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Theorem obselocv 16947
Description: A basis element is in the orthocomplement of a subset of the basis iff it is not in the subset. (Contributed by Mario Carneiro, 23-Oct-2015.)
Hypothesis
Ref Expression
obselocv.o  |-  ._|_  =  ( ocv `  W )
Assertion
Ref Expression
obselocv  |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  ( A  e.  (  ._|_  `  C )  <->  -.  A  e.  C ) )

Proof of Theorem obselocv
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eqid 2435 . . . . . . 7  |-  ( 0g
`  W )  =  ( 0g `  W
)
21obsne0 16944 . . . . . 6  |-  ( ( B  e.  (OBasis `  W )  /\  A  e.  B )  ->  A  =/=  ( 0g `  W
) )
323adant2 976 . . . . 5  |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  A  =/=  ( 0g `  W
) )
4 elin 3522 . . . . . . . 8  |-  ( A  e.  ( C  i^i  (  ._|_  `  C )
)  <->  ( A  e.  C  /\  A  e.  (  ._|_  `  C ) ) )
5 obsrcl 16942 . . . . . . . . . . . . . 14  |-  ( B  e.  (OBasis `  W
)  ->  W  e.  PreHil )
653ad2ant1 978 . . . . . . . . . . . . 13  |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  W  e.  PreHil )
7 phllmod 16853 . . . . . . . . . . . . 13  |-  ( W  e.  PreHil  ->  W  e.  LMod )
86, 7syl 16 . . . . . . . . . . . 12  |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  W  e.  LMod )
9 simp2 958 . . . . . . . . . . . . 13  |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  C  C_  B )
10 eqid 2435 . . . . . . . . . . . . . . 15  |-  ( Base `  W )  =  (
Base `  W )
1110obsss 16943 . . . . . . . . . . . . . 14  |-  ( B  e.  (OBasis `  W
)  ->  B  C_  ( Base `  W ) )
12113ad2ant1 978 . . . . . . . . . . . . 13  |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  B  C_  ( Base `  W
) )
139, 12sstrd 3350 . . . . . . . . . . . 12  |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  C  C_  ( Base `  W
) )
14 eqid 2435 . . . . . . . . . . . . 13  |-  ( LSpan `  W )  =  (
LSpan `  W )
1510, 14lspssid 16053 . . . . . . . . . . . 12  |-  ( ( W  e.  LMod  /\  C  C_  ( Base `  W
) )  ->  C  C_  ( ( LSpan `  W
) `  C )
)
168, 13, 15syl2anc 643 . . . . . . . . . . 11  |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  C  C_  ( ( LSpan `  W
) `  C )
)
17 ssrin 3558 . . . . . . . . . . 11  |-  ( C 
C_  ( ( LSpan `  W ) `  C
)  ->  ( C  i^i  (  ._|_  `  C
) )  C_  (
( ( LSpan `  W
) `  C )  i^i  (  ._|_  `  C
) ) )
1816, 17syl 16 . . . . . . . . . 10  |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  ( C  i^i  (  ._|_  `  C
) )  C_  (
( ( LSpan `  W
) `  C )  i^i  (  ._|_  `  C
) ) )
19 obselocv.o . . . . . . . . . . . . . 14  |-  ._|_  =  ( ocv `  W )
2010, 19, 14ocvlsp 16895 . . . . . . . . . . . . 13  |-  ( ( W  e.  PreHil  /\  C  C_  ( Base `  W
) )  ->  (  ._|_  `  ( ( LSpan `  W ) `  C
) )  =  ( 
._|_  `  C ) )
216, 13, 20syl2anc 643 . . . . . . . . . . . 12  |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  (  ._|_  `  ( ( LSpan `  W ) `  C
) )  =  ( 
._|_  `  C ) )
2221ineq2d 3534 . . . . . . . . . . 11  |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  (
( ( LSpan `  W
) `  C )  i^i  (  ._|_  `  (
( LSpan `  W ) `  C ) ) )  =  ( ( (
LSpan `  W ) `  C )  i^i  (  ._|_  `  C ) ) )
23 eqid 2435 . . . . . . . . . . . . . 14  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
2410, 23, 14lspcl 16044 . . . . . . . . . . . . 13  |-  ( ( W  e.  LMod  /\  C  C_  ( Base `  W
) )  ->  (
( LSpan `  W ) `  C )  e.  (
LSubSp `  W ) )
258, 13, 24syl2anc 643 . . . . . . . . . . . 12  |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  (
( LSpan `  W ) `  C )  e.  (
LSubSp `  W ) )
2619, 23, 1ocvin 16893 . . . . . . . . . . . 12  |-  ( ( W  e.  PreHil  /\  (
( LSpan `  W ) `  C )  e.  (
LSubSp `  W ) )  ->  ( ( (
LSpan `  W ) `  C )  i^i  (  ._|_  `  ( ( LSpan `  W ) `  C
) ) )  =  { ( 0g `  W ) } )
276, 25, 26syl2anc 643 . . . . . . . . . . 11  |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  (
( ( LSpan `  W
) `  C )  i^i  (  ._|_  `  (
( LSpan `  W ) `  C ) ) )  =  { ( 0g
`  W ) } )
2822, 27eqtr3d 2469 . . . . . . . . . 10  |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  (
( ( LSpan `  W
) `  C )  i^i  (  ._|_  `  C
) )  =  {
( 0g `  W
) } )
2918, 28sseqtrd 3376 . . . . . . . . 9  |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  ( C  i^i  (  ._|_  `  C
) )  C_  { ( 0g `  W ) } )
3029sseld 3339 . . . . . . . 8  |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  ( A  e.  ( C  i^i  (  ._|_  `  C
) )  ->  A  e.  { ( 0g `  W ) } ) )
314, 30syl5bir 210 . . . . . . 7  |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  (
( A  e.  C  /\  A  e.  (  ._|_  `  C ) )  ->  A  e.  {
( 0g `  W
) } ) )
32 elsni 3830 . . . . . . 7  |-  ( A  e.  { ( 0g
`  W ) }  ->  A  =  ( 0g `  W ) )
3331, 32syl6 31 . . . . . 6  |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  (
( A  e.  C  /\  A  e.  (  ._|_  `  C ) )  ->  A  =  ( 0g `  W ) ) )
3433necon3ad 2634 . . . . 5  |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  ( A  =/=  ( 0g `  W )  ->  -.  ( A  e.  C  /\  A  e.  (  ._|_  `  C ) ) ) )
353, 34mpd 15 . . . 4  |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  -.  ( A  e.  C  /\  A  e.  (  ._|_  `  C ) ) )
36 imnan 412 . . . 4  |-  ( ( A  e.  C  ->  -.  A  e.  (  ._|_  `  C ) )  <->  -.  ( A  e.  C  /\  A  e.  (  ._|_  `  C ) ) )
3735, 36sylibr 204 . . 3  |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  ( A  e.  C  ->  -.  A  e.  (  ._|_  `  C ) ) )
3837con2d 109 . 2  |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  ( A  e.  (  ._|_  `  C )  ->  -.  A  e.  C )
)
39 simpr 448 . . . . . . 7  |-  ( ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  /\  x  e.  C )  ->  x  e.  C )
40 eleq1 2495 . . . . . . 7  |-  ( A  =  x  ->  ( A  e.  C  <->  x  e.  C ) )
4139, 40syl5ibrcom 214 . . . . . 6  |-  ( ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  /\  x  e.  C )  ->  ( A  =  x  ->  A  e.  C ) )
4241con3d 127 . . . . 5  |-  ( ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  /\  x  e.  C )  ->  ( -.  A  e.  C  ->  -.  A  =  x ) )
43 simpl1 960 . . . . . . 7  |-  ( ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  /\  x  e.  C )  ->  B  e.  (OBasis `  W )
)
44 simpl3 962 . . . . . . 7  |-  ( ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  /\  x  e.  C )  ->  A  e.  B )
459sselda 3340 . . . . . . 7  |-  ( ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  /\  x  e.  C )  ->  x  e.  B )
46 eqid 2435 . . . . . . . 8  |-  ( .i
`  W )  =  ( .i `  W
)
47 eqid 2435 . . . . . . . 8  |-  (Scalar `  W )  =  (Scalar `  W )
48 eqid 2435 . . . . . . . 8  |-  ( 1r
`  (Scalar `  W )
)  =  ( 1r
`  (Scalar `  W )
)
49 eqid 2435 . . . . . . . 8  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
5010, 46, 47, 48, 49obsip 16940 . . . . . . 7  |-  ( ( B  e.  (OBasis `  W )  /\  A  e.  B  /\  x  e.  B )  ->  ( A ( .i `  W ) x )  =  if ( A  =  x ,  ( 1r `  (Scalar `  W ) ) ,  ( 0g `  (Scalar `  W ) ) ) )
5143, 44, 45, 50syl3anc 1184 . . . . . 6  |-  ( ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  /\  x  e.  C )  ->  ( A ( .i `  W ) x )  =  if ( A  =  x ,  ( 1r `  (Scalar `  W ) ) ,  ( 0g `  (Scalar `  W ) ) ) )
52 iffalse 3738 . . . . . . 7  |-  ( -.  A  =  x  ->  if ( A  =  x ,  ( 1r `  (Scalar `  W ) ) ,  ( 0g `  (Scalar `  W ) ) )  =  ( 0g
`  (Scalar `  W )
) )
5352eqeq2d 2446 . . . . . 6  |-  ( -.  A  =  x  -> 
( ( A ( .i `  W ) x )  =  if ( A  =  x ,  ( 1r `  (Scalar `  W ) ) ,  ( 0g `  (Scalar `  W ) ) )  <->  ( A ( .i `  W ) x )  =  ( 0g `  (Scalar `  W ) ) ) )
5451, 53syl5ibcom 212 . . . . 5  |-  ( ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  /\  x  e.  C )  ->  ( -.  A  =  x  ->  ( A ( .i
`  W ) x )  =  ( 0g
`  (Scalar `  W )
) ) )
5542, 54syld 42 . . . 4  |-  ( ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  /\  x  e.  C )  ->  ( -.  A  e.  C  ->  ( A ( .i
`  W ) x )  =  ( 0g
`  (Scalar `  W )
) ) )
5655ralrimdva 2788 . . 3  |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  ( -.  A  e.  C  ->  A. x  e.  C  ( A ( .i `  W ) x )  =  ( 0g `  (Scalar `  W ) ) ) )
57 simp3 959 . . . . 5  |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  A  e.  B )
5812, 57sseldd 3341 . . . 4  |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  A  e.  ( Base `  W
) )
5910, 46, 47, 49, 19elocv 16887 . . . . . 6  |-  ( A  e.  (  ._|_  `  C
)  <->  ( C  C_  ( Base `  W )  /\  A  e.  ( Base `  W )  /\  A. x  e.  C  ( A ( .i `  W ) x )  =  ( 0g `  (Scalar `  W ) ) ) )
60 df-3an 938 . . . . . 6  |-  ( ( C  C_  ( Base `  W )  /\  A  e.  ( Base `  W
)  /\  A. x  e.  C  ( A
( .i `  W
) x )  =  ( 0g `  (Scalar `  W ) ) )  <-> 
( ( C  C_  ( Base `  W )  /\  A  e.  ( Base `  W ) )  /\  A. x  e.  C  ( A ( .i `  W ) x )  =  ( 0g `  (Scalar `  W ) ) ) )
6159, 60bitri 241 . . . . 5  |-  ( A  e.  (  ._|_  `  C
)  <->  ( ( C 
C_  ( Base `  W
)  /\  A  e.  ( Base `  W )
)  /\  A. x  e.  C  ( A
( .i `  W
) x )  =  ( 0g `  (Scalar `  W ) ) ) )
6261baib 872 . . . 4  |-  ( ( C  C_  ( Base `  W )  /\  A  e.  ( Base `  W
) )  ->  ( A  e.  (  ._|_  `  C )  <->  A. x  e.  C  ( A
( .i `  W
) x )  =  ( 0g `  (Scalar `  W ) ) ) )
6313, 58, 62syl2anc 643 . . 3  |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  ( A  e.  (  ._|_  `  C )  <->  A. x  e.  C  ( A
( .i `  W
) x )  =  ( 0g `  (Scalar `  W ) ) ) )
6456, 63sylibrd 226 . 2  |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  ( -.  A  e.  C  ->  A  e.  (  ._|_  `  C ) ) )
6538, 64impbid 184 1  |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  ( A  e.  (  ._|_  `  C )  <->  -.  A  e.  C ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697    i^i cin 3311    C_ wss 3312   ifcif 3731   {csn 3806   ` cfv 5446  (class class class)co 6073   Basecbs 13461  Scalarcsca 13524   .icip 13526   0gc0g 13715   1rcur 15654   LModclmod 15942   LSubSpclss 16000   LSpanclspn 16039   PreHilcphl 16847   ocvcocv 16879  OBasiscobs 16921
This theorem is referenced by:  obs2ss  16948  obslbs  16949
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-tpos 6471  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-plusg 13534  df-mulr 13535  df-sca 13537  df-vsca 13538  df-0g 13719  df-mnd 14682  df-mhm 14730  df-grp 14804  df-minusg 14805  df-sbg 14806  df-ghm 14996  df-mgp 15641  df-rng 15655  df-ur 15657  df-oppr 15720  df-dvdsr 15738  df-unit 15739  df-rnghom 15811  df-drng 15829  df-staf 15925  df-srng 15926  df-lmod 15944  df-lss 16001  df-lsp 16040  df-lmhm 16090  df-lvec 16167  df-sra 16236  df-rgmod 16237  df-phl 16849  df-ocv 16882  df-obs 16924
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