Theorem lspsnat 18001

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 26794 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 3745	. . . . . 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 998	. . . . . . . . . . 11 |- ( ( ( W e. LVec /\ U e. S /\ X e. V ) /\ ( U C_ ( N ` { X } ) /\ x e. ( U \ { .0. } ) ) ) -> W e. LVec )
6		lveclmod 17962	. . . . . . . . . . 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 999	. . . . . . . . . 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 3423	. . . . . . . . . . . 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 17852	. . . . . . . . . . 11 |- ( ( ( W e. LVec /\ U e. S /\ X e. V ) /\ ( U C_ ( N ` { X } ) /\ x e. ( U \ { .0. } ) ) ) -> ( N ` { x } ) C_ U )
12		0ss 3765	. . . . . . . . . . . . . 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 1000	. . . . . . . . . . . . 13 |- ( ( ( W e. LVec /\ U e. S /\ X e. V ) /\ ( U C_ ( N ` { X } ) /\ x e. ( U \ { .0. } ) ) ) -> X e. V )
15		ssdif 3575	. . . . . . . . . . . . . . . 16 |- ( U C_ ( N ` { X } ) -> ( U \ { .0. } ) C_ ( ( N ` { X } ) \ { .0. } ) )
16	15	ad2antrl 726	. . . . . . . . . . . . . . 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 3440	. . . . . . . . . . . . . 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 3584	. . . . . . . . . . . . . . . . . 18 |- ( (/) u. { X } ) = ( { X } u. (/) )
19		un0 3761	. . . . . . . . . . . . . . . . . 18 |- ( { X } u. (/) ) = { X }
20	18, 19	eqtri 2429	. . . . . . . . . . . . . . . . 17 |- ( (/) u. { X } ) = { X }
21	20	fveq2i 5806	. . . . . . . . . . . . . . . 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 17865	. . . . . . . . . . . . . . . 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 3559	. . . . . . . . . . . . . 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 2491	. . . . . . . . . . . . 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 17999	. . . . . . . . . . . . 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 1230	. . . . . . . . . . . 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 3584	. . . . . . . . . . . . . 14 |- ( (/) u. { x } ) = ( { x } u. (/) )
32		un0 3761	. . . . . . . . . . . . . 14 |- ( { x } u. (/) ) = { x }
33	31, 32	eqtri 2429	. . . . . . . . . . . . 13 |- ( (/) u. { x } ) = { x }
34	33	fveq2i 5806	. . . . . . . . . . . 12 |- ( N ` ( (/) u. { x } ) ) = ( N ` { x } )
35	30, 34	syl6eleq 2498	. . . . . . . . . . 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 3440	. . . . . . . . . 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 17852	. . . . . . . . 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 3456	. . . . . . . 8 |- ( ( ( W e. LVec /\ U e. S /\ X e. V ) /\ ( U C_ ( N ` { X } ) /\ x e. ( U \ { .0. } ) ) ) -> U = ( N ` { X } ) )
39	38	expr 613	. . . . . . 7 |- ( ( ( W e. LVec /\ U e. S /\ X e. V ) /\ U C_ ( N ` { X } ) ) -> ( x e. ( U \ { .0. } ) -> U = ( N ` { X } ) ) )
40	39	exlimdv 1743	. . . . . 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 2619	. . . 4 |- ( ( ( W e. LVec /\ U e. S /\ X e. V ) /\ U C_ ( N ` { X } ) ) -> ( -. U = ( N ` { X } ) -> ( U \ { .0. } ) = (/) ) )
43		ssdif0 3825	. . . 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 998	. . . . 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 999	. . . 4 |- ( ( ( W e. LVec /\ U e. S /\ X e. V ) /\ U C_ ( N ` { X } ) ) -> U e. S )
48	23, 3	lssle0 17806	. . . 4 |- ( ( W e. LMod /\ U e. S ) -> ( U C_ { .0. } <-> U = { .0. } ) )
49	46, 47, 48	syl2anc 659	. . 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 376	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 366   /\ wa 367   /\ w3a 972   = wceq 1403   E.wex 1631   e. wcel 1840   =/= wne 2596   \ cdif 3408   u. cun 3409   C_ wss 3411   (/)c0 3735   {csn 3969   ` cfv 5523   Basecbs 14731   0gc0g 14944   LModclmod 17722   LSubSpclss 17788   LSpanclspn 17827   LVecclvec 17958

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528  ax-cnex 9496  ax-resscn 9497  ax-1cn 9498  ax-icn 9499  ax-addcl 9500  ax-addrcl 9501  ax-mulcl 9502  ax-mulrcl 9503  ax-mulcom 9504  ax-addass 9505  ax-mulass 9506  ax-distr 9507  ax-i2m1 9508  ax-1ne0 9509  ax-1rid 9510  ax-rnegex 9511  ax-rrecex 9512  ax-cnre 9513  ax-pre-lttri 9514  ax-pre-lttrn 9515  ax-pre-ltadd 9516  ax-pre-mulgt0 9517

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-nel 2599  df-ral 2756  df-rex 2757  df-reu 2758  df-rmo 2759  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-pss 3427  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-tp 3974  df-op 3976  df-uni 4189  df-int 4225  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4487  df-eprel 4731  df-id 4735  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-riota 6194  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-om 6637  df-1st 6736  df-2nd 6737  df-tpos 6910  df-recs 6997  df-rdg 7031  df-er 7266  df-en 7473  df-dom 7474  df-sdom 7475  df-pnf 9578  df-mnf 9579  df-xr 9580  df-ltxr 9581  df-le 9582  df-sub 9761  df-neg 9762  df-nn 10495  df-2 10553  df-3 10554  df-ndx 14734  df-slot 14735  df-base 14736  df-sets 14737  df-ress 14738  df-plusg 14812  df-mulr 14813  df-0g 14946  df-mgm 16086  df-sgrp 16125  df-mnd 16135  df-grp 16271  df-minusg 16272  df-sbg 16273  df-cmn 17014  df-abl 17015  df-mgp 17352  df-ur 17364  df-ring 17410  df-oppr 17482  df-dvdsr 17500  df-unit 17501  df-invr 17531  df-drng 17608  df-lmod 17724  df-lss 17789  df-lsp 17828  df-lvec 17959

This theorem is referenced by:  lspsncv0  18002  lsatcmp  31985  dihlspsnssN  34316  dihlspsnat  34317