Theorem lbspss 19082

Description: No proper subset of a basis spans the space. (Contributed by Mario Carneiro, 25-Jun-2014.)

Hypotheses

Ref	Expression
lbsind2.j	|- J = (LBasis ` W )
lbsind2.n	|- N = ( LSpan ` W )
lbsind2.f	|- F = (Scalar ` W )
lbsind2.o	|- .1. = ( 1r ` F )
lbsind2.z	|- .0. = ( 0g ` F )
lbspss.v	|- V = ( Base ` W )

Assertion

Ref	Expression
lbspss	|- ( ( ( W e. LMod /\ .1. =/= .0. ) /\ B e. J /\ C C. B ) -> ( N ` C ) =/= V )

Proof of Theorem lbspss

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pssnel 4039	. . 3 |- ( C C. B -> E. x ( x e. B /\ -. x e. C ) )
2	1	3ad2ant3 1084	. 2 |- ( ( ( W e. LMod /\ .1. =/= .0. ) /\ B e. J /\ C C. B ) -> E. x ( x e. B /\ -. x e. C ) )
3		simpl2 1065	. . . . . 6 |- ( ( ( ( W e. LMod /\ .1. =/= .0. ) /\ B e. J /\ C C. B ) /\ ( x e. B /\ -. x e. C ) ) -> B e. J )
4		lbspss.v	. . . . . . 7 |- V = ( Base ` W )
5		lbsind2.j	. . . . . . 7 |- J = (LBasis ` W )
6	4, 5	lbsss 19077	. . . . . 6 |- ( B e. J -> B C_ V )
7	3, 6	syl 17	. . . . 5 |- ( ( ( ( W e. LMod /\ .1. =/= .0. ) /\ B e. J /\ C C. B ) /\ ( x e. B /\ -. x e. C ) ) -> B C_ V )
8		simprl 794	. . . . 5 |- ( ( ( ( W e. LMod /\ .1. =/= .0. ) /\ B e. J /\ C C. B ) /\ ( x e. B /\ -. x e. C ) ) -> x e. B )
9	7, 8	sseldd 3604	. . . 4 |- ( ( ( ( W e. LMod /\ .1. =/= .0. ) /\ B e. J /\ C C. B ) /\ ( x e. B /\ -. x e. C ) ) -> x e. V )
10		simpl1l 1112	. . . . . 6 |- ( ( ( ( W e. LMod /\ .1. =/= .0. ) /\ B e. J /\ C C. B ) /\ ( x e. B /\ -. x e. C ) ) -> W e. LMod )
11	7	ssdifssd 3748	. . . . . 6 |- ( ( ( ( W e. LMod /\ .1. =/= .0. ) /\ B e. J /\ C C. B ) /\ ( x e. B /\ -. x e. C ) ) -> ( B \ { x } ) C_ V )
12		simpl3 1066	. . . . . . . . . . 11 |- ( ( ( ( W e. LMod /\ .1. =/= .0. ) /\ B e. J /\ C C. B ) /\ ( x e. B /\ -. x e. C ) ) -> C C. B )
13	12	pssssd 3704	. . . . . . . . . 10 |- ( ( ( ( W e. LMod /\ .1. =/= .0. ) /\ B e. J /\ C C. B ) /\ ( x e. B /\ -. x e. C ) ) -> C C_ B )
14	13	sseld 3602	. . . . . . . . 9 |- ( ( ( ( W e. LMod /\ .1. =/= .0. ) /\ B e. J /\ C C. B ) /\ ( x e. B /\ -. x e. C ) ) -> ( y e. C -> y e. B ) )
15		simprr 796	. . . . . . . . . . 11 |- ( ( ( ( W e. LMod /\ .1. =/= .0. ) /\ B e. J /\ C C. B ) /\ ( x e. B /\ -. x e. C ) ) -> -. x e. C )
16		eleq1 2689	. . . . . . . . . . . 12 |- ( y = x -> ( y e. C <-> x e. C ) )
17	16	notbid 308	. . . . . . . . . . 11 |- ( y = x -> ( -. y e. C <-> -. x e. C ) )
18	15, 17	syl5ibrcom 237	. . . . . . . . . 10 |- ( ( ( ( W e. LMod /\ .1. =/= .0. ) /\ B e. J /\ C C. B ) /\ ( x e. B /\ -. x e. C ) ) -> ( y = x -> -. y e. C ) )
19	18	necon2ad 2809	. . . . . . . . 9 |- ( ( ( ( W e. LMod /\ .1. =/= .0. ) /\ B e. J /\ C C. B ) /\ ( x e. B /\ -. x e. C ) ) -> ( y e. C -> y =/= x ) )
20	14, 19	jcad 555	. . . . . . . 8 |- ( ( ( ( W e. LMod /\ .1. =/= .0. ) /\ B e. J /\ C C. B ) /\ ( x e. B /\ -. x e. C ) ) -> ( y e. C -> ( y e. B /\ y =/= x ) ) )
21		eldifsn 4317	. . . . . . . 8 |- ( y e. ( B \ { x } ) <-> ( y e. B /\ y =/= x ) )
22	20, 21	syl6ibr 242	. . . . . . 7 |- ( ( ( ( W e. LMod /\ .1. =/= .0. ) /\ B e. J /\ C C. B ) /\ ( x e. B /\ -. x e. C ) ) -> ( y e. C -> y e. ( B \ { x } ) ) )
23	22	ssrdv 3609	. . . . . 6 |- ( ( ( ( W e. LMod /\ .1. =/= .0. ) /\ B e. J /\ C C. B ) /\ ( x e. B /\ -. x e. C ) ) -> C C_ ( B \ { x } ) )
24		lbsind2.n	. . . . . . 7 |- N = ( LSpan ` W )
25	4, 24	lspss 18984	. . . . . 6 |- ( ( W e. LMod /\ ( B \ { x } ) C_ V /\ C C_ ( B \ { x } ) ) -> ( N ` C ) C_ ( N ` ( B \ { x } ) ) )
26	10, 11, 23, 25	syl3anc 1326	. . . . 5 |- ( ( ( ( W e. LMod /\ .1. =/= .0. ) /\ B e. J /\ C C. B ) /\ ( x e. B /\ -. x e. C ) ) -> ( N ` C ) C_ ( N ` ( B \ { x } ) ) )
27		simpl1r 1113	. . . . . 6 |- ( ( ( ( W e. LMod /\ .1. =/= .0. ) /\ B e. J /\ C C. B ) /\ ( x e. B /\ -. x e. C ) ) -> .1. =/= .0. )
28		lbsind2.f	. . . . . . 7 |- F = (Scalar ` W )
29		lbsind2.o	. . . . . . 7 |- .1. = ( 1r ` F )
30		lbsind2.z	. . . . . . 7 |- .0. = ( 0g ` F )
31	5, 24, 28, 29, 30	lbsind2 19081	. . . . . 6 |- ( ( ( W e. LMod /\ .1. =/= .0. ) /\ B e. J /\ x e. B ) -> -. x e. ( N ` ( B \ { x } ) ) )
32	10, 27, 3, 8, 31	syl211anc 1332	. . . . 5 |- ( ( ( ( W e. LMod /\ .1. =/= .0. ) /\ B e. J /\ C C. B ) /\ ( x e. B /\ -. x e. C ) ) -> -. x e. ( N ` ( B \ { x } ) ) )
33	26, 32	ssneldd 3606	. . . 4 |- ( ( ( ( W e. LMod /\ .1. =/= .0. ) /\ B e. J /\ C C. B ) /\ ( x e. B /\ -. x e. C ) ) -> -. x e. ( N ` C ) )
34		nelne1 2890	. . . 4 |- ( ( x e. V /\ -. x e. ( N ` C ) ) -> V =/= ( N ` C ) )
35	9, 33, 34	syl2anc 693	. . 3 |- ( ( ( ( W e. LMod /\ .1. =/= .0. ) /\ B e. J /\ C C. B ) /\ ( x e. B /\ -. x e. C ) ) -> V =/= ( N ` C ) )
36	35	necomd 2849	. 2 |- ( ( ( ( W e. LMod /\ .1. =/= .0. ) /\ B e. J /\ C C. B ) /\ ( x e. B /\ -. x e. C ) ) -> ( N ` C ) =/= V )
37	2, 36	exlimddv 1863	1 |- ( ( ( W e. LMod /\ .1. =/= .0. ) /\ B e. J /\ C C. B ) -> ( N ` C ) =/= V )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   \ cdif 3571   C_ wss 3574   C. wpss 3575   {csn 4177   ` cfv 5888   Basecbs 15857  Scalarcsca 15944   0gc0g 16100   1rcur 18501   LModclmod 18863   LSpanclspn 18971  LBasisclbs 19074

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-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-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-plusg 15954  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-mgp 18490  df-ur 18502  df-ring 18549  df-lmod 18865  df-lss 18933  df-lsp 18972  df-lbs 19075

This theorem is referenced by:  islbs3  19155