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Theorem obslbs 16686
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 16679 . . . . . 6  |-  ( B  e.  (OBasis `  W
)  ->  W  e.  PreHil )
2 eqid 2316 . . . . . . 7  |-  ( Base `  W )  =  (
Base `  W )
32obsss 16680 . . . . . 6  |-  ( B  e.  (OBasis `  W
)  ->  B  C_  ( Base `  W ) )
4 eqid 2316 . . . . . . 7  |-  ( ocv `  W )  =  ( ocv `  W )
5 obslbs.n . . . . . . 7  |-  N  =  ( LSpan `  W )
62, 4, 5ocvlsp 16632 . . . . . 6  |-  ( ( W  e.  PreHil  /\  B  C_  ( Base `  W
) )  ->  (
( ocv `  W
) `  ( N `  B ) )  =  ( ( ocv `  W
) `  B )
)
71, 3, 6syl2anc 642 . . . . 5  |-  ( B  e.  (OBasis `  W
)  ->  ( ( ocv `  W ) `  ( N `  B ) )  =  ( ( ocv `  W ) `
 B ) )
87fveq2d 5567 . . . 4  |-  ( B  e.  (OBasis `  W
)  ->  ( ( ocv `  W ) `  ( ( ocv `  W
) `  ( N `  B ) ) )  =  ( ( ocv `  W ) `  (
( ocv `  W
) `  B )
) )
94, 2obs2ocv 16683 . . . 4  |-  ( B  e.  (OBasis `  W
)  ->  ( ( ocv `  W ) `  ( ( ocv `  W
) `  B )
)  =  ( Base `  W ) )
108, 9eqtrd 2348 . . 3  |-  ( B  e.  (OBasis `  W
)  ->  ( ( ocv `  W ) `  ( ( ocv `  W
) `  ( N `  B ) ) )  =  ( Base `  W
) )
1110eqeq2d 2327 . 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 16639 . . 3  |-  ( W  e.  PreHil  ->  ( ( N `
 B )  e.  C  <->  ( N `  B )  =  ( ( ocv `  W
) `  ( ( ocv `  W ) `  ( N `  B ) ) ) ) )
141, 13syl 15 . 2  |-  ( B  e.  (OBasis `  W
)  ->  ( ( N `  B )  e.  C  <->  ( N `  B )  =  ( ( ocv `  W
) `  ( ( ocv `  W ) `  ( N `  B ) ) ) ) )
15 phllvec 16589 . . . 4  |-  ( W  e.  PreHil  ->  W  e.  LVec )
161, 15syl 15 . . 3  |-  ( B  e.  (OBasis `  W
)  ->  W  e.  LVec )
17 pssnel 3553 . . . . . . 7  |-  ( x 
C.  B  ->  E. y
( y  e.  B  /\  -.  y  e.  x
) )
1817adantl 452 . . . . . 6  |-  ( ( B  e.  (OBasis `  W )  /\  x  C.  B )  ->  E. y
( y  e.  B  /\  -.  y  e.  x
) )
19 simpll 730 . . . . . . . . . . 11  |-  ( ( ( B  e.  (OBasis `  W )  /\  x  C.  B )  /\  y  e.  B )  ->  B  e.  (OBasis `  W )
)
20 pssss 3305 . . . . . . . . . . . 12  |-  ( x 
C.  B  ->  x  C_  B )
2120ad2antlr 707 . . . . . . . . . . 11  |-  ( ( ( B  e.  (OBasis `  W )  /\  x  C.  B )  /\  y  e.  B )  ->  x  C_  B )
22 simpr 447 . . . . . . . . . . 11  |-  ( ( ( B  e.  (OBasis `  W )  /\  x  C.  B )  /\  y  e.  B )  ->  y  e.  B )
234obselocv 16684 . . . . . . . . . . 11  |-  ( ( B  e.  (OBasis `  W )  /\  x  C_  B  /\  y  e.  B )  ->  (
y  e.  ( ( ocv `  W ) `
 x )  <->  -.  y  e.  x ) )
2419, 21, 22, 23syl3anc 1182 . . . . . . . . . 10  |-  ( ( ( B  e.  (OBasis `  W )  /\  x  C.  B )  /\  y  e.  B )  ->  (
y  e.  ( ( ocv `  W ) `
 x )  <->  -.  y  e.  x ) )
25 eqid 2316 . . . . . . . . . . . . . 14  |-  ( 0g
`  W )  =  ( 0g `  W
)
2625obsne0 16681 . . . . . . . . . . . . 13  |-  ( ( B  e.  (OBasis `  W )  /\  y  e.  B )  ->  y  =/=  ( 0g `  W
) )
2719, 22, 26syl2anc 642 . . . . . . . . . . . 12  |-  ( ( ( B  e.  (OBasis `  W )  /\  x  C.  B )  /\  y  e.  B )  ->  y  =/=  ( 0g `  W
) )
28 elsni 3698 . . . . . . . . . . . . 13  |-  ( y  e.  { ( 0g
`  W ) }  ->  y  =  ( 0g `  W ) )
2928necon3ai 2519 . . . . . . . . . . . 12  |-  ( y  =/=  ( 0g `  W )  ->  -.  y  e.  { ( 0g `  W ) } )
3027, 29syl 15 . . . . . . . . . . 11  |-  ( ( ( B  e.  (OBasis `  W )  /\  x  C.  B )  /\  y  e.  B )  ->  -.  y  e.  { ( 0g `  W ) } )
31 nelne1 2568 . . . . . . . . . . . 12  |-  ( ( y  e.  ( ( ocv `  W ) `
 x )  /\  -.  y  e.  { ( 0g `  W ) } )  ->  (
( ocv `  W
) `  x )  =/=  { ( 0g `  W ) } )
3231expcom 424 . . . . . . . . . . 11  |-  ( -.  y  e.  { ( 0g `  W ) }  ->  ( y  e.  ( ( ocv `  W
) `  x )  ->  ( ( ocv `  W
) `  x )  =/=  { ( 0g `  W ) } ) )
3330, 32syl 15 . . . . . . . . . 10  |-  ( ( ( B  e.  (OBasis `  W )  /\  x  C.  B )  /\  y  e.  B )  ->  (
y  e.  ( ( ocv `  W ) `
 x )  -> 
( ( ocv `  W
) `  x )  =/=  { ( 0g `  W ) } ) )
3424, 33sylbird 226 . . . . . . . . 9  |-  ( ( ( B  e.  (OBasis `  W )  /\  x  C.  B )  /\  y  e.  B )  ->  ( -.  y  e.  x  ->  ( ( ocv `  W
) `  x )  =/=  { ( 0g `  W ) } ) )
35 npss 3320 . . . . . . . . . . 11  |-  ( -.  ( N `  x
)  C.  ( Base `  W )  <->  ( ( N `  x )  C_  ( Base `  W
)  ->  ( N `  x )  =  (
Base `  W )
) )
36 phllmod 16590 . . . . . . . . . . . . . . 15  |-  ( W  e.  PreHil  ->  W  e.  LMod )
371, 36syl 15 . . . . . . . . . . . . . 14  |-  ( B  e.  (OBasis `  W
)  ->  W  e.  LMod )
3837ad2antrr 706 . . . . . . . . . . . . 13  |-  ( ( ( B  e.  (OBasis `  W )  /\  x  C.  B )  /\  y  e.  B )  ->  W  e.  LMod )
393ad2antrr 706 . . . . . . . . . . . . . 14  |-  ( ( ( B  e.  (OBasis `  W )  /\  x  C.  B )  /\  y  e.  B )  ->  B  C_  ( Base `  W
) )
4021, 39sstrd 3223 . . . . . . . . . . . . 13  |-  ( ( ( B  e.  (OBasis `  W )  /\  x  C.  B )  /\  y  e.  B )  ->  x  C_  ( Base `  W
) )
412, 5lspssv 15789 . . . . . . . . . . . . 13  |-  ( ( W  e.  LMod  /\  x  C_  ( Base `  W
) )  ->  ( N `  x )  C_  ( Base `  W
) )
4238, 40, 41syl2anc 642 . . . . . . . . . . . 12  |-  ( ( ( B  e.  (OBasis `  W )  /\  x  C.  B )  /\  y  e.  B )  ->  ( N `  x )  C_  ( Base `  W
) )
43 fveq2 5563 . . . . . . . . . . . . 13  |-  ( ( N `  x )  =  ( Base `  W
)  ->  ( ( ocv `  W ) `  ( N `  x ) )  =  ( ( ocv `  W ) `
 ( Base `  W
) ) )
441ad2antrr 706 . . . . . . . . . . . . . . 15  |-  ( ( ( B  e.  (OBasis `  W )  /\  x  C.  B )  /\  y  e.  B )  ->  W  e.  PreHil )
452, 4, 5ocvlsp 16632 . . . . . . . . . . . . . . 15  |-  ( ( W  e.  PreHil  /\  x  C_  ( Base `  W
) )  ->  (
( ocv `  W
) `  ( N `  x ) )  =  ( ( ocv `  W
) `  x )
)
4644, 40, 45syl2anc 642 . . . . . . . . . . . . . 14  |-  ( ( ( B  e.  (OBasis `  W )  /\  x  C.  B )  /\  y  e.  B )  ->  (
( ocv `  W
) `  ( N `  x ) )  =  ( ( ocv `  W
) `  x )
)
472, 4, 25ocv1 16635 . . . . . . . . . . . . . . 15  |-  ( W  e.  PreHil  ->  ( ( ocv `  W ) `  ( Base `  W ) )  =  { ( 0g
`  W ) } )
4844, 47syl 15 . . . . . . . . . . . . . 14  |-  ( ( ( B  e.  (OBasis `  W )  /\  x  C.  B )  /\  y  e.  B )  ->  (
( ocv `  W
) `  ( Base `  W ) )  =  { ( 0g `  W ) } )
4946, 48eqeq12d 2330 . . . . . . . . . . . . 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 210 . . . . . . . . . . . 12  |-  ( ( ( B  e.  (OBasis `  W )  /\  x  C.  B )  /\  y  e.  B )  ->  (
( N `  x
)  =  ( Base `  W )  ->  (
( ocv `  W
) `  x )  =  { ( 0g `  W ) } ) )
5142, 50embantd 50 . . . . . . . . . . 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 208 . . . . . . . . . 10  |-  ( ( ( B  e.  (OBasis `  W )  /\  x  C.  B )  /\  y  e.  B )  ->  ( -.  ( N `  x
)  C.  ( Base `  W )  ->  (
( ocv `  W
) `  x )  =  { ( 0g `  W ) } ) )
5352necon1ad 2546 . . . . . . . . 9  |-  ( ( ( B  e.  (OBasis `  W )  /\  x  C.  B )  /\  y  e.  B )  ->  (
( ( ocv `  W
) `  x )  =/=  { ( 0g `  W ) }  ->  ( N `  x ) 
C.  ( Base `  W
) ) )
5434, 53syld 40 . . . . . . . 8  |-  ( ( ( B  e.  (OBasis `  W )  /\  x  C.  B )  /\  y  e.  B )  ->  ( -.  y  e.  x  ->  ( N `  x
)  C.  ( Base `  W ) ) )
5554expimpd 586 . . . . . . 7  |-  ( ( B  e.  (OBasis `  W )  /\  x  C.  B )  ->  (
( y  e.  B  /\  -.  y  e.  x
)  ->  ( N `  x )  C.  ( Base `  W ) ) )
5655exlimdv 1627 . . . . . 6  |-  ( ( B  e.  (OBasis `  W )  /\  x  C.  B )  ->  ( E. y ( y  e.  B  /\  -.  y  e.  x )  ->  ( N `  x )  C.  ( Base `  W
) ) )
5718, 56mpd 14 . . . . 5  |-  ( ( B  e.  (OBasis `  W )  /\  x  C.  B )  ->  ( N `  x )  C.  ( Base `  W
) )
5857ex 423 . . . 4  |-  ( B  e.  (OBasis `  W
)  ->  ( x  C.  B  ->  ( N `
 x )  C.  ( Base `  W )
) )
5958alrimiv 1622 . . 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 15957 . . . . 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 946 . . . . 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 252 . . . 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 875 . . 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 1180 . 2  |-  ( B  e.  (OBasis `  W
)  ->  ( B  e.  J  <->  ( N `  B )  =  (
Base `  W )
) )
6611, 14, 653bitr4rd 277 1  |-  ( B  e.  (OBasis `  W
)  ->  ( B  e.  J  <->  ( N `  B )  e.  C
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   A.wal 1531   E.wex 1532    = wceq 1633    e. wcel 1701    =/= wne 2479    C_ wss 3186    C. wpss 3187   {csn 3674   ` cfv 5292   Basecbs 13195   0gc0g 13449   LModclmod 15676   LSpanclspn 15777  LBasisclbs 15876   LVecclvec 15904   PreHilcphl 16584   ocvcocv 16616   CSubSpccss 16617  OBasiscobs 16658
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-tpos 6276  df-riota 6346  df-recs 6430  df-rdg 6465  df-er 6702  df-map 6817  df-en 6907  df-dom 6908  df-sdom 6909  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-nn 9792  df-2 9849  df-3 9850  df-4 9851  df-5 9852  df-6 9853  df-ndx 13198  df-slot 13199  df-base 13200  df-sets 13201  df-ress 13202  df-plusg 13268  df-mulr 13269  df-sca 13271  df-vsca 13272  df-0g 13453  df-mnd 14416  df-mhm 14464  df-grp 14538  df-minusg 14539  df-sbg 14540  df-ghm 14730  df-mgp 15375  df-rng 15389  df-ur 15391  df-oppr 15454  df-dvdsr 15472  df-unit 15473  df-invr 15503  df-rnghom 15545  df-drng 15563  df-staf 15659  df-srng 15660  df-lmod 15678  df-lss 15739  df-lsp 15778  df-lmhm 15828  df-lbs 15877  df-lvec 15905  df-sra 15974  df-rgmod 15975  df-phl 16586  df-ocv 16619  df-css 16620  df-obs 16661
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