Theorem l1cvpat 34341

Description: A subspace covered by the set of all vectors, when summed with an atom not under it, equals the set of all vectors. (1cvrjat 34761 analog.) (Contributed by NM, 11-Jan-2015.)

Hypotheses

Ref	Expression
l1cvpat.v	|- V = ( Base ` W )
l1cvpat.s	|- S = ( LSubSp ` W )
l1cvpat.p	|- .(+) = ( LSSum ` W )
l1cvpat.a	|- A = (LSAtoms ` W )
l1cvpat.c	|- C = ( <oLL ` W )
l1cvpat.w	|- ( ph -> W e. LVec )
l1cvpat.u	|- ( ph -> U e. S )
l1cvpat.q	|- ( ph -> Q e. A )
l1cvpat.l	|- ( ph -> U C V )
l1cvpat.m	|- ( ph -> -. Q C_ U )

Assertion

Ref	Expression
l1cvpat	|- ( ph -> ( U .(+) Q ) = V )

Proof of Theorem l1cvpat

Dummy variable v is distinct from all other variables.

Step	Hyp	Ref	Expression
1		l1cvpat.q	. . 3 |- ( ph -> Q e. A )
2		l1cvpat.w	. . . 4 |- ( ph -> W e. LVec )
3		l1cvpat.v	. . . . 5 |- V = ( Base ` W )
4		eqid 2622	. . . . 5 |- ( LSpan ` W ) = ( LSpan ` W )
5		eqid 2622	. . . . 5 |- ( 0g ` W ) = ( 0g ` W )
6		l1cvpat.a	. . . . 5 |- A = (LSAtoms ` W )
7	3, 4, 5, 6	islsat 34278	. . . 4 |- ( W e. LVec -> ( Q e. A <-> E. v e. ( V \ { ( 0g ` W ) } ) Q = ( ( LSpan ` W ) ` { v } ) ) )
8	2, 7	syl 17	. . 3 |- ( ph -> ( Q e. A <-> E. v e. ( V \ { ( 0g ` W ) } ) Q = ( ( LSpan ` W ) ` { v } ) ) )
9	1, 8	mpbid 222	. 2 |- ( ph -> E. v e. ( V \ { ( 0g ` W ) } ) Q = ( ( LSpan ` W ) ` { v } ) )
10		l1cvpat.m	. 2 |- ( ph -> -. Q C_ U )
11		eldifi 3732	. . . 4 |- ( v e. ( V \ { ( 0g ` W ) } ) -> v e. V )
12		l1cvpat.s	. . . . . . . . 9 |- S = ( LSubSp ` W )
13		lveclmod 19106	. . . . . . . . . . 11 |- ( W e. LVec -> W e. LMod )
14	2, 13	syl 17	. . . . . . . . . 10 |- ( ph -> W e. LMod )
15	14	3ad2ant1 1082	. . . . . . . . 9 |- ( ( ph /\ v e. V /\ Q = ( ( LSpan ` W ) ` { v } ) ) -> W e. LMod )
16		l1cvpat.u	. . . . . . . . . 10 |- ( ph -> U e. S )
17	16	3ad2ant1 1082	. . . . . . . . 9 |- ( ( ph /\ v e. V /\ Q = ( ( LSpan ` W ) ` { v } ) ) -> U e. S )
18		simp2 1062	. . . . . . . . 9 |- ( ( ph /\ v e. V /\ Q = ( ( LSpan ` W ) ` { v } ) ) -> v e. V )
19	3, 12, 4, 15, 17, 18	lspsnel5 18995	. . . . . . . 8 |- ( ( ph /\ v e. V /\ Q = ( ( LSpan ` W ) ` { v } ) ) -> ( v e. U <-> ( ( LSpan ` W ) ` { v } ) C_ U ) )
20	19	notbid 308	. . . . . . 7 |- ( ( ph /\ v e. V /\ Q = ( ( LSpan ` W ) ` { v } ) ) -> ( -. v e. U <-> -. ( ( LSpan ` W ) ` { v } ) C_ U ) )
21		l1cvpat.p	. . . . . . . . 9 |- .(+) = ( LSSum ` W )
22		eqid 2622	. . . . . . . . 9 |- (LSHyp ` W ) = (LSHyp ` W )
23	2	3ad2ant1 1082	. . . . . . . . 9 |- ( ( ph /\ v e. V /\ Q = ( ( LSpan ` W ) ` { v } ) ) -> W e. LVec )
24		l1cvpat.l	. . . . . . . . . . 11 |- ( ph -> U C V )
25		l1cvpat.c	. . . . . . . . . . . 12 |- C = ( <oLL ` W )
26	3, 12, 22, 25, 2	islshpcv 34340	. . . . . . . . . . 11 |- ( ph -> ( U e. (LSHyp ` W ) <-> ( U e. S /\ U C V ) ) )
27	16, 24, 26	mpbir2and 957	. . . . . . . . . 10 |- ( ph -> U e. (LSHyp ` W ) )
28	27	3ad2ant1 1082	. . . . . . . . 9 |- ( ( ph /\ v e. V /\ Q = ( ( LSpan ` W ) ` { v } ) ) -> U e. (LSHyp ` W ) )
29	3, 4, 21, 22, 23, 28, 18	lshpnelb 34271	. . . . . . . 8 |- ( ( ph /\ v e. V /\ Q = ( ( LSpan ` W ) ` { v } ) ) -> ( -. v e. U <-> ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = V ) )
30	29	biimpd 219	. . . . . . 7 |- ( ( ph /\ v e. V /\ Q = ( ( LSpan ` W ) ` { v } ) ) -> ( -. v e. U -> ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = V ) )
31	20, 30	sylbird 250	. . . . . 6 |- ( ( ph /\ v e. V /\ Q = ( ( LSpan ` W ) ` { v } ) ) -> ( -. ( ( LSpan ` W ) ` { v } ) C_ U -> ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = V ) )
32		sseq1 3626	. . . . . . . . 9 |- ( Q = ( ( LSpan ` W ) ` { v } ) -> ( Q C_ U <-> ( ( LSpan ` W ) ` { v } ) C_ U ) )
33	32	notbid 308	. . . . . . . 8 |- ( Q = ( ( LSpan ` W ) ` { v } ) -> ( -. Q C_ U <-> -. ( ( LSpan ` W ) ` { v } ) C_ U ) )
34		oveq2 6658	. . . . . . . . 9 |- ( Q = ( ( LSpan ` W ) ` { v } ) -> ( U .(+) Q ) = ( U .(+) ( ( LSpan ` W ) ` { v } ) ) )
35	34	eqeq1d 2624	. . . . . . . 8 |- ( Q = ( ( LSpan ` W ) ` { v } ) -> ( ( U .(+) Q ) = V <-> ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = V ) )
36	33, 35	imbi12d 334	. . . . . . 7 |- ( Q = ( ( LSpan ` W ) ` { v } ) -> ( ( -. Q C_ U -> ( U .(+) Q ) = V ) <-> ( -. ( ( LSpan ` W ) ` { v } ) C_ U -> ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = V ) ) )
37	36	3ad2ant3 1084	. . . . . 6 |- ( ( ph /\ v e. V /\ Q = ( ( LSpan ` W ) ` { v } ) ) -> ( ( -. Q C_ U -> ( U .(+) Q ) = V ) <-> ( -. ( ( LSpan ` W ) ` { v } ) C_ U -> ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = V ) ) )
38	31, 37	mpbird 247	. . . . 5 |- ( ( ph /\ v e. V /\ Q = ( ( LSpan ` W ) ` { v } ) ) -> ( -. Q C_ U -> ( U .(+) Q ) = V ) )
39	38	3exp 1264	. . . 4 |- ( ph -> ( v e. V -> ( Q = ( ( LSpan ` W ) ` { v } ) -> ( -. Q C_ U -> ( U .(+) Q ) = V ) ) ) )
40	11, 39	syl5 34	. . 3 |- ( ph -> ( v e. ( V \ { ( 0g ` W ) } ) -> ( Q = ( ( LSpan ` W ) ` { v } ) -> ( -. Q C_ U -> ( U .(+) Q ) = V ) ) ) )
41	40	rexlimdv 3030	. 2 |- ( ph -> ( E. v e. ( V \ { ( 0g ` W ) } ) Q = ( ( LSpan ` W ) ` { v } ) -> ( -. Q C_ U -> ( U .(+) Q ) = V ) ) )
42	9, 10, 41	mp2d 49	1 |- ( ph -> ( U .(+) Q ) = V )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   \ cdif 3571   C_ wss 3574   {csn 4177   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   0gc0g 16100   LSSumclsm 18049   LModclmod 18863   LSubSpclss 18932   LSpanclspn 18971   LVecclvec 19102  LSAtomsclsa 34261  LSHypclsh 34262   <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-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-subg 17591  df-cntz 17750  df-lsm 18051  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-lshyp 34264  df-lcv 34306

This theorem is referenced by:  l1cvat  34342