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Theorem obselocv 16871
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 2380 . . . . . . 7  |-  ( 0g
`  W )  =  ( 0g `  W
)
21obsne0 16868 . . . . . 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 3466 . . . . . . . 8  |-  ( A  e.  ( C  i^i  (  ._|_  `  C )
)  <->  ( A  e.  C  /\  A  e.  (  ._|_  `  C ) ) )
5 obsrcl 16866 . . . . . . . . . . . . . 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 16777 . . . . . . . . . . . . 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 2380 . . . . . . . . . . . . . . 15  |-  ( Base `  W )  =  (
Base `  W )
1110obsss 16867 . . . . . . . . . . . . . 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 3294 . . . . . . . . . . . 12  |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  C  C_  ( Base `  W
) )
14 eqid 2380 . . . . . . . . . . . . 13  |-  ( LSpan `  W )  =  (
LSpan `  W )
1510, 14lspssid 15981 . . . . . . . . . . . 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 3502 . . . . . . . . . . 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 16819 . . . . . . . . . . . . 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 3478 . . . . . . . . . . 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 2380 . . . . . . . . . . . . . 14  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
2410, 23, 14lspcl 15972 . . . . . . . . . . . . 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 16817 . . . . . . . . . . . 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 2414 . . . . . . . . . 10  |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  (
( ( LSpan `  W
) `  C )  i^i  (  ._|_  `  C
) )  =  {
( 0g `  W
) } )
2918, 28sseqtrd 3320 . . . . . . . . 9  |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  ( C  i^i  (  ._|_  `  C
) )  C_  { ( 0g `  W ) } )
3029sseld 3283 . . . . . . . 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 3774 . . . . . . 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 2579 . . . . 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 2440 . . . . . . 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 3284 . . . . . . 7  |-  ( ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  /\  x  e.  C )  ->  x  e.  B )
46 eqid 2380 . . . . . . . 8  |-  ( .i
`  W )  =  ( .i `  W
)
47 eqid 2380 . . . . . . . 8  |-  (Scalar `  W )  =  (Scalar `  W )
48 eqid 2380 . . . . . . . 8  |-  ( 1r
`  (Scalar `  W )
)  =  ( 1r
`  (Scalar `  W )
)
49 eqid 2380 . . . . . . . 8  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
5010, 46, 47, 48, 49obsip 16864 . . . . . . 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 3682 . . . . . . 7  |-  ( -.  A  =  x  ->  if ( A  =  x ,  ( 1r `  (Scalar `  W ) ) ,  ( 0g `  (Scalar `  W ) ) )  =  ( 0g
`  (Scalar `  W )
) )
5352eqeq2d 2391 . . . . . 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 2732 . . 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 3285 . . . 4  |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  A  e.  ( Base `  W
) )
5910, 46, 47, 49, 19elocv 16811 . . . . . 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 1649    e. wcel 1717    =/= wne 2543   A.wral 2642    i^i cin 3255    C_ wss 3256   ifcif 3675   {csn 3750   ` cfv 5387  (class class class)co 6013   Basecbs 13389  Scalarcsca 13452   .icip 13454   0gc0g 13643   1rcur 15582   LModclmod 15870   LSubSpclss 15928   LSpanclspn 15967   PreHilcphl 16771   ocvcocv 16803  OBasiscobs 16845
This theorem is referenced by:  obs2ss  16872  obslbs  16873
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-tpos 6408  df-riota 6478  df-recs 6562  df-rdg 6597  df-er 6834  df-map 6949  df-en 7039  df-dom 7040  df-sdom 7041  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-nn 9926  df-2 9983  df-3 9984  df-4 9985  df-5 9986  df-6 9987  df-ndx 13392  df-slot 13393  df-base 13394  df-sets 13395  df-plusg 13462  df-mulr 13463  df-sca 13465  df-vsca 13466  df-0g 13647  df-mnd 14610  df-mhm 14658  df-grp 14732  df-minusg 14733  df-sbg 14734  df-ghm 14924  df-mgp 15569  df-rng 15583  df-ur 15585  df-oppr 15648  df-dvdsr 15666  df-unit 15667  df-rnghom 15739  df-drng 15757  df-staf 15853  df-srng 15854  df-lmod 15872  df-lss 15929  df-lsp 15968  df-lmhm 16018  df-lvec 16095  df-sra 16164  df-rgmod 16165  df-phl 16773  df-ocv 16806  df-obs 16848
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