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Theorem atom1d 22879
Description: The 1-dimensional subspaces of Hilbert space are its atoms. Part of Remark 10.3.5 of [BeltramettiCassinelli] p. 107. (Contributed by NM, 4-Jun-2004.) (New usage is discouraged.)
Assertion
Ref Expression
atom1d  |-  ( A  e. HAtoms 
<->  E. x  e.  ~H  ( x  =/=  0h  /\  A  =  ( span `  { x } ) ) )
Distinct variable group:    x, A

Proof of Theorem atom1d
StepHypRef Expression
1 elat2 22866 . . . 4  |-  ( A  e. HAtoms 
<->  ( A  e.  CH  /\  ( A  =/=  0H  /\ 
A. y  e.  CH  ( y  C_  A  ->  ( y  =  A  \/  y  =  0H ) ) ) ) )
2 chne0 22019 . . . . . 6  |-  ( A  e.  CH  ->  ( A  =/=  0H  <->  E. x  e.  A  x  =/=  0h ) )
3 nfv 1629 . . . . . . 7  |-  F/ x  A  e.  CH
4 nfv 1629 . . . . . . . 8  |-  F/ x A. y  e.  CH  (
y  C_  A  ->  ( y  =  A  \/  y  =  0H )
)
5 nfre1 2572 . . . . . . . 8  |-  F/ x E. x  e.  ~H  ( x  =/=  0h  /\  A  =  ( _|_ `  ( _|_ `  {
x } ) ) )
64, 5nfim 1735 . . . . . . 7  |-  F/ x
( A. y  e. 
CH  ( y  C_  A  ->  ( y  =  A  \/  y  =  0H ) )  ->  E. x  e.  ~H  ( x  =/=  0h  /\  A  =  ( _|_ `  ( _|_ `  {
x } ) ) ) )
7 chel 21756 . . . . . . . . . . 11  |-  ( ( A  e.  CH  /\  x  e.  A )  ->  x  e.  ~H )
87adantrr 700 . . . . . . . . . 10  |-  ( ( A  e.  CH  /\  ( x  e.  A  /\  x  =/=  0h )
)  ->  x  e.  ~H )
98adantrr 700 . . . . . . . . 9  |-  ( ( A  e.  CH  /\  ( ( x  e.  A  /\  x  =/= 
0h )  /\  A. y  e.  CH  ( y 
C_  A  ->  (
y  =  A  \/  y  =  0H )
) ) )  ->  x  e.  ~H )
10 simprlr 742 . . . . . . . . 9  |-  ( ( A  e.  CH  /\  ( ( x  e.  A  /\  x  =/= 
0h )  /\  A. y  e.  CH  ( y 
C_  A  ->  (
y  =  A  \/  y  =  0H )
) ) )  ->  x  =/=  0h )
11 h1dn0 22077 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ~H  /\  x  =/=  0h )  -> 
( _|_ `  ( _|_ `  { x }
) )  =/=  0H )
127, 11sylan 459 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CH  /\  x  e.  A )  /\  x  =/=  0h )  ->  ( _|_ `  ( _|_ `  { x }
) )  =/=  0H )
1312anasss 631 . . . . . . . . . . . 12  |-  ( ( A  e.  CH  /\  ( x  e.  A  /\  x  =/=  0h )
)  ->  ( _|_ `  ( _|_ `  {
x } ) )  =/=  0H )
1413adantrr 700 . . . . . . . . . . 11  |-  ( ( A  e.  CH  /\  ( ( x  e.  A  /\  x  =/= 
0h )  /\  A. y  e.  CH  ( y 
C_  A  ->  (
y  =  A  \/  y  =  0H )
) ) )  -> 
( _|_ `  ( _|_ `  { x }
) )  =/=  0H )
15 ch1dle 22878 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  CH  /\  x  e.  A )  ->  ( _|_ `  ( _|_ `  { x }
) )  C_  A
)
16 snssi 3719 . . . . . . . . . . . . . . . . . 18  |-  ( x  e.  ~H  ->  { x }  C_  ~H )
17 occl 21829 . . . . . . . . . . . . . . . . . 18  |-  ( { x }  C_  ~H  ->  ( _|_ `  {
x } )  e. 
CH )
187, 16, 173syl 20 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  CH  /\  x  e.  A )  ->  ( _|_ `  {
x } )  e. 
CH )
19 choccl 21831 . . . . . . . . . . . . . . . . 17  |-  ( ( _|_ `  { x } )  e.  CH  ->  ( _|_ `  ( _|_ `  { x }
) )  e.  CH )
20 sseq1 3160 . . . . . . . . . . . . . . . . . . 19  |-  ( y  =  ( _|_ `  ( _|_ `  { x }
) )  ->  (
y  C_  A  <->  ( _|_ `  ( _|_ `  {
x } ) ) 
C_  A ) )
21 eqeq1 2262 . . . . . . . . . . . . . . . . . . . 20  |-  ( y  =  ( _|_ `  ( _|_ `  { x }
) )  ->  (
y  =  A  <->  ( _|_ `  ( _|_ `  {
x } ) )  =  A ) )
22 eqeq1 2262 . . . . . . . . . . . . . . . . . . . 20  |-  ( y  =  ( _|_ `  ( _|_ `  { x }
) )  ->  (
y  =  0H  <->  ( _|_ `  ( _|_ `  {
x } ) )  =  0H ) )
2321, 22orbi12d 693 . . . . . . . . . . . . . . . . . . 19  |-  ( y  =  ( _|_ `  ( _|_ `  { x }
) )  ->  (
( y  =  A  \/  y  =  0H )  <->  ( ( _|_ `  ( _|_ `  {
x } ) )  =  A  \/  ( _|_ `  ( _|_ `  {
x } ) )  =  0H ) ) )
2420, 23imbi12d 313 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  ( _|_ `  ( _|_ `  { x }
) )  ->  (
( y  C_  A  ->  ( y  =  A  \/  y  =  0H ) )  <->  ( ( _|_ `  ( _|_ `  {
x } ) ) 
C_  A  ->  (
( _|_ `  ( _|_ `  { x }
) )  =  A  \/  ( _|_ `  ( _|_ `  { x }
) )  =  0H ) ) ) )
2524rcla4v 2848 . . . . . . . . . . . . . . . . 17  |-  ( ( _|_ `  ( _|_ `  { x } ) )  e.  CH  ->  ( A. y  e.  CH  ( y  C_  A  ->  ( y  =  A  \/  y  =  0H ) )  ->  (
( _|_ `  ( _|_ `  { x }
) )  C_  A  ->  ( ( _|_ `  ( _|_ `  { x }
) )  =  A  \/  ( _|_ `  ( _|_ `  { x }
) )  =  0H ) ) ) )
2618, 19, 253syl 20 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  CH  /\  x  e.  A )  ->  ( A. y  e. 
CH  ( y  C_  A  ->  ( y  =  A  \/  y  =  0H ) )  -> 
( ( _|_ `  ( _|_ `  { x }
) )  C_  A  ->  ( ( _|_ `  ( _|_ `  { x }
) )  =  A  \/  ( _|_ `  ( _|_ `  { x }
) )  =  0H ) ) ) )
2715, 26mpid 39 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  CH  /\  x  e.  A )  ->  ( A. y  e. 
CH  ( y  C_  A  ->  ( y  =  A  \/  y  =  0H ) )  -> 
( ( _|_ `  ( _|_ `  { x }
) )  =  A  \/  ( _|_ `  ( _|_ `  { x }
) )  =  0H ) ) )
2827impr 605 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CH  /\  ( x  e.  A  /\  A. y  e.  CH  ( y  C_  A  ->  ( y  =  A  \/  y  =  0H ) ) ) )  ->  ( ( _|_ `  ( _|_ `  {
x } ) )  =  A  \/  ( _|_ `  ( _|_ `  {
x } ) )  =  0H ) )
2928adantrlr 706 . . . . . . . . . . . . 13  |-  ( ( A  e.  CH  /\  ( ( x  e.  A  /\  x  =/= 
0h )  /\  A. y  e.  CH  ( y 
C_  A  ->  (
y  =  A  \/  y  =  0H )
) ) )  -> 
( ( _|_ `  ( _|_ `  { x }
) )  =  A  \/  ( _|_ `  ( _|_ `  { x }
) )  =  0H ) )
3029ord 368 . . . . . . . . . . . 12  |-  ( ( A  e.  CH  /\  ( ( x  e.  A  /\  x  =/= 
0h )  /\  A. y  e.  CH  ( y 
C_  A  ->  (
y  =  A  \/  y  =  0H )
) ) )  -> 
( -.  ( _|_ `  ( _|_ `  {
x } ) )  =  A  ->  ( _|_ `  ( _|_ `  {
x } ) )  =  0H ) )
31 nne 2423 . . . . . . . . . . . 12  |-  ( -.  ( _|_ `  ( _|_ `  { x }
) )  =/=  0H  <->  ( _|_ `  ( _|_ `  { x } ) )  =  0H )
3230, 31syl6ibr 220 . . . . . . . . . . 11  |-  ( ( A  e.  CH  /\  ( ( x  e.  A  /\  x  =/= 
0h )  /\  A. y  e.  CH  ( y 
C_  A  ->  (
y  =  A  \/  y  =  0H )
) ) )  -> 
( -.  ( _|_ `  ( _|_ `  {
x } ) )  =  A  ->  -.  ( _|_ `  ( _|_ `  { x } ) )  =/=  0H ) )
3314, 32mt4d 132 . . . . . . . . . 10  |-  ( ( A  e.  CH  /\  ( ( x  e.  A  /\  x  =/= 
0h )  /\  A. y  e.  CH  ( y 
C_  A  ->  (
y  =  A  \/  y  =  0H )
) ) )  -> 
( _|_ `  ( _|_ `  { x }
) )  =  A )
3433eqcomd 2261 . . . . . . . . 9  |-  ( ( A  e.  CH  /\  ( ( x  e.  A  /\  x  =/= 
0h )  /\  A. y  e.  CH  ( y 
C_  A  ->  (
y  =  A  \/  y  =  0H )
) ) )  ->  A  =  ( _|_ `  ( _|_ `  {
x } ) ) )
35 ra4e 2577 . . . . . . . . 9  |-  ( ( x  e.  ~H  /\  ( x  =/=  0h  /\  A  =  ( _|_ `  ( _|_ `  {
x } ) ) ) )  ->  E. x  e.  ~H  ( x  =/= 
0h  /\  A  =  ( _|_ `  ( _|_ `  { x } ) ) ) )
369, 10, 34, 35syl12anc 1185 . . . . . . . 8  |-  ( ( A  e.  CH  /\  ( ( x  e.  A  /\  x  =/= 
0h )  /\  A. y  e.  CH  ( y 
C_  A  ->  (
y  =  A  \/  y  =  0H )
) ) )  ->  E. x  e.  ~H  ( x  =/=  0h  /\  A  =  ( _|_ `  ( _|_ `  {
x } ) ) ) )
3736exp44 599 . . . . . . 7  |-  ( A  e.  CH  ->  (
x  e.  A  -> 
( x  =/=  0h  ->  ( A. y  e. 
CH  ( y  C_  A  ->  ( y  =  A  \/  y  =  0H ) )  ->  E. x  e.  ~H  ( x  =/=  0h  /\  A  =  ( _|_ `  ( _|_ `  {
x } ) ) ) ) ) ) )
383, 6, 37rexlimd 2637 . . . . . 6  |-  ( A  e.  CH  ->  ( E. x  e.  A  x  =/=  0h  ->  ( A. y  e.  CH  (
y  C_  A  ->  ( y  =  A  \/  y  =  0H )
)  ->  E. x  e.  ~H  ( x  =/= 
0h  /\  A  =  ( _|_ `  ( _|_ `  { x } ) ) ) ) ) )
392, 38sylbid 208 . . . . 5  |-  ( A  e.  CH  ->  ( A  =/=  0H  ->  ( A. y  e.  CH  (
y  C_  A  ->  ( y  =  A  \/  y  =  0H )
)  ->  E. x  e.  ~H  ( x  =/= 
0h  /\  A  =  ( _|_ `  ( _|_ `  { x } ) ) ) ) ) )
4039imp32 424 . . . 4  |-  ( ( A  e.  CH  /\  ( A  =/=  0H  /\ 
A. y  e.  CH  ( y  C_  A  ->  ( y  =  A  \/  y  =  0H ) ) ) )  ->  E. x  e.  ~H  ( x  =/=  0h  /\  A  =  ( _|_ `  ( _|_ `  {
x } ) ) ) )
411, 40sylbi 189 . . 3  |-  ( A  e. HAtoms  ->  E. x  e.  ~H  ( x  =/=  0h  /\  A  =  ( _|_ `  ( _|_ `  {
x } ) ) ) )
42 h1da 22875 . . . . . . 7  |-  ( ( x  e.  ~H  /\  x  =/=  0h )  -> 
( _|_ `  ( _|_ `  { x }
) )  e. HAtoms )
43 eleq1 2316 . . . . . . 7  |-  ( A  =  ( _|_ `  ( _|_ `  { x }
) )  ->  ( A  e. HAtoms  <->  ( _|_ `  ( _|_ `  { x }
) )  e. HAtoms )
)
4442, 43syl5ibr 214 . . . . . 6  |-  ( A  =  ( _|_ `  ( _|_ `  { x }
) )  ->  (
( x  e.  ~H  /\  x  =/=  0h )  ->  A  e. HAtoms ) )
4544exp3acom3r 1366 . . . . 5  |-  ( x  e.  ~H  ->  (
x  =/=  0h  ->  ( A  =  ( _|_ `  ( _|_ `  {
x } ) )  ->  A  e. HAtoms )
) )
4645imp3a 422 . . . 4  |-  ( x  e.  ~H  ->  (
( x  =/=  0h  /\  A  =  ( _|_ `  ( _|_ `  {
x } ) ) )  ->  A  e. HAtoms ) )
4746rexlimiv 2634 . . 3  |-  ( E. x  e.  ~H  (
x  =/=  0h  /\  A  =  ( _|_ `  ( _|_ `  {
x } ) ) )  ->  A  e. HAtoms )
4841, 47impbii 182 . 2  |-  ( A  e. HAtoms 
<->  E. x  e.  ~H  ( x  =/=  0h  /\  A  =  ( _|_ `  ( _|_ `  {
x } ) ) ) )
49 spansn 22084 . . . . 5  |-  ( x  e.  ~H  ->  ( span `  { x }
)  =  ( _|_ `  ( _|_ `  {
x } ) ) )
5049eqeq2d 2267 . . . 4  |-  ( x  e.  ~H  ->  ( A  =  ( span `  { x } )  <-> 
A  =  ( _|_ `  ( _|_ `  {
x } ) ) ) )
5150anbi2d 687 . . 3  |-  ( x  e.  ~H  ->  (
( x  =/=  0h  /\  A  =  ( span `  { x } ) )  <->  ( x  =/= 
0h  /\  A  =  ( _|_ `  ( _|_ `  { x } ) ) ) ) )
5251rexbiia 2549 . 2  |-  ( E. x  e.  ~H  (
x  =/=  0h  /\  A  =  ( span `  { x } ) )  <->  E. x  e.  ~H  ( x  =/=  0h  /\  A  =  ( _|_ `  ( _|_ `  {
x } ) ) ) )
5348, 52bitr4i 245 1  |-  ( A  e. HAtoms 
<->  E. x  e.  ~H  ( x  =/=  0h  /\  A  =  ( span `  { x } ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    \/ wo 359    /\ wa 360    = wceq 1619    e. wcel 1621    =/= wne 2419   A.wral 2516   E.wrex 2517    C_ wss 3113   {csn 3600   ` cfv 4659   ~Hchil 21445   0hc0v 21450   CHcch 21455   _|_cort 21456   spancspn 21458   0Hc0h 21461  HAtomscat 21491
This theorem is referenced by:  superpos  22880  chcv1  22881  chjatom  22883
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4091  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470  ax-inf2 7296  ax-cc 8015  ax-cnex 8747  ax-resscn 8748  ax-1cn 8749  ax-icn 8750  ax-addcl 8751  ax-addrcl 8752  ax-mulcl 8753  ax-mulrcl 8754  ax-mulcom 8755  ax-addass 8756  ax-mulass 8757  ax-distr 8758  ax-i2m1 8759  ax-1ne0 8760  ax-1rid 8761  ax-rnegex 8762  ax-rrecex 8763  ax-cnre 8764  ax-pre-lttri 8765  ax-pre-lttrn 8766  ax-pre-ltadd 8767  ax-pre-mulgt0 8768  ax-pre-sup 8769  ax-addf 8770  ax-mulf 8771  ax-hilex 21525  ax-hfvadd 21526  ax-hvcom 21527  ax-hvass 21528  ax-hv0cl 21529  ax-hvaddid 21530  ax-hfvmul 21531  ax-hvmulid 21532  ax-hvmulass 21533  ax-hvdistr1 21534  ax-hvdistr2 21535  ax-hvmul0 21536  ax-hfi 21604  ax-his1 21607  ax-his2 21608  ax-his3 21609  ax-his4 21610  ax-hcompl 21727
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2521  df-rex 2522  df-reu 2523  df-rmo 2524  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pss 3129  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-tp 3608  df-op 3609  df-uni 3788  df-int 3823  df-iun 3867  df-iin 3868  df-br 3984  df-opab 4038  df-mpt 4039  df-tr 4074  df-eprel 4263  df-id 4267  df-po 4272  df-so 4273  df-fr 4310  df-se 4311  df-we 4312  df-ord 4353  df-on 4354  df-lim 4355  df-suc 4356  df-om 4615  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-isom 4676  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-of 5998  df-1st 6042  df-2nd 6043  df-iota 6211  df-riota 6258  df-recs 6342  df-rdg 6377  df-1o 6433  df-2o 6434  df-oadd 6437  df-omul 6438  df-er 6614  df-map 6728  df-pm 6729  df-ixp 6772  df-en 6818  df-dom 6819  df-sdom 6820  df-fin 6821  df-fi 7119  df-sup 7148  df-oi 7179  df-card 7526  df-acn 7529  df-cda 7748  df-pnf 8823  df-mnf 8824  df-xr 8825  df-ltxr 8826  df-le 8827  df-sub 8993  df-neg 8994  df-div 9378  df-n 9701  df-2 9758  df-3 9759  df-4 9760  df-5 9761  df-6 9762  df-7 9763  df-8 9764  df-9 9765  df-10 9766  df-n0 9919  df-z 9978  df-dec 10078  df-uz 10184  df-q 10270  df-rp 10308  df-xneg 10405  df-xadd 10406  df-xmul 10407  df-ioo 10612  df-ico 10614  df-icc 10615  df-fz 10735  df-fzo 10823  df-fl 10877  df-seq 10999  df-exp 11057  df-hash 11290  df-cj 11535  df-re 11536  df-im 11537  df-sqr 11671  df-abs 11672  df-clim 11913  df-rlim 11914  df-sum 12110  df-struct 13098  df-ndx 13099  df-slot 13100  df-base 13101  df-sets 13102  df-ress 13103  df-plusg 13169  df-mulr 13170  df-starv 13171  df-sca 13172  df-vsca 13173  df-tset 13175  df-ple 13176  df-ds 13178  df-hom 13180  df-cco 13181  df-rest 13275  df-topn 13276  df-topgen 13292  df-pt 13293  df-prds 13296  df-xrs 13351  df-0g 13352  df-gsum 13353  df-qtop 13358  df-imas 13359  df-xps 13361  df-mre 13436  df-mrc 13437  df-acs 13439  df-mnd 14315  df-submnd 14364  df-mulg 14440  df-cntz 14741  df-cmn 15039  df-xmet 16321  df-met 16322  df-bl 16323  df-mopn 16324  df-cnfld 16326  df-top 16584  df-bases 16586  df-topon 16587  df-topsp 16588  df-cld 16704  df-ntr 16705  df-cls 16706  df-nei 16783  df-cn 16905  df-cnp 16906  df-lm 16907  df-haus 16991  df-tx 17205  df-hmeo 17394  df-fbas 17468  df-fg 17469  df-fil 17489  df-fm 17581  df-flim 17582  df-flf 17583  df-xms 17833  df-ms 17834  df-tms 17835  df-cfil 18629  df-cau 18630  df-cmet 18631  df-grpo 20804  df-gid 20805  df-ginv 20806  df-gdiv 20807  df-ablo 20895  df-subgo 20915  df-vc 21048  df-nv 21094  df-va 21097  df-ba 21098  df-sm 21099  df-0v 21100  df-vs 21101  df-nmcv 21102  df-ims 21103  df-dip 21220  df-ssp 21244  df-ph 21337  df-cbn 21388  df-hnorm 21494  df-hba 21495  df-hvsub 21497  df-hlim 21498  df-hcau 21499  df-sh 21732  df-ch 21747  df-oc 21777  df-ch0 21778  df-span 21834  df-cv 22805  df-at 22864
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