Theorem lspsnat 18423

Description: There is no subspace strictly between the zero subspace and the span of a vector (i.e. a 1-dimensional subspace is an atom). (h1datomi 27290 analog.) (Contributed by NM, 20-Apr-2014.) (Proof shortened by Mario Carneiro, 22-Jun-2014.)

Hypotheses

Ref	Expression
lspsnat.v	|- V = ( Base ` W )
lspsnat.z	|- .0. = ( 0g ` W )
lspsnat.s	|- S = ( LSubSp ` W )
lspsnat.n	|- N = ( LSpan ` W )

Assertion

Ref	Expression
lspsnat	|- ( ( ( W e. LVec /\ U e. S /\ X e. V ) /\ U C_ ( N ` { X } ) ) -> ( U = ( N ` { X } ) \/ U = { .0. } ) )

Proof of Theorem lspsnat

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		n0 3753	. . . . . 6 |- ( ( U \ { .0. } ) =/= (/) <-> E. x x e. ( U \ { .0. } ) )
2		simprl 769	. . . . . . . . 9 |- ( ( ( W e. LVec /\ U e. S /\ X e. V ) /\ ( U C_ ( N ` { X } ) /\ x e. ( U \ { .0. } ) ) ) -> U C_ ( N ` { X } ) )
3		lspsnat.s	. . . . . . . . . 10 |- S = ( LSubSp ` W )
4		lspsnat.n	. . . . . . . . . 10 |- N = ( LSpan ` W )
5		simpl1 1017	. . . . . . . . . . 11 |- ( ( ( W e. LVec /\ U e. S /\ X e. V ) /\ ( U C_ ( N ` { X } ) /\ x e. ( U \ { .0. } ) ) ) -> W e. LVec )
6		lveclmod 18384	. . . . . . . . . . 11 |- ( W e. LVec -> W e. LMod )
7	5, 6	syl 17	. . . . . . . . . 10 |- ( ( ( W e. LVec /\ U e. S /\ X e. V ) /\ ( U C_ ( N ` { X } ) /\ x e. ( U \ { .0. } ) ) ) -> W e. LMod )
8		simpl2 1018	. . . . . . . . . 10 |- ( ( ( W e. LVec /\ U e. S /\ X e. V ) /\ ( U C_ ( N ` { X } ) /\ x e. ( U \ { .0. } ) ) ) -> U e. S )
9		simprr 771	. . . . . . . . . . . . 13 |- ( ( ( W e. LVec /\ U e. S /\ X e. V ) /\ ( U C_ ( N ` { X } ) /\ x e. ( U \ { .0. } ) ) ) -> x e. ( U \ { .0. } ) )
10	9	eldifad 3428	. . . . . . . . . . . 12 |- ( ( ( W e. LVec /\ U e. S /\ X e. V ) /\ ( U C_ ( N ` { X } ) /\ x e. ( U \ { .0. } ) ) ) -> x e. U )
11	3, 4, 7, 8, 10	lspsnel5a 18274	. . . . . . . . . . 11 |- ( ( ( W e. LVec /\ U e. S /\ X e. V ) /\ ( U C_ ( N ` { X } ) /\ x e. ( U \ { .0. } ) ) ) -> ( N ` { x } ) C_ U )
12		0ss 3775	. . . . . . . . . . . . . 14 |- (/) C_ V
13	12	a1i 11	. . . . . . . . . . . . 13 |- ( ( ( W e. LVec /\ U e. S /\ X e. V ) /\ ( U C_ ( N ` { X } ) /\ x e. ( U \ { .0. } ) ) ) -> (/) C_ V )
14		simpl3 1019	. . . . . . . . . . . . 13 |- ( ( ( W e. LVec /\ U e. S /\ X e. V ) /\ ( U C_ ( N ` { X } ) /\ x e. ( U \ { .0. } ) ) ) -> X e. V )
15		ssdif 3580	. . . . . . . . . . . . . . . 16 |- ( U C_ ( N ` { X } ) -> ( U \ { .0. } ) C_ ( ( N ` { X } ) \ { .0. } ) )
16	15	ad2antrl 739	. . . . . . . . . . . . . . 15 |- ( ( ( W e. LVec /\ U e. S /\ X e. V ) /\ ( U C_ ( N ` { X } ) /\ x e. ( U \ { .0. } ) ) ) -> ( U \ { .0. } ) C_ ( ( N ` { X } ) \ { .0. } ) )
17	16, 9	sseldd 3445	. . . . . . . . . . . . . 14 |- ( ( ( W e. LVec /\ U e. S /\ X e. V ) /\ ( U C_ ( N ` { X } ) /\ x e. ( U \ { .0. } ) ) ) -> x e. ( ( N ` { X } ) \ { .0. } ) )
18		uncom 3590	. . . . . . . . . . . . . . . . . 18 |- ( (/) u. { X } ) = ( { X } u. (/) )
19		un0 3771	. . . . . . . . . . . . . . . . . 18 |- ( { X } u. (/) ) = { X }
20	18, 19	eqtri 2484	. . . . . . . . . . . . . . . . 17 |- ( (/) u. { X } ) = { X }
21	20	fveq2i 5895	. . . . . . . . . . . . . . . 16 |- ( N ` ( (/) u. { X } ) ) = ( N ` { X } )
22	21	a1i 11	. . . . . . . . . . . . . . 15 |- ( ( ( W e. LVec /\ U e. S /\ X e. V ) /\ ( U C_ ( N ` { X } ) /\ x e. ( U \ { .0. } ) ) ) -> ( N ` ( (/) u. { X } ) ) = ( N ` { X } ) )
23		lspsnat.z	. . . . . . . . . . . . . . . . 17 |- .0. = ( 0g ` W )
24	23, 4	lsp0 18287	. . . . . . . . . . . . . . . 16 |- ( W e. LMod -> ( N ` (/) ) = { .0. } )
25	7, 24	syl 17	. . . . . . . . . . . . . . 15 |- ( ( ( W e. LVec /\ U e. S /\ X e. V ) /\ ( U C_ ( N ` { X } ) /\ x e. ( U \ { .0. } ) ) ) -> ( N ` (/) ) = { .0. } )
26	22, 25	difeq12d 3564	. . . . . . . . . . . . . 14 |- ( ( ( W e. LVec /\ U e. S /\ X e. V ) /\ ( U C_ ( N ` { X } ) /\ x e. ( U \ { .0. } ) ) ) -> ( ( N ` ( (/) u. { X } ) ) \ ( N ` (/) ) ) = ( ( N ` { X } ) \ { .0. } ) )
27	17, 26	eleqtrrd 2543	. . . . . . . . . . . . 13 |- ( ( ( W e. LVec /\ U e. S /\ X e. V ) /\ ( U C_ ( N ` { X } ) /\ x e. ( U \ { .0. } ) ) ) -> x e. ( ( N ` ( (/) u. { X } ) ) \ ( N ` (/) ) ) )
28		lspsnat.v	. . . . . . . . . . . . . 14 |- V = ( Base ` W )
29	28, 3, 4	lspsolv 18421	. . . . . . . . . . . . 13 |- ( ( W e. LVec /\ ( (/) C_ V /\ X e. V /\ x e. ( ( N ` ( (/) u. { X } ) ) \ ( N ` (/) ) ) ) ) -> X e. ( N ` ( (/) u. { x } ) ) )
30	5, 13, 14, 27, 29	syl13anc 1278	. . . . . . . . . . . 12 |- ( ( ( W e. LVec /\ U e. S /\ X e. V ) /\ ( U C_ ( N ` { X } ) /\ x e. ( U \ { .0. } ) ) ) -> X e. ( N ` ( (/) u. { x } ) ) )
31		uncom 3590	. . . . . . . . . . . . . 14 |- ( (/) u. { x } ) = ( { x } u. (/) )
32		un0 3771	. . . . . . . . . . . . . 14 |- ( { x } u. (/) ) = { x }
33	31, 32	eqtri 2484	. . . . . . . . . . . . 13 |- ( (/) u. { x } ) = { x }
34	33	fveq2i 5895	. . . . . . . . . . . 12 |- ( N ` ( (/) u. { x } ) ) = ( N ` { x } )
35	30, 34	syl6eleq 2550	. . . . . . . . . . 11 |- ( ( ( W e. LVec /\ U e. S /\ X e. V ) /\ ( U C_ ( N ` { X } ) /\ x e. ( U \ { .0. } ) ) ) -> X e. ( N ` { x } ) )
36	11, 35	sseldd 3445	. . . . . . . . . 10 |- ( ( ( W e. LVec /\ U e. S /\ X e. V ) /\ ( U C_ ( N ` { X } ) /\ x e. ( U \ { .0. } ) ) ) -> X e. U )
37	3, 4, 7, 8, 36	lspsnel5a 18274	. . . . . . . . 9 |- ( ( ( W e. LVec /\ U e. S /\ X e. V ) /\ ( U C_ ( N ` { X } ) /\ x e. ( U \ { .0. } ) ) ) -> ( N ` { X } ) C_ U )
38	2, 37	eqssd 3461	. . . . . . . 8 |- ( ( ( W e. LVec /\ U e. S /\ X e. V ) /\ ( U C_ ( N ` { X } ) /\ x e. ( U \ { .0. } ) ) ) -> U = ( N ` { X } ) )
39	38	expr 624	. . . . . . 7 |- ( ( ( W e. LVec /\ U e. S /\ X e. V ) /\ U C_ ( N ` { X } ) ) -> ( x e. ( U \ { .0. } ) -> U = ( N ` { X } ) ) )
40	39	exlimdv 1790	. . . . . 6 |- ( ( ( W e. LVec /\ U e. S /\ X e. V ) /\ U C_ ( N ` { X } ) ) -> ( E. x x e. ( U \ { .0. } ) -> U = ( N ` { X } ) ) )
41	1, 40	syl5bi 225	. . . . 5 |- ( ( ( W e. LVec /\ U e. S /\ X e. V ) /\ U C_ ( N ` { X } ) ) -> ( ( U \ { .0. } ) =/= (/) -> U = ( N ` { X } ) ) )
42	41	necon1bd 2654	. . . 4 |- ( ( ( W e. LVec /\ U e. S /\ X e. V ) /\ U C_ ( N ` { X } ) ) -> ( -. U = ( N ` { X } ) -> ( U \ { .0. } ) = (/) ) )
43		ssdif0 3835	. . . 4 |- ( U C_ { .0. } <-> ( U \ { .0. } ) = (/) )
44	42, 43	syl6ibr 235	. . 3 |- ( ( ( W e. LVec /\ U e. S /\ X e. V ) /\ U C_ ( N ` { X } ) ) -> ( -. U = ( N ` { X } ) -> U C_ { .0. } ) )
45		simpl1 1017	. . . . 5 |- ( ( ( W e. LVec /\ U e. S /\ X e. V ) /\ U C_ ( N ` { X } ) ) -> W e. LVec )
46	45, 6	syl 17	. . . 4 |- ( ( ( W e. LVec /\ U e. S /\ X e. V ) /\ U C_ ( N ` { X } ) ) -> W e. LMod )
47		simpl2 1018	. . . 4 |- ( ( ( W e. LVec /\ U e. S /\ X e. V ) /\ U C_ ( N ` { X } ) ) -> U e. S )
48	23, 3	lssle0 18228	. . . 4 |- ( ( W e. LMod /\ U e. S ) -> ( U C_ { .0. } <-> U = { .0. } ) )
49	46, 47, 48	syl2anc 671	. . 3 |- ( ( ( W e. LVec /\ U e. S /\ X e. V ) /\ U C_ ( N ` { X } ) ) -> ( U C_ { .0. } <-> U = { .0. } ) )
50	44, 49	sylibd 222	. 2 |- ( ( ( W e. LVec /\ U e. S /\ X e. V ) /\ U C_ ( N ` { X } ) ) -> ( -. U = ( N ` { X } ) -> U = { .0. } ) )
51	50	orrd 384	1 |- ( ( ( W e. LVec /\ U e. S /\ X e. V ) /\ U C_ ( N ` { X } ) ) -> ( U = ( N ` { X } ) \/ U = { .0. } ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 189   \/ wo 374   /\ wa 375   /\ w3a 991   = wceq 1455   E.wex 1674   e. wcel 1898   =/= wne 2633   \ cdif 3413   u. cun 3414   C_ wss 3416   (/)c0 3743   {csn 3980   ` cfv 5605   Basecbs 15176   0gc0g 15393   LModclmod 18146   LSubSpclss 18210   LSpanclspn 18249   LVecclvec 18380

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4531  ax-sep 4541  ax-nul 4550  ax-pow 4598  ax-pr 4656  ax-un 6615  ax-cnex 9626  ax-resscn 9627  ax-1cn 9628  ax-icn 9629  ax-addcl 9630  ax-addrcl 9631  ax-mulcl 9632  ax-mulrcl 9633  ax-mulcom 9634  ax-addass 9635  ax-mulass 9636  ax-distr 9637  ax-i2m1 9638  ax-1ne0 9639  ax-1rid 9640  ax-rnegex 9641  ax-rrecex 9642  ax-cnre 9643  ax-pre-lttri 9644  ax-pre-lttrn 9645  ax-pre-ltadd 9646  ax-pre-mulgt0 9647

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-int 4249  df-iun 4294  df-br 4419  df-opab 4478  df-mpt 4479  df-tr 4514  df-eprel 4767  df-id 4771  df-po 4777  df-so 4778  df-fr 4815  df-we 4817  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-res 4868  df-ima 4869  df-pred 5403  df-ord 5449  df-on 5450  df-lim 5451  df-suc 5452  df-iota 5569  df-fun 5607  df-fn 5608  df-f 5609  df-f1 5610  df-fo 5611  df-f1o 5612  df-fv 5613  df-riota 6282  df-ov 6323  df-oprab 6324  df-mpt2 6325  df-om 6725  df-1st 6825  df-2nd 6826  df-tpos 7004  df-wrecs 7059  df-recs 7121  df-rdg 7159  df-er 7394  df-en 7601  df-dom 7602  df-sdom 7603  df-pnf 9708  df-mnf 9709  df-xr 9710  df-ltxr 9711  df-le 9712  df-sub 9893  df-neg 9894  df-nn 10643  df-2 10701  df-3 10702  df-ndx 15179  df-slot 15180  df-base 15181  df-sets 15182  df-ress 15183  df-plusg 15258  df-mulr 15259  df-0g 15395  df-mgm 16543  df-sgrp 16582  df-mnd 16592  df-grp 16728  df-minusg 16729  df-sbg 16730  df-cmn 17487  df-abl 17488  df-mgp 17779  df-ur 17791  df-ring 17837  df-oppr 17906  df-dvdsr 17924  df-unit 17925  df-invr 17955  df-drng 18032  df-lmod 18148  df-lss 18211  df-lsp 18250  df-lvec 18381

This theorem is referenced by:  lspsncv0  18424  lsatcmp  32615  dihlspsnssN  34946  dihlspsnat  34947