Theorem lspdisj 17898

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 17879	. . . . . . . . . 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 2457	. . . . . . . . . 10 |- (Scalar ` W ) = (Scalar ` W )
6		eqid 2457	. . . . . . . . . 10 |- ( Base ` (Scalar ` W ) ) = ( Base ` (Scalar ` W ) )
7		lspdisj.v	. . . . . . . . . 10 |- V = ( Base ` W )
8		eqid 2457	. . . . . . . . . 10 |- ( .s ` W ) = ( .s ` W )
9		lspdisj.n	. . . . . . . . . 10 |- N = ( LSpan ` W )
10	5, 6, 7, 8, 9	lspsnel 17776	. . . . . . . . 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 757	. . . . . . . . . 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 755	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ v e. U ) /\ ( k e. ( Base ` (Scalar ` W ) ) /\ v = ( k ( .s ` W ) X ) ) ) -> v e. U )
18	14, 17	eqeltrrd 2546	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ v e. U ) /\ ( k e. ( Base ` (Scalar ` W ) ) /\ v = ( k ( .s ` W ) X ) ) ) -> ( k ( .s ` W ) X ) e. U )
19		eqid 2457	. . . . . . . . . . . . . . . 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 756	. . . . . . . . . . . . . . . 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 17883	. . . . . . . . . . . . . . 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 6311	. . . . . . . . . 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 17672	. . . . . . . . . . 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 2502	. . . . . . . . 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 2947	. . . . . 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 3683	. . . 4 |- ( v e. ( ( N ` { X } ) i^i U ) <-> ( v e. ( N ` { X } ) /\ v e. U ) )
43		elsn 4046	. . . 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 3505	. 2 |- ( ph -> ( ( N ` { X } ) i^i U ) C_ { .0. } )
46	7, 20, 9	lspsncl 17750	. . . . 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 17722	. . . 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 17722	. . . 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 3718	. 2 |- ( ph -> { .0. } C_ ( ( N ` { X } ) i^i U ) )
53	45, 52	eqssd 3516	1 |- ( ph -> ( ( N ` { X } ) i^i U ) = { .0. } )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   = wceq 1395   e. wcel 1819   E.wrex 2808   i^i cin 3470   C_ wss 3471   {csn 4032   ` cfv 5594  (class class class)co 6296   Basecbs 14644  Scalarcsca 14715   .scvsca 14716   0gc0g 14857   LModclmod 17639   LSubSpclss 17705   LSpanclspn 17744   LVecclvec 17875

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-tpos 6973  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-ress 14651  df-plusg 14725  df-mulr 14726  df-0g 14859  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-grp 16184  df-minusg 16185  df-sbg 16186  df-mgp 17269  df-ur 17281  df-ring 17327  df-oppr 17399  df-dvdsr 17417  df-unit 17418  df-invr 17448  df-drng 17525  df-lmod 17641  df-lss 17706  df-lsp 17745  df-lvec 17876

This theorem is referenced by:  lspdisjb  17899  lspdisj2  17900  lvecindp  17911