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Theorem obselocv 19289
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 2422 . . . . . . 7  |-  ( 0g
`  W )  =  ( 0g `  W
)
21obsne0 19286 . . . . . 6  |-  ( ( B  e.  (OBasis `  W )  /\  A  e.  B )  ->  A  =/=  ( 0g `  W
) )
323adant2 1024 . . . . 5  |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  A  =/=  ( 0g `  W
) )
4 elin 3649 . . . . . . . 8  |-  ( A  e.  ( C  i^i  (  ._|_  `  C )
)  <->  ( A  e.  C  /\  A  e.  (  ._|_  `  C ) ) )
5 obsrcl 19284 . . . . . . . . . . . . . 14  |-  ( B  e.  (OBasis `  W
)  ->  W  e.  PreHil )
653ad2ant1 1026 . . . . . . . . . . . . 13  |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  W  e.  PreHil )
7 phllmod 19195 . . . . . . . . . . . . 13  |-  ( W  e.  PreHil  ->  W  e.  LMod )
86, 7syl 17 . . . . . . . . . . . 12  |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  W  e.  LMod )
9 simp2 1006 . . . . . . . . . . . . 13  |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  C  C_  B )
10 eqid 2422 . . . . . . . . . . . . . . 15  |-  ( Base `  W )  =  (
Base `  W )
1110obsss 19285 . . . . . . . . . . . . . 14  |-  ( B  e.  (OBasis `  W
)  ->  B  C_  ( Base `  W ) )
12113ad2ant1 1026 . . . . . . . . . . . . 13  |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  B  C_  ( Base `  W
) )
139, 12sstrd 3474 . . . . . . . . . . . 12  |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  C  C_  ( Base `  W
) )
14 eqid 2422 . . . . . . . . . . . . 13  |-  ( LSpan `  W )  =  (
LSpan `  W )
1510, 14lspssid 18207 . . . . . . . . . . . 12  |-  ( ( W  e.  LMod  /\  C  C_  ( Base `  W
) )  ->  C  C_  ( ( LSpan `  W
) `  C )
)
168, 13, 15syl2anc 665 . . . . . . . . . . 11  |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  C  C_  ( ( LSpan `  W
) `  C )
)
17 ssrin 3687 . . . . . . . . . . 11  |-  ( C 
C_  ( ( LSpan `  W ) `  C
)  ->  ( C  i^i  (  ._|_  `  C
) )  C_  (
( ( LSpan `  W
) `  C )  i^i  (  ._|_  `  C
) ) )
1816, 17syl 17 . . . . . . . . . 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 19237 . . . . . . . . . . . . 13  |-  ( ( W  e.  PreHil  /\  C  C_  ( Base `  W
) )  ->  (  ._|_  `  ( ( LSpan `  W ) `  C
) )  =  ( 
._|_  `  C ) )
216, 13, 20syl2anc 665 . . . . . . . . . . . 12  |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  (  ._|_  `  ( ( LSpan `  W ) `  C
) )  =  ( 
._|_  `  C ) )
2221ineq2d 3664 . . . . . . . . . . 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 2422 . . . . . . . . . . . . . 14  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
2410, 23, 14lspcl 18198 . . . . . . . . . . . . 13  |-  ( ( W  e.  LMod  /\  C  C_  ( Base `  W
) )  ->  (
( LSpan `  W ) `  C )  e.  (
LSubSp `  W ) )
258, 13, 24syl2anc 665 . . . . . . . . . . . 12  |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  (
( LSpan `  W ) `  C )  e.  (
LSubSp `  W ) )
2619, 23, 1ocvin 19235 . . . . . . . . . . . 12  |-  ( ( W  e.  PreHil  /\  (
( LSpan `  W ) `  C )  e.  (
LSubSp `  W ) )  ->  ( ( (
LSpan `  W ) `  C )  i^i  (  ._|_  `  ( ( LSpan `  W ) `  C
) ) )  =  { ( 0g `  W ) } )
276, 25, 26syl2anc 665 . . . . . . . . . . 11  |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  (
( ( LSpan `  W
) `  C )  i^i  (  ._|_  `  (
( LSpan `  W ) `  C ) ) )  =  { ( 0g
`  W ) } )
2822, 27eqtr3d 2465 . . . . . . . . . 10  |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  (
( ( LSpan `  W
) `  C )  i^i  (  ._|_  `  C
) )  =  {
( 0g `  W
) } )
2918, 28sseqtrd 3500 . . . . . . . . 9  |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  ( C  i^i  (  ._|_  `  C
) )  C_  { ( 0g `  W ) } )
3029sseld 3463 . . . . . . . 8  |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  ( A  e.  ( C  i^i  (  ._|_  `  C
) )  ->  A  e.  { ( 0g `  W ) } ) )
314, 30syl5bir 221 . . . . . . 7  |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  (
( A  e.  C  /\  A  e.  (  ._|_  `  C ) )  ->  A  e.  {
( 0g `  W
) } ) )
32 elsni 4023 . . . . . . 7  |-  ( A  e.  { ( 0g
`  W ) }  ->  A  =  ( 0g `  W ) )
3331, 32syl6 34 . . . . . 6  |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  (
( A  e.  C  /\  A  e.  (  ._|_  `  C ) )  ->  A  =  ( 0g `  W ) ) )
3433necon3ad 2630 . . . . 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 423 . . . 4  |-  ( ( A  e.  C  ->  -.  A  e.  (  ._|_  `  C ) )  <->  -.  ( A  e.  C  /\  A  e.  (  ._|_  `  C ) ) )
3735, 36sylibr 215 . . 3  |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  ( A  e.  C  ->  -.  A  e.  (  ._|_  `  C ) ) )
3837con2d 118 . 2  |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  ( A  e.  (  ._|_  `  C )  ->  -.  A  e.  C )
)
39 simpr 462 . . . . . . 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 225 . . . . . 6  |-  ( ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  /\  x  e.  C )  ->  ( A  =  x  ->  A  e.  C ) )
4241con3d 138 . . . . 5  |-  ( ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  /\  x  e.  C )  ->  ( -.  A  e.  C  ->  -.  A  =  x ) )
43 simpl1 1008 . . . . . . 7  |-  ( ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  /\  x  e.  C )  ->  B  e.  (OBasis `  W )
)
44 simpl3 1010 . . . . . . 7  |-  ( ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  /\  x  e.  C )  ->  A  e.  B )
459sselda 3464 . . . . . . 7  |-  ( ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  /\  x  e.  C )  ->  x  e.  B )
46 eqid 2422 . . . . . . . 8  |-  ( .i
`  W )  =  ( .i `  W
)
47 eqid 2422 . . . . . . . 8  |-  (Scalar `  W )  =  (Scalar `  W )
48 eqid 2422 . . . . . . . 8  |-  ( 1r
`  (Scalar `  W )
)  =  ( 1r
`  (Scalar `  W )
)
49 eqid 2422 . . . . . . . 8  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
5010, 46, 47, 48, 49obsip 19282 . . . . . . 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 1264 . . . . . 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 3920 . . . . . . 7  |-  ( -.  A  =  x  ->  if ( A  =  x ,  ( 1r `  (Scalar `  W ) ) ,  ( 0g `  (Scalar `  W ) ) )  =  ( 0g
`  (Scalar `  W )
) )
5352eqeq2d 2436 . . . . . 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 223 . . . . 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 45 . . . 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 2840 . . 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 1007 . . . . 5  |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  A  e.  B )
5812, 57sseldd 3465 . . . 4  |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  A  e.  ( Base `  W
) )
5910, 46, 47, 49, 19elocv 19229 . . . . . 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 984 . . . . . 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 252 . . . . 5  |-  ( A  e.  (  ._|_  `  C
)  <->  ( ( C 
C_  ( Base `  W
)  /\  A  e.  ( Base `  W )
)  /\  A. x  e.  C  ( A
( .i `  W
) x )  =  ( 0g `  (Scalar `  W ) ) ) )
6261baib 911 . . . 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 665 . . 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 237 . 2  |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  ( -.  A  e.  C  ->  A  e.  (  ._|_  `  C ) ) )
6538, 64impbid 193 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872    =/= wne 2614   A.wral 2771    i^i cin 3435    C_ wss 3436   ifcif 3911   {csn 3998   ` cfv 5601  (class class class)co 6305   Basecbs 15120  Scalarcsca 15192   .icip 15194   0gc0g 15337   1rcur 17734   LModclmod 18090   LSubSpclss 18154   LSpanclspn 18193   PreHilcphl 19189   ocvcocv 19221  OBasiscobs 19263
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-cnex 9602  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-mulrcl 9609  ax-mulcom 9610  ax-addass 9611  ax-mulass 9612  ax-distr 9613  ax-i2m1 9614  ax-1ne0 9615  ax-1rid 9616  ax-rnegex 9617  ax-rrecex 9618  ax-cnre 9619  ax-pre-lttri 9620  ax-pre-lttrn 9621  ax-pre-ltadd 9622  ax-pre-mulgt0 9623
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 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-int 4256  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  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 6984  df-wrecs 7039  df-recs 7101  df-rdg 7139  df-er 7374  df-map 7485  df-en 7581  df-dom 7582  df-sdom 7583  df-pnf 9684  df-mnf 9685  df-xr 9686  df-ltxr 9687  df-le 9688  df-sub 9869  df-neg 9870  df-nn 10617  df-2 10675  df-3 10676  df-4 10677  df-5 10678  df-6 10679  df-7 10680  df-8 10681  df-ndx 15123  df-slot 15124  df-base 15125  df-sets 15126  df-plusg 15202  df-mulr 15203  df-sca 15205  df-vsca 15206  df-ip 15207  df-0g 15339  df-mgm 16487  df-sgrp 16526  df-mnd 16536  df-mhm 16581  df-grp 16672  df-minusg 16673  df-sbg 16674  df-ghm 16880  df-mgp 17723  df-ur 17735  df-ring 17781  df-oppr 17850  df-dvdsr 17868  df-unit 17869  df-rnghom 17942  df-drng 17976  df-staf 18072  df-srng 18073  df-lmod 18092  df-lss 18155  df-lsp 18194  df-lmhm 18244  df-lvec 18325  df-sra 18394  df-rgmod 18395  df-phl 19191  df-ocv 19224  df-obs 19266
This theorem is referenced by:  obs2ss  19290  obslbs  19291
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