Theorem lsatcv0 34318

Description: An atom covers the zero subspace. (atcv0 29201 analog.) (Contributed by NM, 7-Jan-2015.)

Hypotheses

Ref	Expression
lsatcv0.o	|- .0. = ( 0g ` W )
lsatcv0.a	|- A = (LSAtoms ` W )
lsatcv0.c	|- C = ( <oLL ` W )
lsatcv0.w	|- ( ph -> W e. LVec )
lsatcv0.q	|- ( ph -> Q e. A )

Assertion

Ref	Expression
lsatcv0	|- ( ph -> { .0. } C Q )

Proof of Theorem lsatcv0

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

Step	Hyp	Ref	Expression
1		lsatcv0.w	. . . . 5 |- ( ph -> W e. LVec )
2		lveclmod 19106	. . . . 5 |- ( W e. LVec -> W e. LMod )
3	1, 2	syl 17	. . . 4 |- ( ph -> W e. LMod )
4		eqid 2622	. . . . 5 |- ( LSubSp ` W ) = ( LSubSp ` W )
5		lsatcv0.a	. . . . 5 |- A = (LSAtoms ` W )
6		lsatcv0.q	. . . . 5 |- ( ph -> Q e. A )
7	4, 5, 3, 6	lsatlssel 34284	. . . 4 |- ( ph -> Q e. ( LSubSp ` W ) )
8		lsatcv0.o	. . . . 5 |- .0. = ( 0g ` W )
9	8, 4	lss0ss 18949	. . . 4 |- ( ( W e. LMod /\ Q e. ( LSubSp ` W ) ) -> { .0. } C_ Q )
10	3, 7, 9	syl2anc 693	. . 3 |- ( ph -> { .0. } C_ Q )
11	8, 5, 3, 6	lsatn0 34286	. . . 4 |- ( ph -> Q =/= { .0. } )
12	11	necomd 2849	. . 3 |- ( ph -> { .0. } =/= Q )
13		df-pss 3590	. . 3 |- ( { .0. } C. Q <-> ( { .0. } C_ Q /\ { .0. } =/= Q ) )
14	10, 12, 13	sylanbrc 698	. 2 |- ( ph -> { .0. } C. Q )
15		eqid 2622	. . . . . 6 |- ( Base ` W ) = ( Base ` W )
16		eqid 2622	. . . . . 6 |- ( LSpan ` W ) = ( LSpan ` W )
17	15, 16, 8, 5	islsat 34278	. . . . 5 |- ( W e. LMod -> ( Q e. A <-> E. x e. ( ( Base ` W ) \ { .0. } ) Q = ( ( LSpan ` W ) ` { x } ) ) )
18	3, 17	syl 17	. . . 4 |- ( ph -> ( Q e. A <-> E. x e. ( ( Base ` W ) \ { .0. } ) Q = ( ( LSpan ` W ) ` { x } ) ) )
19	6, 18	mpbid 222	. . 3 |- ( ph -> E. x e. ( ( Base ` W ) \ { .0. } ) Q = ( ( LSpan ` W ) ` { x } ) )
20	1	adantr 481	. . . . . . 7 |- ( ( ph /\ x e. ( ( Base ` W ) \ { .0. } ) ) -> W e. LVec )
21		eldifi 3732	. . . . . . . 8 |- ( x e. ( ( Base ` W ) \ { .0. } ) -> x e. ( Base ` W ) )
22	21	adantl 482	. . . . . . 7 |- ( ( ph /\ x e. ( ( Base ` W ) \ { .0. } ) ) -> x e. ( Base ` W ) )
23		eldifsni 4320	. . . . . . . 8 |- ( x e. ( ( Base ` W ) \ { .0. } ) -> x =/= .0. )
24	23	adantl 482	. . . . . . 7 |- ( ( ph /\ x e. ( ( Base ` W ) \ { .0. } ) ) -> x =/= .0. )
25	15, 8, 4, 16, 20, 22, 24	lspsncv0 19146	. . . . . 6 |- ( ( ph /\ x e. ( ( Base ` W ) \ { .0. } ) ) -> -. E. s e. ( LSubSp ` W ) ( { .0. } C. s /\ s C. ( ( LSpan ` W ) ` { x } ) ) )
26	25	ex 450	. . . . 5 |- ( ph -> ( x e. ( ( Base ` W ) \ { .0. } ) -> -. E. s e. ( LSubSp ` W ) ( { .0. } C. s /\ s C. ( ( LSpan ` W ) ` { x } ) ) ) )
27		psseq2 3695	. . . . . . . . 9 |- ( Q = ( ( LSpan ` W ) ` { x } ) -> ( s C. Q <-> s C. ( ( LSpan ` W ) ` { x } ) ) )
28	27	anbi2d 740	. . . . . . . 8 |- ( Q = ( ( LSpan ` W ) ` { x } ) -> ( ( { .0. } C. s /\ s C. Q ) <-> ( { .0. } C. s /\ s C. ( ( LSpan ` W ) ` { x } ) ) ) )
29	28	rexbidv 3052	. . . . . . 7 |- ( Q = ( ( LSpan ` W ) ` { x } ) -> ( E. s e. ( LSubSp ` W ) ( { .0. } C. s /\ s C. Q ) <-> E. s e. ( LSubSp ` W ) ( { .0. } C. s /\ s C. ( ( LSpan ` W ) ` { x } ) ) ) )
30	29	notbid 308	. . . . . 6 |- ( Q = ( ( LSpan ` W ) ` { x } ) -> ( -. E. s e. ( LSubSp ` W ) ( { .0. } C. s /\ s C. Q ) <-> -. E. s e. ( LSubSp ` W ) ( { .0. } C. s /\ s C. ( ( LSpan ` W ) ` { x } ) ) ) )
31	30	biimprcd 240	. . . . 5 |- ( -. E. s e. ( LSubSp ` W ) ( { .0. } C. s /\ s C. ( ( LSpan ` W ) ` { x } ) ) -> ( Q = ( ( LSpan ` W ) ` { x } ) -> -. E. s e. ( LSubSp ` W ) ( { .0. } C. s /\ s C. Q ) ) )
32	26, 31	syl6 35	. . . 4 |- ( ph -> ( x e. ( ( Base ` W ) \ { .0. } ) -> ( Q = ( ( LSpan ` W ) ` { x } ) -> -. E. s e. ( LSubSp ` W ) ( { .0. } C. s /\ s C. Q ) ) ) )
33	32	rexlimdv 3030	. . 3 |- ( ph -> ( E. x e. ( ( Base ` W ) \ { .0. } ) Q = ( ( LSpan ` W ) ` { x } ) -> -. E. s e. ( LSubSp ` W ) ( { .0. } C. s /\ s C. Q ) ) )
34	19, 33	mpd 15	. 2 |- ( ph -> -. E. s e. ( LSubSp ` W ) ( { .0. } C. s /\ s C. Q ) )
35		lsatcv0.c	. . 3 |- C = ( <oLL ` W )
36	8, 4	lsssn0 18948	. . . 4 |- ( W e. LMod -> { .0. } e. ( LSubSp ` W ) )
37	3, 36	syl 17	. . 3 |- ( ph -> { .0. } e. ( LSubSp ` W ) )
38	4, 35, 1, 37, 7	lcvbr 34308	. 2 |- ( ph -> ( { .0. } C Q <-> ( { .0. } C. Q /\ -. E. s e. ( LSubSp ` W ) ( { .0. } C. s /\ s C. Q ) ) ) )
39	14, 34, 38	mpbir2and 957	1 |- ( ph -> { .0. } C Q )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   \ cdif 3571   C_ wss 3574   C. wpss 3575   {csn 4177   class class class wbr 4653   ` cfv 5888   Basecbs 15857   0gc0g 16100   LModclmod 18863   LSubSpclss 18932   LSpanclspn 18971   LVecclvec 19102  LSAtomsclsa 34261   <oL_L clcv 34305

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  df-lsatoms 34263  df-lcv 34306

This theorem is referenced by:  lsatcveq0  34319  lsat0cv  34320  lsatcv0eq  34334  mapdcnvatN  36955  mapdat  36956