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Theorem obslbs 16959
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 16952 . . . . . 6  |-  ( B  e.  (OBasis `  W
)  ->  W  e.  PreHil )
2 eqid 2438 . . . . . . 7  |-  ( Base `  W )  =  (
Base `  W )
32obsss 16953 . . . . . 6  |-  ( B  e.  (OBasis `  W
)  ->  B  C_  ( Base `  W ) )
4 eqid 2438 . . . . . . 7  |-  ( ocv `  W )  =  ( ocv `  W )
5 obslbs.n . . . . . . 7  |-  N  =  ( LSpan `  W )
62, 4, 5ocvlsp 16905 . . . . . 6  |-  ( ( W  e.  PreHil  /\  B  C_  ( Base `  W
) )  ->  (
( ocv `  W
) `  ( N `  B ) )  =  ( ( ocv `  W
) `  B )
)
71, 3, 6syl2anc 644 . . . . 5  |-  ( B  e.  (OBasis `  W
)  ->  ( ( ocv `  W ) `  ( N `  B ) )  =  ( ( ocv `  W ) `
 B ) )
87fveq2d 5734 . . . 4  |-  ( B  e.  (OBasis `  W
)  ->  ( ( ocv `  W ) `  ( ( ocv `  W
) `  ( N `  B ) ) )  =  ( ( ocv `  W ) `  (
( ocv `  W
) `  B )
) )
94, 2obs2ocv 16956 . . . 4  |-  ( B  e.  (OBasis `  W
)  ->  ( ( ocv `  W ) `  ( ( ocv `  W
) `  B )
)  =  ( Base `  W ) )
108, 9eqtrd 2470 . . 3  |-  ( B  e.  (OBasis `  W
)  ->  ( ( ocv `  W ) `  ( ( ocv `  W
) `  ( N `  B ) ) )  =  ( Base `  W
) )
1110eqeq2d 2449 . 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 16912 . . 3  |-  ( W  e.  PreHil  ->  ( ( N `
 B )  e.  C  <->  ( N `  B )  =  ( ( ocv `  W
) `  ( ( ocv `  W ) `  ( N `  B ) ) ) ) )
141, 13syl 16 . 2  |-  ( B  e.  (OBasis `  W
)  ->  ( ( N `  B )  e.  C  <->  ( N `  B )  =  ( ( ocv `  W
) `  ( ( ocv `  W ) `  ( N `  B ) ) ) ) )
15 phllvec 16862 . . . 4  |-  ( W  e.  PreHil  ->  W  e.  LVec )
161, 15syl 16 . . 3  |-  ( B  e.  (OBasis `  W
)  ->  W  e.  LVec )
17 pssnel 3695 . . . . . . 7  |-  ( x 
C.  B  ->  E. y
( y  e.  B  /\  -.  y  e.  x
) )
1817adantl 454 . . . . . 6  |-  ( ( B  e.  (OBasis `  W )  /\  x  C.  B )  ->  E. y
( y  e.  B  /\  -.  y  e.  x
) )
19 simpll 732 . . . . . . . . . . 11  |-  ( ( ( B  e.  (OBasis `  W )  /\  x  C.  B )  /\  y  e.  B )  ->  B  e.  (OBasis `  W )
)
20 pssss 3444 . . . . . . . . . . . 12  |-  ( x 
C.  B  ->  x  C_  B )
2120ad2antlr 709 . . . . . . . . . . 11  |-  ( ( ( B  e.  (OBasis `  W )  /\  x  C.  B )  /\  y  e.  B )  ->  x  C_  B )
22 simpr 449 . . . . . . . . . . 11  |-  ( ( ( B  e.  (OBasis `  W )  /\  x  C.  B )  /\  y  e.  B )  ->  y  e.  B )
234obselocv 16957 . . . . . . . . . . 11  |-  ( ( B  e.  (OBasis `  W )  /\  x  C_  B  /\  y  e.  B )  ->  (
y  e.  ( ( ocv `  W ) `
 x )  <->  -.  y  e.  x ) )
2419, 21, 22, 23syl3anc 1185 . . . . . . . . . 10  |-  ( ( ( B  e.  (OBasis `  W )  /\  x  C.  B )  /\  y  e.  B )  ->  (
y  e.  ( ( ocv `  W ) `
 x )  <->  -.  y  e.  x ) )
25 eqid 2438 . . . . . . . . . . . . . 14  |-  ( 0g
`  W )  =  ( 0g `  W
)
2625obsne0 16954 . . . . . . . . . . . . 13  |-  ( ( B  e.  (OBasis `  W )  /\  y  e.  B )  ->  y  =/=  ( 0g `  W
) )
2719, 22, 26syl2anc 644 . . . . . . . . . . . 12  |-  ( ( ( B  e.  (OBasis `  W )  /\  x  C.  B )  /\  y  e.  B )  ->  y  =/=  ( 0g `  W
) )
28 elsni 3840 . . . . . . . . . . . . 13  |-  ( y  e.  { ( 0g
`  W ) }  ->  y  =  ( 0g `  W ) )
2928necon3ai 2646 . . . . . . . . . . . 12  |-  ( y  =/=  ( 0g `  W )  ->  -.  y  e.  { ( 0g `  W ) } )
3027, 29syl 16 . . . . . . . . . . 11  |-  ( ( ( B  e.  (OBasis `  W )  /\  x  C.  B )  /\  y  e.  B )  ->  -.  y  e.  { ( 0g `  W ) } )
31 nelne1 2695 . . . . . . . . . . . 12  |-  ( ( y  e.  ( ( ocv `  W ) `
 x )  /\  -.  y  e.  { ( 0g `  W ) } )  ->  (
( ocv `  W
) `  x )  =/=  { ( 0g `  W ) } )
3231expcom 426 . . . . . . . . . . 11  |-  ( -.  y  e.  { ( 0g `  W ) }  ->  ( y  e.  ( ( ocv `  W
) `  x )  ->  ( ( ocv `  W
) `  x )  =/=  { ( 0g `  W ) } ) )
3330, 32syl 16 . . . . . . . . . 10  |-  ( ( ( B  e.  (OBasis `  W )  /\  x  C.  B )  /\  y  e.  B )  ->  (
y  e.  ( ( ocv `  W ) `
 x )  -> 
( ( ocv `  W
) `  x )  =/=  { ( 0g `  W ) } ) )
3424, 33sylbird 228 . . . . . . . . 9  |-  ( ( ( B  e.  (OBasis `  W )  /\  x  C.  B )  /\  y  e.  B )  ->  ( -.  y  e.  x  ->  ( ( ocv `  W
) `  x )  =/=  { ( 0g `  W ) } ) )
35 npss 3459 . . . . . . . . . . 11  |-  ( -.  ( N `  x
)  C.  ( Base `  W )  <->  ( ( N `  x )  C_  ( Base `  W
)  ->  ( N `  x )  =  (
Base `  W )
) )
36 phllmod 16863 . . . . . . . . . . . . . . 15  |-  ( W  e.  PreHil  ->  W  e.  LMod )
371, 36syl 16 . . . . . . . . . . . . . 14  |-  ( B  e.  (OBasis `  W
)  ->  W  e.  LMod )
3837ad2antrr 708 . . . . . . . . . . . . 13  |-  ( ( ( B  e.  (OBasis `  W )  /\  x  C.  B )  /\  y  e.  B )  ->  W  e.  LMod )
393ad2antrr 708 . . . . . . . . . . . . . 14  |-  ( ( ( B  e.  (OBasis `  W )  /\  x  C.  B )  /\  y  e.  B )  ->  B  C_  ( Base `  W
) )
4021, 39sstrd 3360 . . . . . . . . . . . . 13  |-  ( ( ( B  e.  (OBasis `  W )  /\  x  C.  B )  /\  y  e.  B )  ->  x  C_  ( Base `  W
) )
412, 5lspssv 16061 . . . . . . . . . . . . 13  |-  ( ( W  e.  LMod  /\  x  C_  ( Base `  W
) )  ->  ( N `  x )  C_  ( Base `  W
) )
4238, 40, 41syl2anc 644 . . . . . . . . . . . 12  |-  ( ( ( B  e.  (OBasis `  W )  /\  x  C.  B )  /\  y  e.  B )  ->  ( N `  x )  C_  ( Base `  W
) )
43 fveq2 5730 . . . . . . . . . . . . 13  |-  ( ( N `  x )  =  ( Base `  W
)  ->  ( ( ocv `  W ) `  ( N `  x ) )  =  ( ( ocv `  W ) `
 ( Base `  W
) ) )
441ad2antrr 708 . . . . . . . . . . . . . . 15  |-  ( ( ( B  e.  (OBasis `  W )  /\  x  C.  B )  /\  y  e.  B )  ->  W  e.  PreHil )
452, 4, 5ocvlsp 16905 . . . . . . . . . . . . . . 15  |-  ( ( W  e.  PreHil  /\  x  C_  ( Base `  W
) )  ->  (
( ocv `  W
) `  ( N `  x ) )  =  ( ( ocv `  W
) `  x )
)
4644, 40, 45syl2anc 644 . . . . . . . . . . . . . 14  |-  ( ( ( B  e.  (OBasis `  W )  /\  x  C.  B )  /\  y  e.  B )  ->  (
( ocv `  W
) `  ( N `  x ) )  =  ( ( ocv `  W
) `  x )
)
472, 4, 25ocv1 16908 . . . . . . . . . . . . . . 15  |-  ( W  e.  PreHil  ->  ( ( ocv `  W ) `  ( Base `  W ) )  =  { ( 0g
`  W ) } )
4844, 47syl 16 . . . . . . . . . . . . . 14  |-  ( ( ( B  e.  (OBasis `  W )  /\  x  C.  B )  /\  y  e.  B )  ->  (
( ocv `  W
) `  ( Base `  W ) )  =  { ( 0g `  W ) } )
4946, 48eqeq12d 2452 . . . . . . . . . . . . 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 212 . . . . . . . . . . . 12  |-  ( ( ( B  e.  (OBasis `  W )  /\  x  C.  B )  /\  y  e.  B )  ->  (
( N `  x
)  =  ( Base `  W )  ->  (
( ocv `  W
) `  x )  =  { ( 0g `  W ) } ) )
5142, 50embantd 53 . . . . . . . . . . 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 210 . . . . . . . . . 10  |-  ( ( ( B  e.  (OBasis `  W )  /\  x  C.  B )  /\  y  e.  B )  ->  ( -.  ( N `  x
)  C.  ( Base `  W )  ->  (
( ocv `  W
) `  x )  =  { ( 0g `  W ) } ) )
5352necon1ad 2673 . . . . . . . . 9  |-  ( ( ( B  e.  (OBasis `  W )  /\  x  C.  B )  /\  y  e.  B )  ->  (
( ( ocv `  W
) `  x )  =/=  { ( 0g `  W ) }  ->  ( N `  x ) 
C.  ( Base `  W
) ) )
5434, 53syld 43 . . . . . . . 8  |-  ( ( ( B  e.  (OBasis `  W )  /\  x  C.  B )  /\  y  e.  B )  ->  ( -.  y  e.  x  ->  ( N `  x
)  C.  ( Base `  W ) ) )
5554expimpd 588 . . . . . . 7  |-  ( ( B  e.  (OBasis `  W )  /\  x  C.  B )  ->  (
( y  e.  B  /\  -.  y  e.  x
)  ->  ( N `  x )  C.  ( Base `  W ) ) )
5655exlimdv 1647 . . . . . 6  |-  ( ( B  e.  (OBasis `  W )  /\  x  C.  B )  ->  ( E. y ( y  e.  B  /\  -.  y  e.  x )  ->  ( N `  x )  C.  ( Base `  W
) ) )
5718, 56mpd 15 . . . . 5  |-  ( ( B  e.  (OBasis `  W )  /\  x  C.  B )  ->  ( N `  x )  C.  ( Base `  W
) )
5857ex 425 . . . 4  |-  ( B  e.  (OBasis `  W
)  ->  ( x  C.  B  ->  ( N `
 x )  C.  ( Base `  W )
) )
5958alrimiv 1642 . . 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 16229 . . . . 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 949 . . . . 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 254 . . . 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 877 . . 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 1183 . 2  |-  ( B  e.  (OBasis `  W
)  ->  ( B  e.  J  <->  ( N `  B )  =  (
Base `  W )
) )
6611, 14, 653bitr4rd 279 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 178    /\ wa 360    /\ w3a 937   A.wal 1550   E.wex 1551    = wceq 1653    e. wcel 1726    =/= wne 2601    C_ wss 3322    C. wpss 3323   {csn 3816   ` cfv 5456   Basecbs 13471   0gc0g 13725   LModclmod 15952   LSpanclspn 16049  LBasisclbs 16148   LVecclvec 16176   PreHilcphl 16857   ocvcocv 16889   CSubSpccss 16890  OBasiscobs 16931
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-tpos 6481  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-map 7022  df-en 7112  df-dom 7113  df-sdom 7114  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-2 10060  df-3 10061  df-4 10062  df-5 10063  df-6 10064  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-ress 13478  df-plusg 13544  df-mulr 13545  df-sca 13547  df-vsca 13548  df-0g 13729  df-mnd 14692  df-mhm 14740  df-grp 14814  df-minusg 14815  df-sbg 14816  df-ghm 15006  df-mgp 15651  df-rng 15665  df-ur 15667  df-oppr 15730  df-dvdsr 15748  df-unit 15749  df-invr 15779  df-rnghom 15821  df-drng 15839  df-staf 15935  df-srng 15936  df-lmod 15954  df-lss 16011  df-lsp 16050  df-lmhm 16100  df-lbs 16149  df-lvec 16177  df-sra 16246  df-rgmod 16247  df-phl 16859  df-ocv 16892  df-css 16893  df-obs 16934
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