	lss0ss 18250	. . . 4 |- ( ( W e. LMod /\ ( N ` { X } ) e. S ) -> { .0. } C_ ( N ` { X } ) )
49	3, 47, 48	syl2anc 673	. . 3 |- ( ph -> { .0. } C_ ( N ` { X } ) )
50	33, 20	lss0ss 18250	. . . 4 |- ( ( W e. LMod /\ U e. S ) -> { .0. } C_ U )
51	3, 22, 50	syl2anc 673	. . 3 |- ( ph -> { .0. } C_ U )
52	49, 51	ssind 3647	. 2 |- ( ph -> { .0. } C_ ( ( N ` { X } ) i^i U ) )
53	45, 52	eqssd 3435	1 |- ( ph -> ( ( N ` { X } ) i^i U ) = { .0. } )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 189   \/ wo 375   /\ wa 376   = wceq 1452   e. wcel 1904   E.wrex 2757   i^i cin 3389   C_ wss 3390   {csn 3959   ` cfv 5589  (class class class)co 6308   Basecbs 15199  Scalarcsca 15271   .scvsca 15272   0gc0g 15416   LModclmod 18169   LSubSpclss 18233   LSpanclspn 18272   LVecclvec 18403

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-tpos 6991  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-3 10691  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-0g 15418  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-grp 16751  df-minusg 16752  df-sbg 16753  df-mgp 17802  df-ur 17814  df-ring 17860  df-oppr 17929  df-dvdsr 17947  df-unit 17948  df-invr 17978  df-drng 18055  df-lmod 18171  df-lss 18234  df-lsp 18273  df-lvec 18404

This theorem is referenced by:  lspdisjb  18427  lspdisj2  18428  lvecindp  18439