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Theorem obselocv 18886
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 2457 . . . . . . 7  |-  ( 0g
`  W )  =  ( 0g `  W
)
21obsne0 18883 . . . . . 6  |-  ( ( B  e.  (OBasis `  W )  /\  A  e.  B )  ->  A  =/=  ( 0g `  W
) )
323adant2 1015 . . . . 5  |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  A  =/=  ( 0g `  W
) )
4 elin 3683 . . . . . . . 8  |-  ( A  e.  ( C  i^i  (  ._|_  `  C )
)  <->  ( A  e.  C  /\  A  e.  (  ._|_  `  C ) ) )
5 obsrcl 18881 . . . . . . . . . . . . . 14  |-  ( B  e.  (OBasis `  W
)  ->  W  e.  PreHil )
653ad2ant1 1017 . . . . . . . . . . . . 13  |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  W  e.  PreHil )
7 phllmod 18792 . . . . . . . . . . . . 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 997 . . . . . . . . . . . . 13  |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  C  C_  B )
10 eqid 2457 . . . . . . . . . . . . . . 15  |-  ( Base `  W )  =  (
Base `  W )
1110obsss 18882 . . . . . . . . . . . . . 14  |-  ( B  e.  (OBasis `  W
)  ->  B  C_  ( Base `  W ) )
12113ad2ant1 1017 . . . . . . . . . . . . 13  |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  B  C_  ( Base `  W
) )
139, 12sstrd 3509 . . . . . . . . . . . 12  |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  C  C_  ( Base `  W
) )
14 eqid 2457 . . . . . . . . . . . . 13  |-  ( LSpan `  W )  =  (
LSpan `  W )
1510, 14lspssid 17758 . . . . . . . . . . . 12  |-  ( ( W  e.  LMod  /\  C  C_  ( Base `  W
) )  ->  C  C_  ( ( LSpan `  W
) `  C )
)
168, 13, 15syl2anc 661 . . . . . . . . . . 11  |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  C  C_  ( ( LSpan `  W
) `  C )
)
17 ssrin 3719 . . . . . . . . . . 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 18834 . . . . . . . . . . . . 13  |-  ( ( W  e.  PreHil  /\  C  C_  ( Base `  W
) )  ->  (  ._|_  `  ( ( LSpan `  W ) `  C
) )  =  ( 
._|_  `  C ) )
216, 13, 20syl2anc 661 . . . . . . . . . . . 12  |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  (  ._|_  `  ( ( LSpan `  W ) `  C
) )  =  ( 
._|_  `  C ) )
2221ineq2d 3696 . . . . . . . . . . 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 2457 . . . . . . . . . . . . . 14  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
2410, 23, 14lspcl 17749 . . . . . . . . . . . . 13  |-  ( ( W  e.  LMod  /\  C  C_  ( Base `  W
) )  ->  (
( LSpan `  W ) `  C )  e.  (
LSubSp `  W ) )
258, 13, 24syl2anc 661 . . . . . . . . . . . 12  |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  (
( LSpan `  W ) `  C )  e.  (
LSubSp `  W ) )
2619, 23, 1ocvin 18832 . . . . . . . . . . . 12  |-  ( ( W  e.  PreHil  /\  (
( LSpan `  W ) `  C )  e.  (
LSubSp `  W ) )  ->  ( ( (
LSpan `  W ) `  C )  i^i  (  ._|_  `  ( ( LSpan `  W ) `  C
) ) )  =  { ( 0g `  W ) } )
276, 25, 26syl2anc 661 . . . . . . . . . . 11  |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  (
( ( LSpan `  W
) `  C )  i^i  (  ._|_  `  (
( LSpan `  W ) `  C ) ) )  =  { ( 0g
`  W ) } )
2822, 27eqtr3d 2500 . . . . . . . . . 10  |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  (
( ( LSpan `  W
) `  C )  i^i  (  ._|_  `  C
) )  =  {
( 0g `  W
) } )
2918, 28sseqtrd 3535 . . . . . . . . 9  |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  ( C  i^i  (  ._|_  `  C
) )  C_  { ( 0g `  W ) } )
3029sseld 3498 . . . . . . . 8  |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  ( A  e.  ( C  i^i  (  ._|_  `  C
) )  ->  A  e.  { ( 0g `  W ) } ) )
314, 30syl5bir 218 . . . . . . 7  |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  (
( A  e.  C  /\  A  e.  (  ._|_  `  C ) )  ->  A  e.  {
( 0g `  W
) } ) )
32 elsni 4057 . . . . . . 7  |-  ( A  e.  { ( 0g
`  W ) }  ->  A  =  ( 0g `  W ) )
3331, 32syl6 33 . . . . . 6  |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  (
( A  e.  C  /\  A  e.  (  ._|_  `  C ) )  ->  A  =  ( 0g `  W ) ) )
3433necon3ad 2667 . . . . 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 422 . . . 4  |-  ( ( A  e.  C  ->  -.  A  e.  (  ._|_  `  C ) )  <->  -.  ( A  e.  C  /\  A  e.  (  ._|_  `  C ) ) )
3735, 36sylibr 212 . . 3  |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  ( A  e.  C  ->  -.  A  e.  (  ._|_  `  C ) ) )
3837con2d 115 . 2  |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  ( A  e.  (  ._|_  `  C )  ->  -.  A  e.  C )
)
39 simpr 461 . . . . . . 7  |-  ( ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  /\  x  e.  C )  ->  x  e.  C )
40 eleq1 2529 . . . . . . 7  |-  ( A  =  x  ->  ( A  e.  C  <->  x  e.  C ) )
4139, 40syl5ibrcom 222 . . . . . 6  |-  ( ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  /\  x  e.  C )  ->  ( A  =  x  ->  A  e.  C ) )
4241con3d 133 . . . . 5  |-  ( ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  /\  x  e.  C )  ->  ( -.  A  e.  C  ->  -.  A  =  x ) )
43 simpl1 999 . . . . . . 7  |-  ( ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  /\  x  e.  C )  ->  B  e.  (OBasis `  W )
)
44 simpl3 1001 . . . . . . 7  |-  ( ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  /\  x  e.  C )  ->  A  e.  B )
459sselda 3499 . . . . . . 7  |-  ( ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  /\  x  e.  C )  ->  x  e.  B )
46 eqid 2457 . . . . . . . 8  |-  ( .i
`  W )  =  ( .i `  W
)
47 eqid 2457 . . . . . . . 8  |-  (Scalar `  W )  =  (Scalar `  W )
48 eqid 2457 . . . . . . . 8  |-  ( 1r
`  (Scalar `  W )
)  =  ( 1r
`  (Scalar `  W )
)
49 eqid 2457 . . . . . . . 8  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
5010, 46, 47, 48, 49obsip 18879 . . . . . . 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 1228 . . . . . 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 3953 . . . . . . 7  |-  ( -.  A  =  x  ->  if ( A  =  x ,  ( 1r `  (Scalar `  W ) ) ,  ( 0g `  (Scalar `  W ) ) )  =  ( 0g
`  (Scalar `  W )
) )
5352eqeq2d 2471 . . . . . 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 220 . . . . 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 44 . . . 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 2875 . . 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 998 . . . . 5  |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  A  e.  B )
5812, 57sseldd 3500 . . . 4  |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  A  e.  ( Base `  W
) )
5910, 46, 47, 49, 19elocv 18826 . . . . . 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 975 . . . . . 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 249 . . . . 5  |-  ( A  e.  (  ._|_  `  C
)  <->  ( ( C 
C_  ( Base `  W
)  /\  A  e.  ( Base `  W )
)  /\  A. x  e.  C  ( A
( .i `  W
) x )  =  ( 0g `  (Scalar `  W ) ) ) )
6261baib 903 . . . 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 661 . . 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 234 . 2  |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  ( -.  A  e.  C  ->  A  e.  (  ._|_  `  C ) ) )
6538, 64impbid 191 1  |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  ( A  e.  (  ._|_  `  C )  <->  -.  A  e.  C ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    =/= wne 2652   A.wral 2807    i^i cin 3470    C_ wss 3471   ifcif 3944   {csn 4032   ` cfv 5594  (class class class)co 6296   Basecbs 14644  Scalarcsca 14715   .icip 14717   0gc0g 14857   1rcur 17280   LModclmod 17639   LSubSpclss 17705   LSpanclspn 17744   PreHilcphl 18786   ocvcocv 18818  OBasiscobs 18860
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-tpos 6973  df-recs 7060  df-rdg 7094  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-plusg 14725  df-mulr 14726  df-sca 14728  df-vsca 14729  df-ip 14730  df-0g 14859  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-mhm 16093  df-grp 16184  df-minusg 16185  df-sbg 16186  df-ghm 16392  df-mgp 17269  df-ur 17281  df-ring 17327  df-oppr 17399  df-dvdsr 17417  df-unit 17418  df-rnghom 17491  df-drng 17525  df-staf 17621  df-srng 17622  df-lmod 17641  df-lss 17706  df-lsp 17745  df-lmhm 17795  df-lvec 17876  df-sra 17945  df-rgmod 17946  df-phl 18788  df-ocv 18821  df-obs 18863
This theorem is referenced by:  obs2ss  18887  obslbs  18888
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