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Theorem obslbs 16630
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 16623 . . . . . 6  |-  ( B  e.  (OBasis `  W
)  ->  W  e.  PreHil )
2 eqid 2283 . . . . . . 7  |-  ( Base `  W )  =  (
Base `  W )
32obsss 16624 . . . . . 6  |-  ( B  e.  (OBasis `  W
)  ->  B  C_  ( Base `  W ) )
4 eqid 2283 . . . . . . 7  |-  ( ocv `  W )  =  ( ocv `  W )
5 obslbs.n . . . . . . 7  |-  N  =  ( LSpan `  W )
62, 4, 5ocvlsp 16576 . . . . . 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 5529 . . . 4  |-  ( B  e.  (OBasis `  W
)  ->  ( ( ocv `  W ) `  ( ( ocv `  W
) `  ( N `  B ) ) )  =  ( ( ocv `  W ) `  (
( ocv `  W
) `  B )
) )
94, 2obs2ocv 16627 . . . 4  |-  ( B  e.  (OBasis `  W
)  ->  ( ( ocv `  W ) `  ( ( ocv `  W
) `  B )
)  =  ( Base `  W ) )
108, 9eqtrd 2315 . . 3  |-  ( B  e.  (OBasis `  W
)  ->  ( ( ocv `  W ) `  ( ( ocv `  W
) `  ( N `  B ) ) )  =  ( Base `  W
) )
1110eqeq2d 2294 . 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 16583 . . 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 16533 . . . 4  |-  ( W  e.  PreHil  ->  W  e.  LVec )
161, 15syl 15 . . 3  |-  ( B  e.  (OBasis `  W
)  ->  W  e.  LVec )
17 pssnel 3519 . . . . . . 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 3271 . . . . . . . . . . . 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 16628 . . . . . . . . . . 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 2283 . . . . . . . . . . . . . 14  |-  ( 0g
`  W )  =  ( 0g `  W
)
2625obsne0 16625 . . . . . . . . . . . . 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 3664 . . . . . . . . . . . . 13  |-  ( y  e.  { ( 0g
`  W ) }  ->  y  =  ( 0g `  W ) )
2928necon3ai 2486 . . . . . . . . . . . 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 2535 . . . . . . . . . . . 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 3286 . . . . . . . . . . 11  |-  ( -.  ( N `  x
)  C.  ( Base `  W )  <->  ( ( N `  x )  C_  ( Base `  W
)  ->  ( N `  x )  =  (
Base `  W )
) )
36 phllmod 16534 . . . . . . . . . . . . . . 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 3189 . . . . . . . . . . . . 13  |-  ( ( ( B  e.  (OBasis `  W )  /\  x  C.  B )  /\  y  e.  B )  ->  x  C_  ( Base `  W
) )
412, 5lspssv 15740 . . . . . . . . . . . . 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 5525 . . . . . . . . . . . . 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 16576 . . . . . . . . . . . . . . 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 16579 . . . . . . . . . . . . . . 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 2297 . . . . . . . . . . . . 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 2513 . . . . . . . . 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 1664 . . . . . 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 1617 . . 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 15908 . . . . 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 1527   E.wex 1528    = wceq 1623    e. wcel 1684    =/= wne 2446    C_ wss 3152    C. wpss 3153   {csn 3640   ` cfv 5255   Basecbs 13148   0gc0g 13400   LModclmod 15627   LSpanclspn 15728  LBasisclbs 15827   LVecclvec 15855   PreHilcphl 16528   ocvcocv 16560   CSubSpccss 16561  OBasiscobs 16602
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-0g 13404  df-mnd 14367  df-mhm 14415  df-grp 14489  df-minusg 14490  df-sbg 14491  df-ghm 14681  df-mgp 15326  df-rng 15340  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-rnghom 15496  df-drng 15514  df-staf 15610  df-srng 15611  df-lmod 15629  df-lss 15690  df-lsp 15729  df-lmhm 15779  df-lbs 15828  df-lvec 15856  df-sra 15925  df-rgmod 15926  df-phl 16530  df-ocv 16563  df-css 16564  df-obs 16605
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