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Theorem obslbs 19224
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.
StepHypRef Expression
1 obsrcl 19217 . . . . . 6  |-  ( B  e.  (OBasis `  W
)  ->  W  e.  PreHil )
2 eqid 2429 . . . . . . 7  |-  ( Base `  W )  =  (
Base `  W )
32obsss 19218 . . . . . 6  |-  ( B  e.  (OBasis `  W
)  ->  B  C_  ( Base `  W ) )
4 eqid 2429 . . . . . . 7  |-  ( ocv `  W )  =  ( ocv `  W )
5 obslbs.n . . . . . . 7  |-  N  =  ( LSpan `  W )
62, 4, 5ocvlsp 19170 . . . . . 6  |-  ( ( W  e.  PreHil  /\  B  C_  ( Base `  W
) )  ->  (
( ocv `  W
) `  ( N `  B ) )  =  ( ( ocv `  W
) `  B )
)
71, 3, 6syl2anc 665 . . . . 5  |-  ( B  e.  (OBasis `  W
)  ->  ( ( ocv `  W ) `  ( N `  B ) )  =  ( ( ocv `  W ) `
 B ) )
87fveq2d 5885 . . . 4  |-  ( B  e.  (OBasis `  W
)  ->  ( ( ocv `  W ) `  ( ( ocv `  W
) `  ( N `  B ) ) )  =  ( ( ocv `  W ) `  (
( ocv `  W
) `  B )
) )
94, 2obs2ocv 19221 . . . 4  |-  ( B  e.  (OBasis `  W
)  ->  ( ( ocv `  W ) `  ( ( ocv `  W
) `  B )
)  =  ( Base `  W ) )
108, 9eqtrd 2470 . . 3  |-  ( B  e.  (OBasis `  W
)  ->  ( ( ocv `  W ) `  ( ( ocv `  W
) `  ( N `  B ) ) )  =  ( Base `  W
) )
1110eqeq2d 2443 . 2  |-  ( B  e.  (OBasis `  W
)  ->  ( ( N `  B )  =  ( ( ocv `  W ) `  (
( ocv `  W
) `  ( N `  B ) ) )  <-> 
( N `  B
)  =  ( Base `  W ) ) )
12 obslbs.c . . . 4  |-  C  =  ( CSubSp `  W )
134, 12iscss 19177 . . 3  |-  ( W  e.  PreHil  ->  ( ( N `
 B )  e.  C  <->  ( N `  B )  =  ( ( ocv `  W
) `  ( ( ocv `  W ) `  ( N `  B ) ) ) ) )
141, 13syl 17 . 2  |-  ( B  e.  (OBasis `  W
)  ->  ( ( N `  B )  e.  C  <->  ( N `  B )  =  ( ( ocv `  W
) `  ( ( ocv `  W ) `  ( N `  B ) ) ) ) )
15 phllvec 19127 . . . 4  |-  ( W  e.  PreHil  ->  W  e.  LVec )
161, 15syl 17 . . 3  |-  ( B  e.  (OBasis `  W
)  ->  W  e.  LVec )
17 pssnel 3866 . . . . . . 7  |-  ( x 
C.  B  ->  E. y
( y  e.  B  /\  -.  y  e.  x
) )
1817adantl 467 . . . . . 6  |-  ( ( B  e.  (OBasis `  W )  /\  x  C.  B )  ->  E. y
( y  e.  B  /\  -.  y  e.  x
) )
19 simpll 758 . . . . . . . . . . 11  |-  ( ( ( B  e.  (OBasis `  W )  /\  x  C.  B )  /\  y  e.  B )  ->  B  e.  (OBasis `  W )
)
20 pssss 3566 . . . . . . . . . . . 12  |-  ( x 
C.  B  ->  x  C_  B )
2120ad2antlr 731 . . . . . . . . . . 11  |-  ( ( ( B  e.  (OBasis `  W )  /\  x  C.  B )  /\  y  e.  B )  ->  x  C_  B )
22 simpr 462 . . . . . . . . . . 11  |-  ( ( ( B  e.  (OBasis `  W )  /\  x  C.  B )  /\  y  e.  B )  ->  y  e.  B )
234obselocv 19222 . . . . . . . . . . 11  |-  ( ( B  e.  (OBasis `  W )  /\  x  C_  B  /\  y  e.  B )  ->  (
y  e.  ( ( ocv `  W ) `
 x )  <->  -.  y  e.  x ) )
2419, 21, 22, 23syl3anc 1264 . . . . . . . . . 10  |-  ( ( ( B  e.  (OBasis `  W )  /\  x  C.  B )  /\  y  e.  B )  ->  (
y  e.  ( ( ocv `  W ) `
 x )  <->  -.  y  e.  x ) )
25 eqid 2429 . . . . . . . . . . . . . 14  |-  ( 0g
`  W )  =  ( 0g `  W
)
2625obsne0 19219 . . . . . . . . . . . . 13  |-  ( ( B  e.  (OBasis `  W )  /\  y  e.  B )  ->  y  =/=  ( 0g `  W
) )
2719, 22, 26syl2anc 665 . . . . . . . . . . . 12  |-  ( ( ( B  e.  (OBasis `  W )  /\  x  C.  B )  /\  y  e.  B )  ->  y  =/=  ( 0g `  W
) )
28 elsni 4027 . . . . . . . . . . . . 13  |-  ( y  e.  { ( 0g
`  W ) }  ->  y  =  ( 0g `  W ) )
2928necon3ai 2659 . . . . . . . . . . . 12  |-  ( y  =/=  ( 0g `  W )  ->  -.  y  e.  { ( 0g `  W ) } )
3027, 29syl 17 . . . . . . . . . . 11  |-  ( ( ( B  e.  (OBasis `  W )  /\  x  C.  B )  /\  y  e.  B )  ->  -.  y  e.  { ( 0g `  W ) } )
31 nelne1 2760 . . . . . . . . . . . 12  |-  ( ( y  e.  ( ( ocv `  W ) `
 x )  /\  -.  y  e.  { ( 0g `  W ) } )  ->  (
( ocv `  W
) `  x )  =/=  { ( 0g `  W ) } )
3231expcom 436 . . . . . . . . . . 11  |-  ( -.  y  e.  { ( 0g `  W ) }  ->  ( y  e.  ( ( ocv `  W
) `  x )  ->  ( ( ocv `  W
) `  x )  =/=  { ( 0g `  W ) } ) )
3330, 32syl 17 . . . . . . . . . 10  |-  ( ( ( B  e.  (OBasis `  W )  /\  x  C.  B )  /\  y  e.  B )  ->  (
y  e.  ( ( ocv `  W ) `
 x )  -> 
( ( ocv `  W
) `  x )  =/=  { ( 0g `  W ) } ) )
3424, 33sylbird 238 . . . . . . . . 9  |-  ( ( ( B  e.  (OBasis `  W )  /\  x  C.  B )  /\  y  e.  B )  ->  ( -.  y  e.  x  ->  ( ( ocv `  W
) `  x )  =/=  { ( 0g `  W ) } ) )
35 npss 3581 . . . . . . . . . . 11  |-  ( -.  ( N `  x
)  C.  ( Base `  W )  <->  ( ( N `  x )  C_  ( Base `  W
)  ->  ( N `  x )  =  (
Base `  W )
) )
36 phllmod 19128 . . . . . . . . . . . . . . 15  |-  ( W  e.  PreHil  ->  W  e.  LMod )
371, 36syl 17 . . . . . . . . . . . . . 14  |-  ( B  e.  (OBasis `  W
)  ->  W  e.  LMod )
3837ad2antrr 730 . . . . . . . . . . . . 13  |-  ( ( ( B  e.  (OBasis `  W )  /\  x  C.  B )  /\  y  e.  B )  ->  W  e.  LMod )
393ad2antrr 730 . . . . . . . . . . . . . 14  |-  ( ( ( B  e.  (OBasis `  W )  /\  x  C.  B )  /\  y  e.  B )  ->  B  C_  ( Base `  W
) )
4021, 39sstrd 3480 . . . . . . . . . . . . 13  |-  ( ( ( B  e.  (OBasis `  W )  /\  x  C.  B )  /\  y  e.  B )  ->  x  C_  ( Base `  W
) )
412, 5lspssv 18141 . . . . . . . . . . . . 13  |-  ( ( W  e.  LMod  /\  x  C_  ( Base `  W
) )  ->  ( N `  x )  C_  ( Base `  W
) )
4238, 40, 41syl2anc 665 . . . . . . . . . . . 12  |-  ( ( ( B  e.  (OBasis `  W )  /\  x  C.  B )  /\  y  e.  B )  ->  ( N `  x )  C_  ( Base `  W
) )
43 fveq2 5881 . . . . . . . . . . . . 13  |-  ( ( N `  x )  =  ( Base `  W
)  ->  ( ( ocv `  W ) `  ( N `  x ) )  =  ( ( ocv `  W ) `
 ( Base `  W
) ) )
441ad2antrr 730 . . . . . . . . . . . . . . 15  |-  ( ( ( B  e.  (OBasis `  W )  /\  x  C.  B )  /\  y  e.  B )  ->  W  e.  PreHil )
452, 4, 5ocvlsp 19170 . . . . . . . . . . . . . . 15  |-  ( ( W  e.  PreHil  /\  x  C_  ( Base `  W
) )  ->  (
( ocv `  W
) `  ( N `  x ) )  =  ( ( ocv `  W
) `  x )
)
4644, 40, 45syl2anc 665 . . . . . . . . . . . . . 14  |-  ( ( ( B  e.  (OBasis `  W )  /\  x  C.  B )  /\  y  e.  B )  ->  (
( ocv `  W
) `  ( N `  x ) )  =  ( ( ocv `  W
) `  x )
)
472, 4, 25ocv1 19173 . . . . . . . . . . . . . . 15  |-  ( W  e.  PreHil  ->  ( ( ocv `  W ) `  ( Base `  W ) )  =  { ( 0g
`  W ) } )
4844, 47syl 17 . . . . . . . . . . . . . 14  |-  ( ( ( B  e.  (OBasis `  W )  /\  x  C.  B )  /\  y  e.  B )  ->  (
( ocv `  W
) `  ( Base `  W ) )  =  { ( 0g `  W ) } )
4946, 48eqeq12d 2451 . . . . . . . . . . . . 13  |-  ( ( ( B  e.  (OBasis `  W )  /\  x  C.  B )  /\  y  e.  B )  ->  (
( ( ocv `  W
) `  ( N `  x ) )  =  ( ( ocv `  W
) `  ( Base `  W ) )  <->  ( ( ocv `  W ) `  x )  =  {
( 0g `  W
) } ) )
5043, 49syl5ib 222 . . . . . . . . . . . 12  |-  ( ( ( B  e.  (OBasis `  W )  /\  x  C.  B )  /\  y  e.  B )  ->  (
( N `  x
)  =  ( Base `  W )  ->  (
( ocv `  W
) `  x )  =  { ( 0g `  W ) } ) )
5142, 50embantd 56 . . . . . . . . . . 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 ) } ) )
5235, 51syl5bi 220 . . . . . . . . . 10  |-  ( ( ( B  e.  (OBasis `  W )  /\  x  C.  B )  /\  y  e.  B )  ->  ( -.  ( N `  x
)  C.  ( Base `  W )  ->  (
( ocv `  W
) `  x )  =  { ( 0g `  W ) } ) )
5352necon1ad 2647 . . . . . . . . 9  |-  ( ( ( B  e.  (OBasis `  W )  /\  x  C.  B )  /\  y  e.  B )  ->  (
( ( ocv `  W
) `  x )  =/=  { ( 0g `  W ) }  ->  ( N `  x ) 
C.  ( Base `  W
) ) )
5434, 53syld 45 . . . . . . . 8  |-  ( ( ( B  e.  (OBasis `  W )  /\  x  C.  B )  /\  y  e.  B )  ->  ( -.  y  e.  x  ->  ( N `  x
)  C.  ( Base `  W ) ) )
5554expimpd 606 . . . . . . 7  |-  ( ( B  e.  (OBasis `  W )  /\  x  C.  B )  ->  (
( y  e.  B  /\  -.  y  e.  x
)  ->  ( N `  x )  C.  ( Base `  W ) ) )
5655exlimdv 1771 . . . . . 6  |-  ( ( B  e.  (OBasis `  W )  /\  x  C.  B )  ->  ( E. y ( y  e.  B  /\  -.  y  e.  x )  ->  ( N `  x )  C.  ( Base `  W
) ) )
5718, 56mpd 15 . . . . 5  |-  ( ( B  e.  (OBasis `  W )  /\  x  C.  B )  ->  ( N `  x )  C.  ( Base `  W
) )
5857ex 435 . . . 4  |-  ( B  e.  (OBasis `  W
)  ->  ( x  C.  B  ->  ( N `  x )  C.  ( Base `  W ) ) )
5958alrimiv 1766 . . 3  |-  ( B  e.  (OBasis `  W
)  ->  A. x
( x  C.  B  ->  ( N `  x
)  C.  ( Base `  W ) ) )
60 obslbs.j . . . . . 6  |-  J  =  (LBasis `  W )
612, 60, 5islbs3 18313 . . . . 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
) ) ) ) )
62 3anan32 994 . . . . 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
) ) )
6361, 62syl6bb 264 . . . 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 )
) ) )
6463baibd 917 . . 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 )
) )
6516, 3, 59, 64syl12anc 1262 . 2  |-  ( B  e.  (OBasis `  W
)  ->  ( B  e.  J  <->  ( N `  B )  =  (
Base `  W )
) )
6611, 14, 653bitr4rd 289 1  |-  ( B  e.  (OBasis `  W
)  ->  ( B  e.  J  <->  ( N `  B )  e.  C
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982   A.wal 1435    = wceq 1437   E.wex 1659    e. wcel 1870    =/= wne 2625    C_ wss 3442    C. wpss 3443   {csn 4002   ` cfv 5601   Basecbs 15084   0gc0g 15297   LModclmod 18026   LSpanclspn 18129  LBasisclbs 18232   LVecclvec 18260   PreHilcphl 19122   ocvcocv 19154   CSubSpccss 19155  OBasiscobs 19196
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-tpos 6981  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-er 7371  df-map 7482  df-en 7578  df-dom 7579  df-sdom 7580  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15165  df-mulr 15166  df-sca 15168  df-vsca 15169  df-ip 15170  df-0g 15299  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-mhm 16533  df-grp 16624  df-minusg 16625  df-sbg 16626  df-ghm 16832  df-mgp 17659  df-ur 17671  df-ring 17717  df-oppr 17786  df-dvdsr 17804  df-unit 17805  df-invr 17835  df-rnghom 17878  df-drng 17912  df-staf 18008  df-srng 18009  df-lmod 18028  df-lss 18091  df-lsp 18130  df-lmhm 18180  df-lbs 18233  df-lvec 18261  df-sra 18330  df-rgmod 18331  df-phl 19124  df-ocv 19157  df-css 19158  df-obs 19199
This theorem is referenced by: (None)
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