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Theorem atom1d 26948
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 26935 . . . 4  |-  ( A  e. HAtoms 
<->  ( A  e.  CH  /\  ( A  =/=  0H  /\ 
A. y  e.  CH  ( y  C_  A  ->  ( y  =  A  \/  y  =  0H ) ) ) ) )
2 chne0 26088 . . . . . 6  |-  ( A  e.  CH  ->  ( A  =/=  0H  <->  E. x  e.  A  x  =/=  0h ) )
3 nfv 1683 . . . . . . 7  |-  F/ x  A  e.  CH
4 nfv 1683 . . . . . . . 8  |-  F/ x A. y  e.  CH  (
y  C_  A  ->  ( y  =  A  \/  y  =  0H )
)
5 nfre1 2925 . . . . . . . 8  |-  F/ x E. x  e.  ~H  ( x  =/=  0h  /\  A  =  ( _|_ `  ( _|_ `  {
x } ) ) )
64, 5nfim 1867 . . . . . . 7  |-  F/ x
( A. y  e. 
CH  ( y  C_  A  ->  ( y  =  A  \/  y  =  0H ) )  ->  E. x  e.  ~H  ( x  =/=  0h  /\  A  =  ( _|_ `  ( _|_ `  {
x } ) ) ) )
7 chel 25824 . . . . . . . . . . 11  |-  ( ( A  e.  CH  /\  x  e.  A )  ->  x  e.  ~H )
87adantrr 716 . . . . . . . . . 10  |-  ( ( A  e.  CH  /\  ( x  e.  A  /\  x  =/=  0h )
)  ->  x  e.  ~H )
98adantrr 716 . . . . . . . . 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 762 . . . . . . . . 9  |-  ( ( A  e.  CH  /\  ( ( x  e.  A  /\  x  =/= 
0h )  /\  A. y  e.  CH  ( y 
C_  A  ->  (
y  =  A  \/  y  =  0H )
) ) )  ->  x  =/=  0h )
11 h1dn0 26146 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ~H  /\  x  =/=  0h )  -> 
( _|_ `  ( _|_ `  { x }
) )  =/=  0H )
127, 11sylan 471 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CH  /\  x  e.  A )  /\  x  =/=  0h )  ->  ( _|_ `  ( _|_ `  { x }
) )  =/=  0H )
1312anasss 647 . . . . . . . . . . . 12  |-  ( ( A  e.  CH  /\  ( x  e.  A  /\  x  =/=  0h )
)  ->  ( _|_ `  ( _|_ `  {
x } ) )  =/=  0H )
1413adantrr 716 . . . . . . . . . . 11  |-  ( ( A  e.  CH  /\  ( ( x  e.  A  /\  x  =/= 
0h )  /\  A. y  e.  CH  ( y 
C_  A  ->  (
y  =  A  \/  y  =  0H )
) ) )  -> 
( _|_ `  ( _|_ `  { x }
) )  =/=  0H )
15 ch1dle 26947 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  CH  /\  x  e.  A )  ->  ( _|_ `  ( _|_ `  { x }
) )  C_  A
)
16 snssi 4171 . . . . . . . . . . . . . . . . . 18  |-  ( x  e.  ~H  ->  { x }  C_  ~H )
17 occl 25898 . . . . . . . . . . . . . . . . . 18  |-  ( { x }  C_  ~H  ->  ( _|_ `  {
x } )  e. 
CH )
187, 16, 173syl 20 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  CH  /\  x  e.  A )  ->  ( _|_ `  {
x } )  e. 
CH )
19 choccl 25900 . . . . . . . . . . . . . . . . 17  |-  ( ( _|_ `  { x } )  e.  CH  ->  ( _|_ `  ( _|_ `  { x }
) )  e.  CH )
20 sseq1 3525 . . . . . . . . . . . . . . . . . . 19  |-  ( y  =  ( _|_ `  ( _|_ `  { x }
) )  ->  (
y  C_  A  <->  ( _|_ `  ( _|_ `  {
x } ) ) 
C_  A ) )
21 eqeq1 2471 . . . . . . . . . . . . . . . . . . . 20  |-  ( y  =  ( _|_ `  ( _|_ `  { x }
) )  ->  (
y  =  A  <->  ( _|_ `  ( _|_ `  {
x } ) )  =  A ) )
22 eqeq1 2471 . . . . . . . . . . . . . . . . . . . 20  |-  ( y  =  ( _|_ `  ( _|_ `  { x }
) )  ->  (
y  =  0H  <->  ( _|_ `  ( _|_ `  {
x } ) )  =  0H ) )
2321, 22orbi12d 709 . . . . . . . . . . . . . . . . . . 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 3210 . . . . . . . . . . . . . . . . 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 619 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CH  /\  ( x  e.  A  /\  A. y  e.  CH  ( y  C_  A  ->  ( y  =  A  \/  y  =  0H ) ) ) )  ->  ( ( _|_ `  ( _|_ `  {
x } ) )  =  A  \/  ( _|_ `  ( _|_ `  {
x } ) )  =  0H ) )
2928adantrlr 722 . . . . . . . . . . . . 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 2668 . . . . . . . . . . . 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 2475 . . . . . . . . 9  |-  ( ( A  e.  CH  /\  ( ( x  e.  A  /\  x  =/= 
0h )  /\  A. y  e.  CH  ( y 
C_  A  ->  (
y  =  A  \/  y  =  0H )
) ) )  ->  A  =  ( _|_ `  ( _|_ `  {
x } ) ) )
35 rspe 2922 . . . . . . . . 9  |-  ( ( x  e.  ~H  /\  ( x  =/=  0h  /\  A  =  ( _|_ `  ( _|_ `  {
x } ) ) ) )  ->  E. x  e.  ~H  ( x  =/= 
0h  /\  A  =  ( _|_ `  ( _|_ `  { x } ) ) ) )
369, 10, 34, 35syl12anc 1226 . . . . . . . 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 613 . . . . . . 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 2947 . . . . . 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 26944 . . . . . . 7  |-  ( ( x  e.  ~H  /\  x  =/=  0h )  -> 
( _|_ `  ( _|_ `  { x }
) )  e. HAtoms )
43 eleq1 2539 . . . . . . 7  |-  ( A  =  ( _|_ `  ( _|_ `  { x }
) )  ->  ( A  e. HAtoms  <->  ( _|_ `  ( _|_ `  { x }
) )  e. HAtoms )
)
4442, 43syl5ibr 221 . . . . . 6  |-  ( A  =  ( _|_ `  ( _|_ `  { x }
) )  ->  (
( x  e.  ~H  /\  x  =/=  0h )  ->  A  e. HAtoms ) )
4544expdcom 439 . . . . 5  |-  ( x  e.  ~H  ->  (
x  =/=  0h  ->  ( A  =  ( _|_ `  ( _|_ `  {
x } ) )  ->  A  e. HAtoms )
) )
4645impd 431 . . . 4  |-  ( x  e.  ~H  ->  (
( x  =/=  0h  /\  A  =  ( _|_ `  ( _|_ `  {
x } ) ) )  ->  A  e. HAtoms ) )
4746rexlimiv 2949 . . 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 26153 . . . . 5  |-  ( x  e.  ~H  ->  ( span `  { x }
)  =  ( _|_ `  ( _|_ `  {
x } ) ) )
5049eqeq2d 2481 . . . 4  |-  ( x  e.  ~H  ->  ( A  =  ( span `  { x } )  <-> 
A  =  ( _|_ `  ( _|_ `  {
x } ) ) ) )
5150anbi2d 703 . . 3  |-  ( x  e.  ~H  ->  (
( x  =/=  0h  /\  A  =  ( span `  { x } ) )  <->  ( x  =/= 
0h  /\  A  =  ( _|_ `  ( _|_ `  { x } ) ) ) ) )
5251rexbiia 2964 . 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 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   E.wrex 2815    C_ wss 3476   {csn 4027   ` cfv 5586   ~Hchil 25512   0hc0v 25517   CHcch 25522   _|_cort 25523   spancspn 25525   0Hc0h 25528  HAtomscat 25558
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cc 8811  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566  ax-addf 9567  ax-mulf 9568  ax-hilex 25592  ax-hfvadd 25593  ax-hvcom 25594  ax-hvass 25595  ax-hv0cl 25596  ax-hvaddid 25597  ax-hfvmul 25598  ax-hvmulid 25599  ax-hvmulass 25600  ax-hvdistr1 25601  ax-hvdistr2 25602  ax-hvmul0 25603  ax-hfi 25672  ax-his1 25675  ax-his2 25676  ax-his3 25677  ax-his4 25678  ax-hcompl 25795
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-of 6522  df-om 6679  df-1st 6781  df-2nd 6782  df-supp 6899  df-recs 7039  df-rdg 7073  df-1o 7127  df-2o 7128  df-oadd 7131  df-omul 7132  df-er 7308  df-map 7419  df-pm 7420  df-ixp 7467  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-fsupp 7826  df-fi 7867  df-sup 7897  df-oi 7931  df-card 8316  df-acn 8319  df-cda 8544  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10973  df-uz 11079  df-q 11179  df-rp 11217  df-xneg 11314  df-xadd 11315  df-xmul 11316  df-ioo 11529  df-ico 11531  df-icc 11532  df-fz 11669  df-fzo 11789  df-fl 11893  df-seq 12072  df-exp 12131  df-hash 12370  df-cj 12891  df-re 12892  df-im 12893  df-sqrt 13027  df-abs 13028  df-clim 13270  df-rlim 13271  df-sum 13468  df-struct 14488  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-ress 14493  df-plusg 14564  df-mulr 14565  df-starv 14566  df-sca 14567  df-vsca 14568  df-ip 14569  df-tset 14570  df-ple 14571  df-ds 14573  df-unif 14574  df-hom 14575  df-cco 14576  df-rest 14674  df-topn 14675  df-0g 14693  df-gsum 14694  df-topgen 14695  df-pt 14696  df-prds 14699  df-xrs 14753  df-qtop 14758  df-imas 14759  df-xps 14761  df-mre 14837  df-mrc 14838  df-acs 14840  df-mnd 15728  df-submnd 15778  df-mulg 15861  df-cntz 16150  df-cmn 16596  df-psmet 18182  df-xmet 18183  df-met 18184  df-bl 18185  df-mopn 18186  df-fbas 18187  df-fg 18188  df-cnfld 18192  df-top 19166  df-bases 19168  df-topon 19169  df-topsp 19170  df-cld 19286  df-ntr 19287  df-cls 19288  df-nei 19365  df-cn 19494  df-cnp 19495  df-lm 19496  df-haus 19582  df-tx 19798  df-hmeo 19991  df-fil 20082  df-fm 20174  df-flim 20175  df-flf 20176  df-xms 20558  df-ms 20559  df-tms 20560  df-cfil 21429  df-cau 21430  df-cmet 21431  df-grpo 24869  df-gid 24870  df-ginv 24871  df-gdiv 24872  df-ablo 24960  df-subgo 24980  df-vc 25115  df-nv 25161  df-va 25164  df-ba 25165  df-sm 25166  df-0v 25167  df-vs 25168  df-nmcv 25169  df-ims 25170  df-dip 25287  df-ssp 25311  df-ph 25404  df-cbn 25455  df-hnorm 25561  df-hba 25562  df-hvsub 25564  df-hlim 25565  df-hcau 25566  df-sh 25800  df-ch 25815  df-oc 25846  df-ch0 25847  df-span 25903  df-cv 26874  df-at 26933
This theorem is referenced by:  superpos  26949  chcv1  26950  chjatom  26952
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