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Theorem atom1d 25692
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 25679 . . . 4  |-  ( A  e. HAtoms 
<->  ( A  e.  CH  /\  ( A  =/=  0H  /\ 
A. y  e.  CH  ( y  C_  A  ->  ( y  =  A  \/  y  =  0H ) ) ) ) )
2 chne0 24832 . . . . . 6  |-  ( A  e.  CH  ->  ( A  =/=  0H  <->  E. x  e.  A  x  =/=  0h ) )
3 nfv 1678 . . . . . . 7  |-  F/ x  A  e.  CH
4 nfv 1678 . . . . . . . 8  |-  F/ x A. y  e.  CH  (
y  C_  A  ->  ( y  =  A  \/  y  =  0H )
)
5 nfre1 2770 . . . . . . . 8  |-  F/ x E. x  e.  ~H  ( x  =/=  0h  /\  A  =  ( _|_ `  ( _|_ `  {
x } ) ) )
64, 5nfim 1857 . . . . . . 7  |-  F/ x
( A. y  e. 
CH  ( y  C_  A  ->  ( y  =  A  \/  y  =  0H ) )  ->  E. x  e.  ~H  ( x  =/=  0h  /\  A  =  ( _|_ `  ( _|_ `  {
x } ) ) ) )
7 chel 24568 . . . . . . . . . . 11  |-  ( ( A  e.  CH  /\  x  e.  A )  ->  x  e.  ~H )
87adantrr 711 . . . . . . . . . 10  |-  ( ( A  e.  CH  /\  ( x  e.  A  /\  x  =/=  0h )
)  ->  x  e.  ~H )
98adantrr 711 . . . . . . . . 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 757 . . . . . . . . 9  |-  ( ( A  e.  CH  /\  ( ( x  e.  A  /\  x  =/= 
0h )  /\  A. y  e.  CH  ( y 
C_  A  ->  (
y  =  A  \/  y  =  0H )
) ) )  ->  x  =/=  0h )
11 h1dn0 24890 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ~H  /\  x  =/=  0h )  -> 
( _|_ `  ( _|_ `  { x }
) )  =/=  0H )
127, 11sylan 468 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CH  /\  x  e.  A )  /\  x  =/=  0h )  ->  ( _|_ `  ( _|_ `  { x }
) )  =/=  0H )
1312anasss 642 . . . . . . . . . . . 12  |-  ( ( A  e.  CH  /\  ( x  e.  A  /\  x  =/=  0h )
)  ->  ( _|_ `  ( _|_ `  {
x } ) )  =/=  0H )
1413adantrr 711 . . . . . . . . . . 11  |-  ( ( A  e.  CH  /\  ( ( x  e.  A  /\  x  =/= 
0h )  /\  A. y  e.  CH  ( y 
C_  A  ->  (
y  =  A  \/  y  =  0H )
) ) )  -> 
( _|_ `  ( _|_ `  { x }
) )  =/=  0H )
15 ch1dle 25691 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  CH  /\  x  e.  A )  ->  ( _|_ `  ( _|_ `  { x }
) )  C_  A
)
16 snssi 4014 . . . . . . . . . . . . . . . . . 18  |-  ( x  e.  ~H  ->  { x }  C_  ~H )
17 occl 24642 . . . . . . . . . . . . . . . . . 18  |-  ( { x }  C_  ~H  ->  ( _|_ `  {
x } )  e. 
CH )
187, 16, 173syl 20 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  CH  /\  x  e.  A )  ->  ( _|_ `  {
x } )  e. 
CH )
19 choccl 24644 . . . . . . . . . . . . . . . . 17  |-  ( ( _|_ `  { x } )  e.  CH  ->  ( _|_ `  ( _|_ `  { x }
) )  e.  CH )
20 sseq1 3374 . . . . . . . . . . . . . . . . . . 19  |-  ( y  =  ( _|_ `  ( _|_ `  { x }
) )  ->  (
y  C_  A  <->  ( _|_ `  ( _|_ `  {
x } ) ) 
C_  A ) )
21 eqeq1 2447 . . . . . . . . . . . . . . . . . . . 20  |-  ( y  =  ( _|_ `  ( _|_ `  { x }
) )  ->  (
y  =  A  <->  ( _|_ `  ( _|_ `  {
x } ) )  =  A ) )
22 eqeq1 2447 . . . . . . . . . . . . . . . . . . . 20  |-  ( y  =  ( _|_ `  ( _|_ `  { x }
) )  ->  (
y  =  0H  <->  ( _|_ `  ( _|_ `  {
x } ) )  =  0H ) )
2321, 22orbi12d 704 . . . . . . . . . . . . . . . . . . 19  |-  ( y  =  ( _|_ `  ( _|_ `  { x }
) )  ->  (
( y  =  A  \/  y  =  0H )  <->  ( ( _|_ `  ( _|_ `  {
x } ) )  =  A  \/  ( _|_ `  ( _|_ `  {
x } ) )  =  0H ) ) )
2420, 23imbi12d 320 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  ( _|_ `  ( _|_ `  { x }
) )  ->  (
( y  C_  A  ->  ( y  =  A  \/  y  =  0H ) )  <->  ( ( _|_ `  ( _|_ `  {
x } ) ) 
C_  A  ->  (
( _|_ `  ( _|_ `  { x }
) )  =  A  \/  ( _|_ `  ( _|_ `  { x }
) )  =  0H ) ) ) )
2524rspcv 3066 . . . . . . . . . . . . . . . . 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 41 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  CH  /\  x  e.  A )  ->  ( A. y  e. 
CH  ( y  C_  A  ->  ( y  =  A  \/  y  =  0H ) )  -> 
( ( _|_ `  ( _|_ `  { x }
) )  =  A  \/  ( _|_ `  ( _|_ `  { x }
) )  =  0H ) ) )
2827impr 616 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CH  /\  ( x  e.  A  /\  A. y  e.  CH  ( y  C_  A  ->  ( y  =  A  \/  y  =  0H ) ) ) )  ->  ( ( _|_ `  ( _|_ `  {
x } ) )  =  A  \/  ( _|_ `  ( _|_ `  {
x } ) )  =  0H ) )
2928adantrlr 717 . . . . . . . . . . . . 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 377 . . . . . . . . . . . 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 2610 . . . . . . . . . . . 12  |-  ( -.  ( _|_ `  ( _|_ `  { x }
) )  =/=  0H  <->  ( _|_ `  ( _|_ `  { x } ) )  =  0H )
3230, 31syl6ibr 227 . . . . . . . . . . 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 138 . . . . . . . . . 10  |-  ( ( A  e.  CH  /\  ( ( x  e.  A  /\  x  =/= 
0h )  /\  A. y  e.  CH  ( y 
C_  A  ->  (
y  =  A  \/  y  =  0H )
) ) )  -> 
( _|_ `  ( _|_ `  { x }
) )  =  A )
3433eqcomd 2446 . . . . . . . . 9  |-  ( ( A  e.  CH  /\  ( ( x  e.  A  /\  x  =/= 
0h )  /\  A. y  e.  CH  ( y 
C_  A  ->  (
y  =  A  \/  y  =  0H )
) ) )  ->  A  =  ( _|_ `  ( _|_ `  {
x } ) ) )
35 rspe 2775 . . . . . . . . 9  |-  ( ( x  e.  ~H  /\  ( x  =/=  0h  /\  A  =  ( _|_ `  ( _|_ `  {
x } ) ) ) )  ->  E. x  e.  ~H  ( x  =/= 
0h  /\  A  =  ( _|_ `  ( _|_ `  { x } ) ) ) )
369, 10, 34, 35syl12anc 1211 . . . . . . . 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 610 . . . . . . 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 2836 . . . . . 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 215 . . . . 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 433 . . . 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 195 . . 3  |-  ( A  e. HAtoms  ->  E. x  e.  ~H  ( x  =/=  0h  /\  A  =  ( _|_ `  ( _|_ `  {
x } ) ) ) )
42 h1da 25688 . . . . . . 7  |-  ( ( x  e.  ~H  /\  x  =/=  0h )  -> 
( _|_ `  ( _|_ `  { x }
) )  e. HAtoms )
43 eleq1 2501 . . . . . . 7  |-  ( A  =  ( _|_ `  ( _|_ `  { x }
) )  ->  ( A  e. HAtoms  <->  ( _|_ `  ( _|_ `  { x }
) )  e. HAtoms )
)
4442, 43syl5ibr 221 . . . . . 6  |-  ( A  =  ( _|_ `  ( _|_ `  { x }
) )  ->  (
( x  e.  ~H  /\  x  =/=  0h )  ->  A  e. HAtoms ) )
4544exp3acom3r 1419 . . . . 5  |-  ( x  e.  ~H  ->  (
x  =/=  0h  ->  ( A  =  ( _|_ `  ( _|_ `  {
x } ) )  ->  A  e. HAtoms )
) )
4645imp3a 431 . . . 4  |-  ( x  e.  ~H  ->  (
( x  =/=  0h  /\  A  =  ( _|_ `  ( _|_ `  {
x } ) ) )  ->  A  e. HAtoms ) )
4746rexlimiv 2833 . . 3  |-  ( E. x  e.  ~H  (
x  =/=  0h  /\  A  =  ( _|_ `  ( _|_ `  {
x } ) ) )  ->  A  e. HAtoms )
4841, 47impbii 188 . 2  |-  ( A  e. HAtoms 
<->  E. x  e.  ~H  ( x  =/=  0h  /\  A  =  ( _|_ `  ( _|_ `  {
x } ) ) ) )
49 spansn 24897 . . . . 5  |-  ( x  e.  ~H  ->  ( span `  { x }
)  =  ( _|_ `  ( _|_ `  {
x } ) ) )
5049eqeq2d 2452 . . . 4  |-  ( x  e.  ~H  ->  ( A  =  ( span `  { x } )  <-> 
A  =  ( _|_ `  ( _|_ `  {
x } ) ) ) )
5150anbi2d 698 . . 3  |-  ( x  e.  ~H  ->  (
( x  =/=  0h  /\  A  =  ( span `  { x } ) )  <->  ( x  =/= 
0h  /\  A  =  ( _|_ `  ( _|_ `  { x } ) ) ) ) )
5251rexbiia 2746 . 2  |-  ( E. x  e.  ~H  (
x  =/=  0h  /\  A  =  ( span `  { x } ) )  <->  E. x  e.  ~H  ( x  =/=  0h  /\  A  =  ( _|_ `  ( _|_ `  {
x } ) ) ) )
5348, 52bitr4i 252 1  |-  ( A  e. HAtoms 
<->  E. x  e.  ~H  ( x  =/=  0h  /\  A  =  ( span `  { x } ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1364    e. wcel 1761    =/= wne 2604   A.wral 2713   E.wrex 2714    C_ wss 3325   {csn 3874   ` cfv 5415   ~Hchil 24256   0hc0v 24261   CHcch 24266   _|_cort 24267   spancspn 24269   0Hc0h 24272  HAtomscat 24302
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cc 8600  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356  ax-addf 9357  ax-mulf 9358  ax-hilex 24336  ax-hfvadd 24337  ax-hvcom 24338  ax-hvass 24339  ax-hv0cl 24340  ax-hvaddid 24341  ax-hfvmul 24342  ax-hvmulid 24343  ax-hvmulass 24344  ax-hvdistr1 24345  ax-hvdistr2 24346  ax-hvmul0 24347  ax-hfi 24416  ax-his1 24419  ax-his2 24420  ax-his3 24421  ax-his4 24422  ax-hcompl 24539
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-omul 6921  df-er 7097  df-map 7212  df-pm 7213  df-ixp 7260  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fsupp 7617  df-fi 7657  df-sup 7687  df-oi 7720  df-card 8105  df-acn 8108  df-cda 8333  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-q 10950  df-rp 10988  df-xneg 11085  df-xadd 11086  df-xmul 11087  df-ioo 11300  df-ico 11302  df-icc 11303  df-fz 11434  df-fzo 11545  df-fl 11638  df-seq 11803  df-exp 11862  df-hash 12100  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-clim 12962  df-rlim 12963  df-sum 13160  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-starv 14249  df-sca 14250  df-vsca 14251  df-ip 14252  df-tset 14253  df-ple 14254  df-ds 14256  df-unif 14257  df-hom 14258  df-cco 14259  df-rest 14357  df-topn 14358  df-0g 14376  df-gsum 14377  df-topgen 14378  df-pt 14379  df-prds 14382  df-xrs 14436  df-qtop 14441  df-imas 14442  df-xps 14444  df-mre 14520  df-mrc 14521  df-acs 14523  df-mnd 15411  df-submnd 15461  df-mulg 15541  df-cntz 15828  df-cmn 16272  df-psmet 17768  df-xmet 17769  df-met 17770  df-bl 17771  df-mopn 17772  df-fbas 17773  df-fg 17774  df-cnfld 17778  df-top 18462  df-bases 18464  df-topon 18465  df-topsp 18466  df-cld 18582  df-ntr 18583  df-cls 18584  df-nei 18661  df-cn 18790  df-cnp 18791  df-lm 18792  df-haus 18878  df-tx 19094  df-hmeo 19287  df-fil 19378  df-fm 19470  df-flim 19471  df-flf 19472  df-xms 19854  df-ms 19855  df-tms 19856  df-cfil 20725  df-cau 20726  df-cmet 20727  df-grpo 23613  df-gid 23614  df-ginv 23615  df-gdiv 23616  df-ablo 23704  df-subgo 23724  df-vc 23859  df-nv 23905  df-va 23908  df-ba 23909  df-sm 23910  df-0v 23911  df-vs 23912  df-nmcv 23913  df-ims 23914  df-dip 24031  df-ssp 24055  df-ph 24148  df-cbn 24199  df-hnorm 24305  df-hba 24306  df-hvsub 24308  df-hlim 24309  df-hcau 24310  df-sh 24544  df-ch 24559  df-oc 24590  df-ch0 24591  df-span 24647  df-cv 25618  df-at 25677
This theorem is referenced by:  superpos  25693  chcv1  25694  chjatom  25696
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