Theorem lspsnat 17352

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 25156 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 3757	. . . . . 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 17313	. . . . . . . . . . 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 3451	. . . . . . . . . . . 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 17203	. . . . . . . . . . 11 |- ( ( ( W e. LVec /\ U e. S /\ X e. V ) /\ ( U C_ ( N ` { X } ) /\ x e. ( U \ { .0. } ) ) ) -> ( N ` { x } ) C_ U )
12		0ss 3777	. . . . . . . . . . . . . 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 3602	. . . . . . . . . . . . . . . 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 3468	. . . . . . . . . . . . . 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 3611	. . . . . . . . . . . . . . . . . 18 |- ( (/) u. { X } ) = ( { X } u. (/) )
19		un0 3773	. . . . . . . . . . . . . . . . . 18 |- ( { X } u. (/) ) = { X }
20	18, 19	eqtri 2483	. . . . . . . . . . . . . . . . 17 |- ( (/) u. { X } ) = { X }
21	20	fveq2i 5805	. . . . . . . . . . . . . . . 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 17216	. . . . . . . . . . . . . . . 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 3586	. . . . . . . . . . . . . 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 2545	. . . . . . . . . . . . 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 17350	. . . . . . . . . . . . 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 1221	. . . . . . . . . . . 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 3611	. . . . . . . . . . . . . 14 |- ( (/) u. { x } ) = ( { x } u. (/) )
32		un0 3773	. . . . . . . . . . . . . 14 |- ( { x } u. (/) ) = { x }
33	31, 32	eqtri 2483	. . . . . . . . . . . . 13 |- ( (/) u. { x } ) = { x }
34	33	fveq2i 5805	. . . . . . . . . . . 12 |- ( N ` ( (/) u. { x } ) ) = ( N ` { x } )
35	30, 34	syl6eleq 2552	. . . . . . . . . . 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 3468	. . . . . . . . . 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 17203	. . . . . . . . 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 3484	. . . . . . . 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 1691	. . . . . 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 2670	. . . 4 |- ( ( ( W e. LVec /\ U e. S /\ X e. V ) /\ U C_ ( N ` { X } ) ) -> ( -. U = ( N ` { X } ) -> ( U \ { .0. } ) = (/) ) )
43		ssdif0 3848	. . . 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 17157	. . . 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 1370   E.wex 1587   e. wcel 1758   =/= wne 2648   \ cdif 3436   u. cun 3437   C_ wss 3439   (/)c0 3748   {csn 3988   ` cfv 5529   Basecbs 14295   0gc0g 14500   LModclmod 17074   LSubSpclss 17139   LSpanclspn 17178   LVecclvec 17309

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-tpos 6858  df-recs 6945  df-rdg 6979  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-nn 10437  df-2 10494  df-3 10495  df-ndx 14298  df-slot 14299  df-base 14300  df-sets 14301  df-ress 14302  df-plusg 14373  df-mulr 14374  df-0g 14502  df-mnd 15537  df-grp 15667  df-minusg 15668  df-sbg 15669  df-cmn 16403  df-abl 16404  df-mgp 16717  df-ur 16729  df-rng 16773  df-oppr 16841  df-dvdsr 16859  df-unit 16860  df-invr 16890  df-drng 16960  df-lmod 17076  df-lss 17140  df-lsp 17179  df-lvec 17310

This theorem is referenced by:  lspsncv0  17353  lsatcmp  33006  dihlspsnssN  35335  dihlspsnat  35336