Theorem lspsnat 19145

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 28440 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 3931	. . . . . 6 |- ( ( U \ { .0. } ) =/= (/) <-> E. x x e. ( U \ { .0. } ) )
2		simprl 794	. . . . . . . . 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 1064	. . . . . . . . . . 11 |- ( ( ( W e. LVec /\ U e. S /\ X e. V ) /\ ( U C_ ( N ` { X } ) /\ x e. ( U \ { .0. } ) ) ) -> W e. LVec )
6		lveclmod 19106	. . . . . . . . . . 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 1065	. . . . . . . . . 10 |- ( ( ( W e. LVec /\ U e. S /\ X e. V ) /\ ( U C_ ( N ` { X } ) /\ x e. ( U \ { .0. } ) ) ) -> U e. S )
9		simprr 796	. . . . . . . . . . . . 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 3586	. . . . . . . . . . . 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 18996	. . . . . . . . . . 11 |- ( ( ( W e. LVec /\ U e. S /\ X e. V ) /\ ( U C_ ( N ` { X } ) /\ x e. ( U \ { .0. } ) ) ) -> ( N ` { x } ) C_ U )
12		0ss 3972	. . . . . . . . . . . . . 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 1066	. . . . . . . . . . . . 13 |- ( ( ( W e. LVec /\ U e. S /\ X e. V ) /\ ( U C_ ( N ` { X } ) /\ x e. ( U \ { .0. } ) ) ) -> X e. V )
15		ssdif 3745	. . . . . . . . . . . . . . . 16 |- ( U C_ ( N ` { X } ) -> ( U \ { .0. } ) C_ ( ( N ` { X } ) \ { .0. } ) )
16	15	ad2antrl 764	. . . . . . . . . . . . . . 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 3604	. . . . . . . . . . . . . 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 3757	. . . . . . . . . . . . . . . . . 18 |- ( (/) u. { X } ) = ( { X } u. (/) )
19		un0 3967	. . . . . . . . . . . . . . . . . 18 |- ( { X } u. (/) ) = { X }
20	18, 19	eqtri 2644	. . . . . . . . . . . . . . . . 17 |- ( (/) u. { X } ) = { X }
21	20	fveq2i 6194	. . . . . . . . . . . . . . . 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 19009	. . . . . . . . . . . . . . . 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 3729	. . . . . . . . . . . . . 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 2704	. . . . . . . . . . . . 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 19143	. . . . . . . . . . . . 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 1328	. . . . . . . . . . . 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 3757	. . . . . . . . . . . . . 14 |- ( (/) u. { x } ) = ( { x } u. (/) )
32		un0 3967	. . . . . . . . . . . . . 14 |- ( { x } u. (/) ) = { x }
33	31, 32	eqtri 2644	. . . . . . . . . . . . 13 |- ( (/) u. { x } ) = { x }
34	33	fveq2i 6194	. . . . . . . . . . . 12 |- ( N ` ( (/) u. { x } ) ) = ( N ` { x } )
35	30, 34	syl6eleq 2711	. . . . . . . . . . 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 3604	. . . . . . . . . 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 18996	. . . . . . . . 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 3620	. . . . . . . 8 |- ( ( ( W e. LVec /\ U e. S /\ X e. V ) /\ ( U C_ ( N ` { X } ) /\ x e. ( U \ { .0. } ) ) ) -> U = ( N ` { X } ) )
39	38	expr 643	. . . . . . 7 |- ( ( ( W e. LVec /\ U e. S /\ X e. V ) /\ U C_ ( N ` { X } ) ) -> ( x e. ( U \ { .0. } ) -> U = ( N ` { X } ) ) )
40	39	exlimdv 1861	. . . . . 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 232	. . . . 5 |- ( ( ( W e. LVec /\ U e. S /\ X e. V ) /\ U C_ ( N ` { X } ) ) -> ( ( U \ { .0. } ) =/= (/) -> U = ( N ` { X } ) ) )
42	41	necon1bd 2812	. . . 4 |- ( ( ( W e. LVec /\ U e. S /\ X e. V ) /\ U C_ ( N ` { X } ) ) -> ( -. U = ( N ` { X } ) -> ( U \ { .0. } ) = (/) ) )
43		ssdif0 3942	. . . 4 |- ( U C_ { .0. } <-> ( U \ { .0. } ) = (/) )
44	42, 43	syl6ibr 242	. . 3 |- ( ( ( W e. LVec /\ U e. S /\ X e. V ) /\ U C_ ( N ` { X } ) ) -> ( -. U = ( N ` { X } ) -> U C_ { .0. } ) )
45		simpl1 1064	. . . . 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 1065	. . . 4 |- ( ( ( W e. LVec /\ U e. S /\ X e. V ) /\ U C_ ( N ` { X } ) ) -> U e. S )
48	23, 3	lssle0 18950	. . . 4 |- ( ( W e. LMod /\ U e. S ) -> ( U C_ { .0. } <-> U = { .0. } ) )
49	46, 47, 48	syl2anc 693	. . 3 |- ( ( ( W e. LVec /\ U e. S /\ X e. V ) /\ U C_ ( N ` { X } ) ) -> ( U C_ { .0. } <-> U = { .0. } ) )
50	44, 49	sylibd 229	. 2 |- ( ( ( W e. LVec /\ U e. S /\ X e. V ) /\ U C_ ( N ` { X } ) ) -> ( -. U = ( N ` { X } ) -> U = { .0. } ) )
51	50	orrd 393	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 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   \ cdif 3571   u. cun 3572   C_ wss 3574   (/)c0 3915   {csn 4177   ` cfv 5888   Basecbs 15857   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-cmn 18195  df-abl 18196  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:  lspsncv0  19146  lsatcmp  34290  dihlspsnssN  36621  dihlspsnat  36622