Theorem lspdisj 19125

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 19106	. . . . . . . . . 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 2622	. . . . . . . . . 10 |- (Scalar ` W ) = (Scalar ` W )
6		eqid 2622	. . . . . . . . . 10 |- ( Base ` (Scalar ` W ) ) = ( Base ` (Scalar ` W ) )
7		lspdisj.v	. . . . . . . . . 10 |- V = ( Base ` W )
8		eqid 2622	. . . . . . . . . 10 |- ( .s ` W ) = ( .s ` W )
9		lspdisj.n	. . . . . . . . . 10 |- N = ( LSpan ` W )
10	5, 6, 7, 8, 9	lspsnel 19003	. . . . . . . . 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 693	. . . . . . . 8 |- ( ph -> ( v e. ( N ` { X } ) <-> E. k e. ( Base ` (Scalar ` W ) ) v = ( k ( .s ` W ) X ) ) )
12	11	biimpa 501	. . . . . . 7 |- ( ( ph /\ v e. ( N ` { X } ) ) -> E. k e. ( Base ` (Scalar ` W ) ) v = ( k ( .s ` W ) X ) )
13	12	adantrr 753	. . . . . 6 |- ( ( ph /\ ( v e. ( N ` { X } ) /\ v e. U ) ) -> E. k e. ( Base ` (Scalar ` W ) ) v = ( k ( .s ` W ) X ) )
14		simprr 796	. . . . . . . . . 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 762	. . . . . . . . . . . 12 |- ( ( ( ph /\ v e. U ) /\ ( k e. ( Base ` (Scalar ` W ) ) /\ v = ( k ( .s ` W ) X ) ) ) -> -. X e. U )
17		simplr 792	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ v e. U ) /\ ( k e. ( Base ` (Scalar ` W ) ) /\ v = ( k ( .s ` W ) X ) ) ) -> v e. U )
18	14, 17	eqeltrrd 2702	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ v e. U ) /\ ( k e. ( Base ` (Scalar ` W ) ) /\ v = ( k ( .s ` W ) X ) ) ) -> ( k ( .s ` W ) X ) e. U )
19		eqid 2622	. . . . . . . . . . . . . . . 16 |- ( 0g ` (Scalar ` W ) ) = ( 0g ` (Scalar ` W ) )
20		lspdisj.s	. . . . . . . . . . . . . . . 16 |- S = ( LSubSp ` W )
21	1	ad2antrr 762	. . . . . . . . . . . . . . . 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 762	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ v e. U ) /\ ( k e. ( Base ` (Scalar ` W ) ) /\ v = ( k ( .s ` W ) X ) ) ) -> U e. S )
24	4	ad2antrr 762	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ v e. U ) /\ ( k e. ( Base ` (Scalar ` W ) ) /\ v = ( k ( .s ` W ) X ) ) ) -> X e. V )
25		simprl 794	. . . . . . . . . . . . . . . 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 19110	. . . . . . . . . . . . . . 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 222	. . . . . . . . . . . . . 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 403	. . . . . . . . . . . . 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 392	. . . . . . . . . . . 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 6665	. . . . . . . . . 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 762	. . . . . . . . . . 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 18896	. . . . . . . . . . 11 |- ( ( W e. LMod /\ X e. V ) -> ( ( 0g ` (Scalar ` W ) ) ( .s ` W ) X ) = .0. )
35	32, 24, 34	syl2anc 693	. . . . . . . . . 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 2660	. . . . . . . . 9 |- ( ( ( ph /\ v e. U ) /\ ( k e. ( Base ` (Scalar ` W ) ) /\ v = ( k ( .s ` W ) X ) ) ) -> v = .0. )
37	36	exp32 631	. . . . . . . 8 |- ( ( ph /\ v e. U ) -> ( k e. ( Base ` (Scalar ` W ) ) -> ( v = ( k ( .s ` W ) X ) -> v = .0. ) ) )
38	37	adantrl 752	. . . . . . 7 |- ( ( ph /\ ( v e. ( N ` { X } ) /\ v e. U ) ) -> ( k e. ( Base ` (Scalar ` W ) ) -> ( v = ( k ( .s ` W ) X ) -> v = .0. ) ) )
39	38	rexlimdv 3030	. . . . . 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 450	. . . 4 |- ( ph -> ( ( v e. ( N ` { X } ) /\ v e. U ) -> v = .0. ) )
42		elin 3796	. . . 4 |- ( v e. ( ( N ` { X } ) i^i U ) <-> ( v e. ( N ` { X } ) /\ v e. U ) )
43		velsn 4193	. . . 4 |- ( v e. { .0. } <-> v = .0. )
44	41, 42, 43	3imtr4g 285	. . 3 |- ( ph -> ( v e. ( ( N ` { X } ) i^i U ) -> v e. { .0. } ) )
45	44	ssrdv 3609	. 2 |- ( ph -> ( ( N ` { X } ) i^i U ) C_ { .0. } )
46	7, 20, 9	lspsncl 18977	. . . . 5 |- ( ( W e. LMod /\ X e. V ) -> ( N ` { X } ) e. S )
47	3, 4, 46	syl2anc 693	. . . 4 |- ( ph -> ( N ` { X } ) e. S )
48	33, 20	lss0ss 18949	. . . 4 |- ( ( W e. LMod /\ ( N ` { X } ) e. S ) -> { .0. } C_ ( N ` { X } ) )
49	3, 47, 48	syl2anc 693	. . 3 |- ( ph -> { .0. } C_ ( N ` { X } ) )
50	33, 20	lss0ss 18949	. . . 4 |- ( ( W e. LMod /\ U e. S ) -> { .0. } C_ U )
51	3, 22, 50	syl2anc 693	. . 3 |- ( ph -> { .0. } C_ U )
52	49, 51	ssind 3837	. 2 |- ( ph -> { .0. } C_ ( ( N ` { X } ) i^i U ) )
53	45, 52	eqssd 3620	1 |- ( ph -> ( ( N ` { X } ) i^i U ) = { .0. } )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   i^i cin 3573   C_ wss 3574   {csn 4177   ` cfv 5888  (class class class)co 6650   Basecbs 15857  Scalarcsca 15944   .scvsca 15945   0gc0g 16100   LModclmod 18863   LSubSpclss 18932   LSpanclspn 18971   LVecclvec 19102

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mgp 18490  df-ur 18502  df-ring 18549  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-drng 18749  df-lmod 18865  df-lss 18933  df-lsp 18972  df-lvec 19103

This theorem is referenced by:  lspdisjb  19126  lspdisj2  19127  lvecindp  19138