Theorem obslbs 20074

Description: An orthogonal basis is a linear basis iff the span of the basis elements is closed (which is usually not true). (Contributed by Mario Carneiro, 29-Oct-2015.)

Hypotheses

Ref	Expression
obslbs.j	|- J = (LBasis ` W )
obslbs.n	|- N = ( LSpan ` W )
obslbs.c	|- C = ( CSubSp ` W )

Assertion

Ref	Expression
obslbs	|- ( B e. (OBasis ` W ) -> ( B e. J <-> ( N ` B ) e. C ) )

Proof of Theorem obslbs

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

Step	Hyp	Ref	Expression
1		obsrcl 20067	. . . . . 6 |- ( B e. (OBasis ` W ) -> W e. PreHil )
2		eqid 2622	. . . . . . 7 |- ( Base ` W ) = ( Base ` W )
3	2	obsss 20068	. . . . . 6 |- ( B e. (OBasis ` W ) -> B C_ ( Base ` W ) )
4		eqid 2622	. . . . . . 7 |- ( ocv ` W ) = ( ocv ` W )
5		obslbs.n	. . . . . . 7 |- N = ( LSpan ` W )
6	2, 4, 5	ocvlsp 20020	. . . . . 6 |- ( ( W e. PreHil /\ B C_ ( Base ` W ) ) -> ( ( ocv ` W ) ` ( N ` B ) ) = ( ( ocv ` W ) ` B ) )
7	1, 3, 6	syl2anc 693	. . . . 5 |- ( B e. (OBasis ` W ) -> ( ( ocv ` W ) ` ( N ` B ) ) = ( ( ocv ` W ) ` B ) )
8	7	fveq2d 6195	. . . 4 |- ( B e. (OBasis ` W ) -> ( ( ocv ` W ) ` ( ( ocv ` W ) ` ( N ` B ) ) ) = ( ( ocv ` W ) ` ( ( ocv ` W ) ` B ) ) )
9	4, 2	obs2ocv 20071	. . . 4 |- ( B e. (OBasis ` W ) -> ( ( ocv ` W ) ` ( ( ocv ` W ) ` B ) ) = ( Base ` W ) )
10	8, 9	eqtrd 2656	. . 3 |- ( B e. (OBasis ` W ) -> ( ( ocv ` W ) ` ( ( ocv ` W ) ` ( N ` B ) ) ) = ( Base ` W ) )
11	10	eqeq2d 2632	. 2 |- ( B e. (OBasis ` W ) -> ( ( N ` B ) = ( ( ocv ` W ) ` ( ( ocv ` W ) ` ( N ` B ) ) ) <-> ( N ` B ) = ( Base ` W ) ) )
12		obslbs.c	. . . 4 |- C = ( CSubSp ` W )
13	4, 12	iscss 20027	. . 3 |- ( W e. PreHil -> ( ( N ` B ) e. C <-> ( N ` B ) = ( ( ocv ` W ) ` ( ( ocv ` W ) ` ( N ` B ) ) ) ) )
14	1, 13	syl 17	. 2 |- ( B e. (OBasis ` W ) -> ( ( N ` B ) e. C <-> ( N ` B ) = ( ( ocv ` W ) ` ( ( ocv ` W ) ` ( N ` B ) ) ) ) )
15		phllvec 19974	. . . 4 |- ( W e. PreHil -> W e. LVec )
16	1, 15	syl 17	. . 3 |- ( B e. (OBasis ` W ) -> W e. LVec )
17		pssnel 4039	. . . . . . 7 |- ( x C. B -> E. y ( y e. B /\ -. y e. x ) )
18	17	adantl 482	. . . . . 6 |- ( ( B e. (OBasis ` W ) /\ x C. B ) -> E. y ( y e. B /\ -. y e. x ) )
19		simpll 790	. . . . . . . . . . 11 |- ( ( ( B e. (OBasis ` W ) /\ x C. B ) /\ y e. B ) -> B e. (OBasis ` W ) )
20		pssss 3702	. . . . . . . . . . . 12 |- ( x C. B -> x C_ B )
21	20	ad2antlr 763	. . . . . . . . . . 11 |- ( ( ( B e. (OBasis ` W ) /\ x C. B ) /\ y e. B ) -> x C_ B )
22		simpr 477	. . . . . . . . . . 11 |- ( ( ( B e. (OBasis ` W ) /\ x C. B ) /\ y e. B ) -> y e. B )
23	4	obselocv 20072	. . . . . . . . . . 11 |- ( ( B e. (OBasis ` W ) /\ x C_ B /\ y e. B ) -> ( y e. ( ( ocv ` W ) ` x ) <-> -. y e. x ) )
24	19, 21, 22, 23	syl3anc 1326	. . . . . . . . . 10 |- ( ( ( B e. (OBasis ` W ) /\ x C. B ) /\ y e. B ) -> ( y e. ( ( ocv ` W ) ` x ) <-> -. y e. x ) )
25		eqid 2622	. . . . . . . . . . . . . 14 |- ( 0g ` W ) = ( 0g ` W )
26	25	obsne0 20069	. . . . . . . . . . . . 13 |- ( ( B e. (OBasis ` W ) /\ y e. B ) -> y =/= ( 0g ` W ) )
27	19, 22, 26	syl2anc 693	. . . . . . . . . . . 12 |- ( ( ( B e. (OBasis ` W ) /\ x C. B ) /\ y e. B ) -> y =/= ( 0g ` W ) )
28		nelsn 4212	. . . . . . . . . . . 12 |- ( y =/= ( 0g ` W ) -> -. y e. { ( 0g ` W ) } )
29	27, 28	syl 17	. . . . . . . . . . 11 |- ( ( ( B e. (OBasis ` W ) /\ x C. B ) /\ y e. B ) -> -. y e. { ( 0g ` W ) } )
30		nelne1 2890	. . . . . . . . . . . 12 |- ( ( y e. ( ( ocv ` W ) ` x ) /\ -. y e. { ( 0g ` W ) } ) -> ( ( ocv ` W ) ` x ) =/= { ( 0g ` W ) } )
31	30	expcom 451	. . . . . . . . . . 11 |- ( -. y e. { ( 0g ` W ) } -> ( y e. ( ( ocv ` W ) ` x ) -> ( ( ocv ` W ) ` x ) =/= { ( 0g ` W ) } ) )
32	29, 31	syl 17	. . . . . . . . . 10 |- ( ( ( B e. (OBasis ` W ) /\ x C. B ) /\ y e. B ) -> ( y e. ( ( ocv ` W ) ` x ) -> ( ( ocv ` W ) ` x ) =/= { ( 0g ` W ) } ) )
33	24, 32	sylbird 250	. . . . . . . . 9 |- ( ( ( B e. (OBasis ` W ) /\ x C. B ) /\ y e. B ) -> ( -. y e. x -> ( ( ocv ` W ) ` x ) =/= { ( 0g ` W ) } ) )
34		npss 3717	. . . . . . . . . . 11 |- ( -. ( N ` x ) C. ( Base ` W ) <-> ( ( N ` x ) C_ ( Base ` W ) -> ( N ` x ) = ( Base ` W ) ) )
35		phllmod 19975	. . . . . . . . . . . . . . 15 |- ( W e. PreHil -> W e. LMod )
36	1, 35	syl 17	. . . . . . . . . . . . . 14 |- ( B e. (OBasis ` W ) -> W e. LMod )
37	36	ad2antrr 762	. . . . . . . . . . . . 13 |- ( ( ( B e. (OBasis ` W ) /\ x C. B ) /\ y e. B ) -> W e. LMod )
38	3	ad2antrr 762	. . . . . . . . . . . . . 14 |- ( ( ( B e. (OBasis ` W ) /\ x C. B ) /\ y e. B ) -> B C_ ( Base ` W ) )
39	21, 38	sstrd 3613	. . . . . . . . . . . . 13 |- ( ( ( B e. (OBasis ` W ) /\ x C. B ) /\ y e. B ) -> x C_ ( Base ` W ) )
40	2, 5	lspssv 18983	. . . . . . . . . . . . 13 |- ( ( W e. LMod /\ x C_ ( Base ` W ) ) -> ( N ` x ) C_ ( Base ` W ) )
41	37, 39, 40	syl2anc 693	. . . . . . . . . . . 12 |- ( ( ( B e. (OBasis ` W ) /\ x C. B ) /\ y e. B ) -> ( N ` x ) C_ ( Base ` W ) )
42		fveq2 6191	. . . . . . . . . . . . 13 |- ( ( N ` x ) = ( Base ` W ) -> ( ( ocv ` W ) ` ( N ` x ) ) = ( ( ocv ` W ) ` ( Base ` W ) ) )
43	1	ad2antrr 762	. . . . . . . . . . . . . . 15 |- ( ( ( B e. (OBasis ` W ) /\ x C. B ) /\ y e. B ) -> W e. PreHil )
44	2, 4, 5	ocvlsp 20020	. . . . . . . . . . . . . . 15 |- ( ( W e. PreHil /\ x C_ ( Base ` W ) ) -> ( ( ocv ` W ) ` ( N ` x ) ) = ( ( ocv ` W ) ` x ) )
45	43, 39, 44	syl2anc 693	. . . . . . . . . . . . . 14 |- ( ( ( B e. (OBasis ` W ) /\ x C. B ) /\ y e. B ) -> ( ( ocv ` W ) ` ( N ` x ) ) = ( ( ocv ` W ) ` x ) )
46	2, 4, 25	ocv1 20023	. . . . . . . . . . . . . . 15 |- ( W e. PreHil -> ( ( ocv ` W ) ` ( Base ` W ) ) = { ( 0g ` W ) } )
47	43, 46	syl 17	. . . . . . . . . . . . . 14 |- ( ( ( B e. (OBasis ` W ) /\ x C. B ) /\ y e. B ) -> ( ( ocv ` W ) ` ( Base ` W ) ) = { ( 0g ` W ) } )
48	45, 47	eqeq12d 2637	. . . . . . . . . . . . 13 |- ( ( ( B e. (OBasis ` W ) /\ x C. B ) /\ y e. B ) -> ( ( ( ocv ` W ) ` ( N ` x ) ) = ( ( ocv ` W ) ` ( Base ` W ) ) <-> ( ( ocv ` W ) ` x ) = { ( 0g ` W ) } ) )
49	42, 48	syl5ib 234	. . . . . . . . . . . 12 |- ( ( ( B e. (OBasis ` W ) /\ x C. B ) /\ y e. B ) -> ( ( N ` x ) = ( Base ` W ) -> ( ( ocv ` W ) ` x ) = { ( 0g ` W ) } ) )
50	41, 49	embantd 59	. . . . . . . . . . 11 |- ( ( ( B e. (OBasis ` W ) /\ x C. B ) /\ y e. B ) -> ( ( ( N ` x ) C_ ( Base ` W ) -> ( N ` x ) = ( Base ` W ) ) -> ( ( ocv ` W ) ` x ) = { ( 0g ` W ) } ) )
51	34, 50	syl5bi 232	. . . . . . . . . 10 |- ( ( ( B e. (OBasis ` W ) /\ x C. B ) /\ y e. B ) -> ( -. ( N ` x ) C. ( Base ` W ) -> ( ( ocv ` W ) ` x ) = { ( 0g ` W ) } ) )
52	51	necon1ad 2811	. . . . . . . . 9 |- ( ( ( B e. (OBasis ` W ) /\ x C. B ) /\ y e. B ) -> ( ( ( ocv ` W ) ` x ) =/= { ( 0g ` W ) } -> ( N ` x ) C. ( Base ` W ) ) )
53	33, 52	syld 47	. . . . . . . 8 |- ( ( ( B e. (OBasis ` W ) /\ x C. B ) /\ y e. B ) -> ( -. y e. x -> ( N ` x ) C. ( Base ` W ) ) )
54	53	expimpd 629	. . . . . . 7 |- ( ( B e. (OBasis ` W ) /\ x C. B ) -> ( ( y e. B /\ -. y e. x ) -> ( N ` x ) C. ( Base ` W ) ) )
55	54	exlimdv 1861	. . . . . 6 |- ( ( B e. (OBasis ` W ) /\ x C. B ) -> ( E. y ( y e. B /\ -. y e. x ) -> ( N ` x ) C. ( Base ` W ) ) )
56	18, 55	mpd 15	. . . . 5 |- ( ( B e. (OBasis ` W ) /\ x C. B ) -> ( N ` x ) C. ( Base ` W ) )
57	56	ex 450	. . . 4 |- ( B e. (OBasis ` W ) -> ( x C. B -> ( N ` x ) C. ( Base ` W ) ) )
58	57	alrimiv 1855	. . 3 |- ( B e. (OBasis ` W ) -> A. x ( x C. B -> ( N ` x ) C. ( Base ` W ) ) )
59		obslbs.j	. . . . . 6 |- J = (LBasis ` W )
60	2, 59, 5	islbs3 19155	. . . . 5 |- ( W e. LVec -> ( B e. J <-> ( B C_ ( Base ` W ) /\ ( N ` B ) = ( Base ` W ) /\ A. x ( x C. B -> ( N ` x ) C. ( Base ` W ) ) ) ) )
61		3anan32 1050	. . . . 5 |- ( ( B C_ ( Base ` W ) /\ ( N ` B ) = ( Base ` W ) /\ A. x ( x C. B -> ( N ` x ) C. ( Base ` W ) ) ) <-> ( ( B C_ ( Base ` W ) /\ A. x ( x C. B -> ( N ` x ) C. ( Base ` W ) ) ) /\ ( N ` B ) = ( Base ` W ) ) )
62	60, 61	syl6bb 276	. . . 4 |- ( W e. LVec -> ( B e. J <-> ( ( B C_ ( Base ` W ) /\ A. x ( x C. B -> ( N ` x ) C. ( Base ` W ) ) ) /\ ( N ` B ) = ( Base ` W ) ) ) )
63	62	baibd 948	. . 3 |- ( ( W e. LVec /\ ( B C_ ( Base ` W ) /\ A. x ( x C. B -> ( N ` x ) C. ( Base ` W ) ) ) ) -> ( B e. J <-> ( N ` B ) = ( Base ` W ) ) )
64	16, 3, 58, 63	syl12anc 1324	. 2 |- ( B e. (OBasis ` W ) -> ( B e. J <-> ( N ` B ) = ( Base ` W ) ) )
65	11, 14, 64	3bitr4rd 301	1 |- ( B e. (OBasis ` W ) -> ( B e. J <-> ( N ` B ) e. C ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   C_ wss 3574   C. wpss 3575   {csn 4177   ` cfv 5888   Basecbs 15857   0gc0g 16100   LModclmod 18863   LSpanclspn 18971  LBasisclbs 19074   LVecclvec 19102   PreHilcphl 19969   ocvcocv 20004   CSubSpccss 20005  OBasiscobs 20046

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-map 7859  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-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-ip 15959  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-grp 17425  df-minusg 17426  df-sbg 17427  df-ghm 17658  df-mgp 18490  df-ur 18502  df-ring 18549  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-rnghom 18715  df-drng 18749  df-staf 18845  df-srng 18846  df-lmod 18865  df-lss 18933  df-lsp 18972  df-lmhm 19022  df-lbs 19075  df-lvec 19103  df-sra 19172  df-rgmod 19173  df-phl 19971  df-ocv 20007  df-css 20008  df-obs 20049

This theorem is referenced by: (None)