Theorem lspsnat 17203

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 24935 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 3641	. . . . . 6 |- ( ( U \ { .0. } ) =/= (/) <-> E. x x e. ( U \ { .0. } ) )
2		simprl 755	. . . . . . . . 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 991	. . . . . . . . . . 11 |- ( ( ( W e. LVec /\ U e. S /\ X e. V ) /\ ( U C_ ( N ` { X } ) /\ x e. ( U \ { .0. } ) ) ) -> W e. LVec )
6		lveclmod 17164	. . . . . . . . . . 11 |- ( W e. LVec -> W e. LMod )
7	5, 6	syl 16	. . . . . . . . . 10 |- ( ( ( W e. LVec /\ U e. S /\ X e. V ) /\ ( U C_ ( N ` { X } ) /\ x e. ( U \ { .0. } ) ) ) -> W e. LMod )
8		simpl2 992	. . . . . . . . . 10 |- ( ( ( W e. LVec /\ U e. S /\ X e. V ) /\ ( U C_ ( N ` { X } ) /\ x e. ( U \ { .0. } ) ) ) -> U e. S )
9		simprr 756	. . . . . . . . . . . . 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 3335	. . . . . . . . . . . 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 17054	. . . . . . . . . . 11 |- ( ( ( W e. LVec /\ U e. S /\ X e. V ) /\ ( U C_ ( N ` { X } ) /\ x e. ( U \ { .0. } ) ) ) -> ( N ` { x } ) C_ U )
12		0ss 3661	. . . . . . . . . . . . . 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 993	. . . . . . . . . . . . 13 |- ( ( ( W e. LVec /\ U e. S /\ X e. V ) /\ ( U C_ ( N ` { X } ) /\ x e. ( U \ { .0. } ) ) ) -> X e. V )
15		ssdif 3486	. . . . . . . . . . . . . . . 16 |- ( U C_ ( N ` { X } ) -> ( U \ { .0. } ) C_ ( ( N ` { X } ) \ { .0. } ) )
16	15	ad2antrl 727	. . . . . . . . . . . . . . 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 3352	. . . . . . . . . . . . . 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 3495	. . . . . . . . . . . . . . . . . 18 |- ( (/) u. { X } ) = ( { X } u. (/) )
19		un0 3657	. . . . . . . . . . . . . . . . . 18 |- ( { X } u. (/) ) = { X }
20	18, 19	eqtri 2458	. . . . . . . . . . . . . . . . 17 |- ( (/) u. { X } ) = { X }
21	20	fveq2i 5689	. . . . . . . . . . . . . . . 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 17067	. . . . . . . . . . . . . . . 16 |- ( W e. LMod -> ( N ` (/) ) = { .0. } )
25	7, 24	syl 16	. . . . . . . . . . . . . . 15 |- ( ( ( W e. LVec /\ U e. S /\ X e. V ) /\ ( U C_ ( N ` { X } ) /\ x e. ( U \ { .0. } ) ) ) -> ( N ` (/) ) = { .0. } )
26	22, 25	difeq12d 3470	. . . . . . . . . . . . . 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 2515	. . . . . . . . . . . . 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 17201	. . . . . . . . . . . . 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 1220	. . . . . . . . . . . 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 3495	. . . . . . . . . . . . . 14 |- ( (/) u. { x } ) = ( { x } u. (/) )
32		un0 3657	. . . . . . . . . . . . . 14 |- ( { x } u. (/) ) = { x }
33	31, 32	eqtri 2458	. . . . . . . . . . . . 13 |- ( (/) u. { x } ) = { x }
34	33	fveq2i 5689	. . . . . . . . . . . 12 |- ( N ` ( (/) u. { x } ) ) = ( N ` { x } )
35	30, 34	syl6eleq 2528	. . . . . . . . . . 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 3352	. . . . . . . . . 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 17054	. . . . . . . . 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 3368	. . . . . . . 8 |- ( ( ( W e. LVec /\ U e. S /\ X e. V ) /\ ( U C_ ( N ` { X } ) /\ x e. ( U \ { .0. } ) ) ) -> U = ( N ` { X } ) )
39	38	expr 615	. . . . . . 7 |- ( ( ( W e. LVec /\ U e. S /\ X e. V ) /\ U C_ ( N ` { X } ) ) -> ( x e. ( U \ { .0. } ) -> U = ( N ` { X } ) ) )
40	39	exlimdv 1690	. . . . . 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 217	. . . . 5 |- ( ( ( W e. LVec /\ U e. S /\ X e. V ) /\ U C_ ( N ` { X } ) ) -> ( ( U \ { .0. } ) =/= (/) -> U = ( N ` { X } ) ) )
42	41	necon1bd 2674	. . . 4 |- ( ( ( W e. LVec /\ U e. S /\ X e. V ) /\ U C_ ( N ` { X } ) ) -> ( -. U = ( N ` { X } ) -> ( U \ { .0. } ) = (/) ) )
43		ssdif0 3732	. . . 4 |- ( U C_ { .0. } <-> ( U \ { .0. } ) = (/) )
44	42, 43	syl6ibr 227	. . 3 |- ( ( ( W e. LVec /\ U e. S /\ X e. V ) /\ U C_ ( N ` { X } ) ) -> ( -. U = ( N ` { X } ) -> U C_ { .0. } ) )
45		simpl1 991	. . . . 5 |- ( ( ( W e. LVec /\ U e. S /\ X e. V ) /\ U C_ ( N ` { X } ) ) -> W e. LVec )
46	45, 6	syl 16	. . . 4 |- ( ( ( W e. LVec /\ U e. S /\ X e. V ) /\ U C_ ( N ` { X } ) ) -> W e. LMod )
47		simpl2 992	. . . 4 |- ( ( ( W e. LVec /\ U e. S /\ X e. V ) /\ U C_ ( N ` { X } ) ) -> U e. S )
48	23, 3	lssle0 17008	. . . 4 |- ( ( W e. LMod /\ U e. S ) -> ( U C_ { .0. } <-> U = { .0. } ) )
49	46, 47, 48	syl2anc 661	. . 3 |- ( ( ( W e. LVec /\ U e. S /\ X e. V ) /\ U C_ ( N ` { X } ) ) -> ( U C_ { .0. } <-> U = { .0. } ) )
50	44, 49	sylibd 214	. 2 |- ( ( ( W e. LVec /\ U e. S /\ X e. V ) /\ U C_ ( N ` { X } ) ) -> ( -. U = ( N ` { X } ) -> U = { .0. } ) )
51	50	orrd 378	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 184   \/ wo 368   /\ wa 369   /\ w3a 965   = wceq 1369   E.wex 1586   e. wcel 1756   =/= wne 2601   \ cdif 3320   u. cun 3321   C_ wss 3323   (/)c0 3632   {csn 3872   ` cfv 5413   Basecbs 14166   0gc0g 14370   LModclmod 16926   LSubSpclss 16990   LSpanclspn 17029   LVecclvec 17160

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-tpos 6740  df-recs 6824  df-rdg 6858  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-2 10372  df-3 10373  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-mulr 14244  df-0g 14372  df-mnd 15407  df-grp 15536  df-minusg 15537  df-sbg 15538  df-cmn 16270  df-abl 16271  df-mgp 16580  df-ur 16592  df-rng 16635  df-oppr 16703  df-dvdsr 16721  df-unit 16722  df-invr 16752  df-drng 16812  df-lmod 16928  df-lss 16991  df-lsp 17030  df-lvec 17161

This theorem is referenced by:  lspsncv0  17204  lsatcmp  32488  dihlspsnssN  34817  dihlspsnat  34818