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Theorem atom1d 23844
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
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 elat2 23831 . . . 4  |-  ( A  e. HAtoms 
<->  ( A  e.  CH  /\  ( A  =/=  0H  /\ 
A. y  e.  CH  ( y  C_  A  ->  ( y  =  A  \/  y  =  0H ) ) ) ) )
2 chne0 22984 . . . . . 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 2754 . . . . . . . 8  |-  F/ x E. x  e.  ~H  ( x  =/=  0h  /\  A  =  ( _|_ `  ( _|_ `  {
x } ) ) )
64, 5nfim 1832 . . . . . . 7  |-  F/ x
( A. y  e. 
CH  ( y  C_  A  ->  ( y  =  A  \/  y  =  0H ) )  ->  E. x  e.  ~H  ( x  =/=  0h  /\  A  =  ( _|_ `  ( _|_ `  {
x } ) ) ) )
7 chel 22721 . . . . . . . . . . 11  |-  ( ( A  e.  CH  /\  x  e.  A )  ->  x  e.  ~H )
87adantrr 698 . . . . . . . . . 10  |-  ( ( A  e.  CH  /\  ( x  e.  A  /\  x  =/=  0h )
)  ->  x  e.  ~H )
98adantrr 698 . . . . . . . . 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 740 . . . . . . . . 9  |-  ( ( A  e.  CH  /\  ( ( x  e.  A  /\  x  =/= 
0h )  /\  A. y  e.  CH  ( y 
C_  A  ->  (
y  =  A  \/  y  =  0H )
) ) )  ->  x  =/=  0h )
11 h1dn0 23042 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ~H  /\  x  =/=  0h )  -> 
( _|_ `  ( _|_ `  { x }
) )  =/=  0H )
127, 11sylan 458 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CH  /\  x  e.  A )  /\  x  =/=  0h )  ->  ( _|_ `  ( _|_ `  { x }
) )  =/=  0H )
1312anasss 629 . . . . . . . . . . . 12  |-  ( ( A  e.  CH  /\  ( x  e.  A  /\  x  =/=  0h )
)  ->  ( _|_ `  ( _|_ `  {
x } ) )  =/=  0H )
1413adantrr 698 . . . . . . . . . . 11  |-  ( ( A  e.  CH  /\  ( ( x  e.  A  /\  x  =/= 
0h )  /\  A. y  e.  CH  ( y 
C_  A  ->  (
y  =  A  \/  y  =  0H )
) ) )  -> 
( _|_ `  ( _|_ `  { x }
) )  =/=  0H )
15 ch1dle 23843 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  CH  /\  x  e.  A )  ->  ( _|_ `  ( _|_ `  { x }
) )  C_  A
)
16 snssi 3934 . . . . . . . . . . . . . . . . . 18  |-  ( x  e.  ~H  ->  { x }  C_  ~H )
17 occl 22794 . . . . . . . . . . . . . . . . . 18  |-  ( { x }  C_  ~H  ->  ( _|_ `  {
x } )  e. 
CH )
187, 16, 173syl 19 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  CH  /\  x  e.  A )  ->  ( _|_ `  {
x } )  e. 
CH )
19 choccl 22796 . . . . . . . . . . . . . . . . 17  |-  ( ( _|_ `  { x } )  e.  CH  ->  ( _|_ `  ( _|_ `  { x }
) )  e.  CH )
20 sseq1 3361 . . . . . . . . . . . . . . . . . . 19  |-  ( y  =  ( _|_ `  ( _|_ `  { x }
) )  ->  (
y  C_  A  <->  ( _|_ `  ( _|_ `  {
x } ) ) 
C_  A ) )
21 eqeq1 2441 . . . . . . . . . . . . . . . . . . . 20  |-  ( y  =  ( _|_ `  ( _|_ `  { x }
) )  ->  (
y  =  A  <->  ( _|_ `  ( _|_ `  {
x } ) )  =  A ) )
22 eqeq1 2441 . . . . . . . . . . . . . . . . . . . 20  |-  ( y  =  ( _|_ `  ( _|_ `  { x }
) )  ->  (
y  =  0H  <->  ( _|_ `  ( _|_ `  {
x } ) )  =  0H ) )
2321, 22orbi12d 691 . . . . . . . . . . . . . . . . . . 19  |-  ( y  =  ( _|_ `  ( _|_ `  { x }
) )  ->  (
( y  =  A  \/  y  =  0H )  <->  ( ( _|_ `  ( _|_ `  {
x } ) )  =  A  \/  ( _|_ `  ( _|_ `  {
x } ) )  =  0H ) ) )
2420, 23imbi12d 312 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  ( _|_ `  ( _|_ `  { x }
) )  ->  (
( y  C_  A  ->  ( y  =  A  \/  y  =  0H ) )  <->  ( ( _|_ `  ( _|_ `  {
x } ) ) 
C_  A  ->  (
( _|_ `  ( _|_ `  { x }
) )  =  A  \/  ( _|_ `  ( _|_ `  { x }
) )  =  0H ) ) ) )
2524rspcv 3040 . . . . . . . . . . . . . . . . 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 19 . . . . . . . . . . . . . . . 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 603 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CH  /\  ( x  e.  A  /\  A. y  e.  CH  ( y  C_  A  ->  ( y  =  A  \/  y  =  0H ) ) ) )  ->  ( ( _|_ `  ( _|_ `  {
x } ) )  =  A  \/  ( _|_ `  ( _|_ `  {
x } ) )  =  0H ) )
2928adantrlr 704 . . . . . . . . . . . . 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 367 . . . . . . . . . . . 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 2602 . . . . . . . . . . . 12  |-  ( -.  ( _|_ `  ( _|_ `  { x }
) )  =/=  0H  <->  ( _|_ `  ( _|_ `  { x } ) )  =  0H )
3230, 31syl6ibr 219 . . . . . . . . . . 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 2440 . . . . . . . . 9  |-  ( ( A  e.  CH  /\  ( ( x  e.  A  /\  x  =/= 
0h )  /\  A. y  e.  CH  ( y 
C_  A  ->  (
y  =  A  \/  y  =  0H )
) ) )  ->  A  =  ( _|_ `  ( _|_ `  {
x } ) ) )
35 rspe 2759 . . . . . . . . 9  |-  ( ( x  e.  ~H  /\  ( x  =/=  0h  /\  A  =  ( _|_ `  ( _|_ `  {
x } ) ) ) )  ->  E. x  e.  ~H  ( x  =/= 
0h  /\  A  =  ( _|_ `  ( _|_ `  { x } ) ) ) )
369, 10, 34, 35syl12anc 1182 . . . . . . . 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 597 . . . . . . 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 2819 . . . . . 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 207 . . . . 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 423 . . . 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 188 . . 3  |-  ( A  e. HAtoms  ->  E. x  e.  ~H  ( x  =/=  0h  /\  A  =  ( _|_ `  ( _|_ `  {
x } ) ) ) )
42 h1da 23840 . . . . . . 7  |-  ( ( x  e.  ~H  /\  x  =/=  0h )  -> 
( _|_ `  ( _|_ `  { x }
) )  e. HAtoms )
43 eleq1 2495 . . . . . . 7  |-  ( A  =  ( _|_ `  ( _|_ `  { x }
) )  ->  ( A  e. HAtoms  <->  ( _|_ `  ( _|_ `  { x }
) )  e. HAtoms )
)
4442, 43syl5ibr 213 . . . . . 6  |-  ( A  =  ( _|_ `  ( _|_ `  { x }
) )  ->  (
( x  e.  ~H  /\  x  =/=  0h )  ->  A  e. HAtoms ) )
4544exp3acom3r 1379 . . . . 5  |-  ( x  e.  ~H  ->  (
x  =/=  0h  ->  ( A  =  ( _|_ `  ( _|_ `  {
x } ) )  ->  A  e. HAtoms )
) )
4645imp3a 421 . . . 4  |-  ( x  e.  ~H  ->  (
( x  =/=  0h  /\  A  =  ( _|_ `  ( _|_ `  {
x } ) ) )  ->  A  e. HAtoms ) )
4746rexlimiv 2816 . . 3  |-  ( E. x  e.  ~H  (
x  =/=  0h  /\  A  =  ( _|_ `  ( _|_ `  {
x } ) ) )  ->  A  e. HAtoms )
4841, 47impbii 181 . 2  |-  ( A  e. HAtoms 
<->  E. x  e.  ~H  ( x  =/=  0h  /\  A  =  ( _|_ `  ( _|_ `  {
x } ) ) ) )
49 spansn 23049 . . . . 5  |-  ( x  e.  ~H  ->  ( span `  { x }
)  =  ( _|_ `  ( _|_ `  {
x } ) ) )
5049eqeq2d 2446 . . . 4  |-  ( x  e.  ~H  ->  ( A  =  ( span `  { x } )  <-> 
A  =  ( _|_ `  ( _|_ `  {
x } ) ) ) )
5150anbi2d 685 . . 3  |-  ( x  e.  ~H  ->  (
( x  =/=  0h  /\  A  =  ( span `  { x } ) )  <->  ( x  =/= 
0h  /\  A  =  ( _|_ `  ( _|_ `  { x } ) ) ) ) )
5251rexbiia 2730 . 2  |-  ( E. x  e.  ~H  (
x  =/=  0h  /\  A  =  ( span `  { x } ) )  <->  E. x  e.  ~H  ( x  =/=  0h  /\  A  =  ( _|_ `  ( _|_ `  {
x } ) ) ) )
5348, 52bitr4i 244 1  |-  ( A  e. HAtoms 
<->  E. x  e.  ~H  ( x  =/=  0h  /\  A  =  ( span `  { x } ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   E.wrex 2698    C_ wss 3312   {csn 3806   ` cfv 5445   ~Hchil 22410   0hc0v 22415   CHcch 22420   _|_cort 22421   spancspn 22423   0Hc0h 22426  HAtomscat 22456
This theorem is referenced by:  superpos  23845  chcv1  23846  chjatom  23848
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-inf2 7585  ax-cc 8304  ax-cnex 9035  ax-resscn 9036  ax-1cn 9037  ax-icn 9038  ax-addcl 9039  ax-addrcl 9040  ax-mulcl 9041  ax-mulrcl 9042  ax-mulcom 9043  ax-addass 9044  ax-mulass 9045  ax-distr 9046  ax-i2m1 9047  ax-1ne0 9048  ax-1rid 9049  ax-rnegex 9050  ax-rrecex 9051  ax-cnre 9052  ax-pre-lttri 9053  ax-pre-lttrn 9054  ax-pre-ltadd 9055  ax-pre-mulgt0 9056  ax-pre-sup 9057  ax-addf 9058  ax-mulf 9059  ax-hilex 22490  ax-hfvadd 22491  ax-hvcom 22492  ax-hvass 22493  ax-hv0cl 22494  ax-hvaddid 22495  ax-hfvmul 22496  ax-hvmulid 22497  ax-hvmulass 22498  ax-hvdistr1 22499  ax-hvdistr2 22500  ax-hvmul0 22501  ax-hfi 22569  ax-his1 22572  ax-his2 22573  ax-his3 22574  ax-his4 22575  ax-hcompl 22692
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-isom 5454  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-of 6296  df-1st 6340  df-2nd 6341  df-riota 6540  df-recs 6624  df-rdg 6659  df-1o 6715  df-2o 6716  df-oadd 6719  df-omul 6720  df-er 6896  df-map 7011  df-pm 7012  df-ixp 7055  df-en 7101  df-dom 7102  df-sdom 7103  df-fin 7104  df-fi 7407  df-sup 7437  df-oi 7468  df-card 7815  df-acn 7818  df-cda 8037  df-pnf 9111  df-mnf 9112  df-xr 9113  df-ltxr 9114  df-le 9115  df-sub 9282  df-neg 9283  df-div 9667  df-nn 9990  df-2 10047  df-3 10048  df-4 10049  df-5 10050  df-6 10051  df-7 10052  df-8 10053  df-9 10054  df-10 10055  df-n0 10211  df-z 10272  df-dec 10372  df-uz 10478  df-q 10564  df-rp 10602  df-xneg 10699  df-xadd 10700  df-xmul 10701  df-ioo 10909  df-ico 10911  df-icc 10912  df-fz 11033  df-fzo 11124  df-fl 11190  df-seq 11312  df-exp 11371  df-hash 11607  df-cj 11892  df-re 11893  df-im 11894  df-sqr 12028  df-abs 12029  df-clim 12270  df-rlim 12271  df-sum 12468  df-struct 13459  df-ndx 13460  df-slot 13461  df-base 13462  df-sets 13463  df-ress 13464  df-plusg 13530  df-mulr 13531  df-starv 13532  df-sca 13533  df-vsca 13534  df-tset 13536  df-ple 13537  df-ds 13539  df-unif 13540  df-hom 13541  df-cco 13542  df-rest 13638  df-topn 13639  df-topgen 13655  df-pt 13656  df-prds 13659  df-xrs 13714  df-0g 13715  df-gsum 13716  df-qtop 13721  df-imas 13722  df-xps 13724  df-mre 13799  df-mrc 13800  df-acs 13802  df-mnd 14678  df-submnd 14727  df-mulg 14803  df-cntz 15104  df-cmn 15402  df-psmet 16682  df-xmet 16683  df-met 16684  df-bl 16685  df-mopn 16686  df-fbas 16687  df-fg 16688  df-cnfld 16692  df-top 16951  df-bases 16953  df-topon 16954  df-topsp 16955  df-cld 17071  df-ntr 17072  df-cls 17073  df-nei 17150  df-cn 17279  df-cnp 17280  df-lm 17281  df-haus 17367  df-tx 17582  df-hmeo 17775  df-fil 17866  df-fm 17958  df-flim 17959  df-flf 17960  df-xms 18338  df-ms 18339  df-tms 18340  df-cfil 19196  df-cau 19197  df-cmet 19198  df-grpo 21767  df-gid 21768  df-ginv 21769  df-gdiv 21770  df-ablo 21858  df-subgo 21878  df-vc 22013  df-nv 22059  df-va 22062  df-ba 22063  df-sm 22064  df-0v 22065  df-vs 22066  df-nmcv 22067  df-ims 22068  df-dip 22185  df-ssp 22209  df-ph 22302  df-cbn 22353  df-hnorm 22459  df-hba 22460  df-hvsub 22462  df-hlim 22463  df-hcau 22464  df-sh 22697  df-ch 22712  df-oc 22742  df-ch0 22743  df-span 22799  df-cv 23770  df-at 23829
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