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Theorem atom1d 28004
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 27991 . . . 4  |-  ( A  e. HAtoms 
<->  ( A  e.  CH  /\  ( A  =/=  0H  /\ 
A. y  e.  CH  ( y  C_  A  ->  ( y  =  A  \/  y  =  0H ) ) ) ) )
2 chne0 27145 . . . . . 6  |-  ( A  e.  CH  ->  ( A  =/=  0H  <->  E. x  e.  A  x  =/=  0h ) )
3 nfv 1755 . . . . . . 7  |-  F/ x  A  e.  CH
4 nfv 1755 . . . . . . . 8  |-  F/ x A. y  e.  CH  (
y  C_  A  ->  ( y  =  A  \/  y  =  0H )
)
5 nfre1 2883 . . . . . . . 8  |-  F/ x E. x  e.  ~H  ( x  =/=  0h  /\  A  =  ( _|_ `  ( _|_ `  {
x } ) ) )
64, 5nfim 1980 . . . . . . 7  |-  F/ x
( A. y  e. 
CH  ( y  C_  A  ->  ( y  =  A  \/  y  =  0H ) )  ->  E. x  e.  ~H  ( x  =/=  0h  /\  A  =  ( _|_ `  ( _|_ `  {
x } ) ) ) )
7 chel 26881 . . . . . . . . . . 11  |-  ( ( A  e.  CH  /\  x  e.  A )  ->  x  e.  ~H )
87adantrr 721 . . . . . . . . . 10  |-  ( ( A  e.  CH  /\  ( x  e.  A  /\  x  =/=  0h )
)  ->  x  e.  ~H )
98adantrr 721 . . . . . . . . 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 771 . . . . . . . . 9  |-  ( ( A  e.  CH  /\  ( ( x  e.  A  /\  x  =/= 
0h )  /\  A. y  e.  CH  ( y 
C_  A  ->  (
y  =  A  \/  y  =  0H )
) ) )  ->  x  =/=  0h )
11 h1dn0 27203 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ~H  /\  x  =/=  0h )  -> 
( _|_ `  ( _|_ `  { x }
) )  =/=  0H )
127, 11sylan 473 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CH  /\  x  e.  A )  /\  x  =/=  0h )  ->  ( _|_ `  ( _|_ `  { x }
) )  =/=  0H )
1312anasss 651 . . . . . . . . . . . 12  |-  ( ( A  e.  CH  /\  ( x  e.  A  /\  x  =/=  0h )
)  ->  ( _|_ `  ( _|_ `  {
x } ) )  =/=  0H )
1413adantrr 721 . . . . . . . . . . 11  |-  ( ( A  e.  CH  /\  ( ( x  e.  A  /\  x  =/= 
0h )  /\  A. y  e.  CH  ( y 
C_  A  ->  (
y  =  A  \/  y  =  0H )
) ) )  -> 
( _|_ `  ( _|_ `  { x }
) )  =/=  0H )
15 ch1dle 28003 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  CH  /\  x  e.  A )  ->  ( _|_ `  ( _|_ `  { x }
) )  C_  A
)
16 snssi 4144 . . . . . . . . . . . . . . . . . 18  |-  ( x  e.  ~H  ->  { x }  C_  ~H )
17 occl 26955 . . . . . . . . . . . . . . . . . 18  |-  ( { x }  C_  ~H  ->  ( _|_ `  {
x } )  e. 
CH )
187, 16, 173syl 18 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  CH  /\  x  e.  A )  ->  ( _|_ `  {
x } )  e. 
CH )
19 choccl 26957 . . . . . . . . . . . . . . . . 17  |-  ( ( _|_ `  { x } )  e.  CH  ->  ( _|_ `  ( _|_ `  { x }
) )  e.  CH )
20 sseq1 3485 . . . . . . . . . . . . . . . . . . 19  |-  ( y  =  ( _|_ `  ( _|_ `  { x }
) )  ->  (
y  C_  A  <->  ( _|_ `  ( _|_ `  {
x } ) ) 
C_  A ) )
21 eqeq1 2426 . . . . . . . . . . . . . . . . . . . 20  |-  ( y  =  ( _|_ `  ( _|_ `  { x }
) )  ->  (
y  =  A  <->  ( _|_ `  ( _|_ `  {
x } ) )  =  A ) )
22 eqeq1 2426 . . . . . . . . . . . . . . . . . . . 20  |-  ( y  =  ( _|_ `  ( _|_ `  { x }
) )  ->  (
y  =  0H  <->  ( _|_ `  ( _|_ `  {
x } ) )  =  0H ) )
2321, 22orbi12d 714 . . . . . . . . . . . . . . . . . . 19  |-  ( y  =  ( _|_ `  ( _|_ `  { x }
) )  ->  (
( y  =  A  \/  y  =  0H )  <->  ( ( _|_ `  ( _|_ `  {
x } ) )  =  A  \/  ( _|_ `  ( _|_ `  {
x } ) )  =  0H ) ) )
2420, 23imbi12d 321 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  ( _|_ `  ( _|_ `  { x }
) )  ->  (
( y  C_  A  ->  ( y  =  A  \/  y  =  0H ) )  <->  ( ( _|_ `  ( _|_ `  {
x } ) ) 
C_  A  ->  (
( _|_ `  ( _|_ `  { x }
) )  =  A  \/  ( _|_ `  ( _|_ `  { x }
) )  =  0H ) ) ) )
2524rspcv 3178 . . . . . . . . . . . . . . . . 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 18 . . . . . . . . . . . . . . . 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 42 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  CH  /\  x  e.  A )  ->  ( A. y  e. 
CH  ( y  C_  A  ->  ( y  =  A  \/  y  =  0H ) )  -> 
( ( _|_ `  ( _|_ `  { x }
) )  =  A  \/  ( _|_ `  ( _|_ `  { x }
) )  =  0H ) ) )
2827impr 623 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CH  /\  ( x  e.  A  /\  A. y  e.  CH  ( y  C_  A  ->  ( y  =  A  \/  y  =  0H ) ) ) )  ->  ( ( _|_ `  ( _|_ `  {
x } ) )  =  A  \/  ( _|_ `  ( _|_ `  {
x } ) )  =  0H ) )
2928adantrlr 727 . . . . . . . . . . . . 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 378 . . . . . . . . . . . 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 2620 . . . . . . . . . . . 12  |-  ( -.  ( _|_ `  ( _|_ `  { x }
) )  =/=  0H  <->  ( _|_ `  ( _|_ `  { x } ) )  =  0H )
3230, 31syl6ibr 230 . . . . . . . . . . 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 143 . . . . . . . . . 10  |-  ( ( A  e.  CH  /\  ( ( x  e.  A  /\  x  =/= 
0h )  /\  A. y  e.  CH  ( y 
C_  A  ->  (
y  =  A  \/  y  =  0H )
) ) )  -> 
( _|_ `  ( _|_ `  { x }
) )  =  A )
3433eqcomd 2430 . . . . . . . . 9  |-  ( ( A  e.  CH  /\  ( ( x  e.  A  /\  x  =/= 
0h )  /\  A. y  e.  CH  ( y 
C_  A  ->  (
y  =  A  \/  y  =  0H )
) ) )  ->  A  =  ( _|_ `  ( _|_ `  {
x } ) ) )
35 rspe 2880 . . . . . . . . 9  |-  ( ( x  e.  ~H  /\  ( x  =/=  0h  /\  A  =  ( _|_ `  ( _|_ `  {
x } ) ) ) )  ->  E. x  e.  ~H  ( x  =/= 
0h  /\  A  =  ( _|_ `  ( _|_ `  { x } ) ) ) )
369, 10, 34, 35syl12anc 1262 . . . . . . . 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 616 . . . . . . 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 2906 . . . . . 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 218 . . . . 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 434 . . . 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 198 . . 3  |-  ( A  e. HAtoms  ->  E. x  e.  ~H  ( x  =/=  0h  /\  A  =  ( _|_ `  ( _|_ `  {
x } ) ) ) )
42 h1da 28000 . . . . . . 7  |-  ( ( x  e.  ~H  /\  x  =/=  0h )  -> 
( _|_ `  ( _|_ `  { x }
) )  e. HAtoms )
43 eleq1 2495 . . . . . . 7  |-  ( A  =  ( _|_ `  ( _|_ `  { x }
) )  ->  ( A  e. HAtoms  <->  ( _|_ `  ( _|_ `  { x }
) )  e. HAtoms )
)
4442, 43syl5ibr 224 . . . . . 6  |-  ( A  =  ( _|_ `  ( _|_ `  { x }
) )  ->  (
( x  e.  ~H  /\  x  =/=  0h )  ->  A  e. HAtoms ) )
4544expdcom 440 . . . . 5  |-  ( x  e.  ~H  ->  (
x  =/=  0h  ->  ( A  =  ( _|_ `  ( _|_ `  {
x } ) )  ->  A  e. HAtoms )
) )
4645impd 432 . . . 4  |-  ( x  e.  ~H  ->  (
( x  =/=  0h  /\  A  =  ( _|_ `  ( _|_ `  {
x } ) ) )  ->  A  e. HAtoms ) )
4746rexlimiv 2908 . . 3  |-  ( E. x  e.  ~H  (
x  =/=  0h  /\  A  =  ( _|_ `  ( _|_ `  {
x } ) ) )  ->  A  e. HAtoms )
4841, 47impbii 190 . 2  |-  ( A  e. HAtoms 
<->  E. x  e.  ~H  ( x  =/=  0h  /\  A  =  ( _|_ `  ( _|_ `  {
x } ) ) ) )
49 spansn 27210 . . . . 5  |-  ( x  e.  ~H  ->  ( span `  { x }
)  =  ( _|_ `  ( _|_ `  {
x } ) ) )
5049eqeq2d 2436 . . . 4  |-  ( x  e.  ~H  ->  ( A  =  ( span `  { x } )  <-> 
A  =  ( _|_ `  ( _|_ `  {
x } ) ) ) )
5150anbi2d 708 . . 3  |-  ( x  e.  ~H  ->  (
( x  =/=  0h  /\  A  =  ( span `  { x } ) )  <->  ( x  =/= 
0h  /\  A  =  ( _|_ `  ( _|_ `  { x } ) ) ) ) )
5251rexbiia 2923 . 2  |-  ( E. x  e.  ~H  (
x  =/=  0h  /\  A  =  ( span `  { x } ) )  <->  E. x  e.  ~H  ( x  =/=  0h  /\  A  =  ( _|_ `  ( _|_ `  {
x } ) ) ) )
5348, 52bitr4i 255 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 187    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1872    =/= wne 2614   A.wral 2771   E.wrex 2772    C_ wss 3436   {csn 3998   ` cfv 5601   ~Hchil 26570   0hc0v 26575   CHcch 26580   _|_cort 26581   spancspn 26583   0Hc0h 26586  HAtomscat 26616
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-inf2 8155  ax-cc 8872  ax-cnex 9602  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-mulrcl 9609  ax-mulcom 9610  ax-addass 9611  ax-mulass 9612  ax-distr 9613  ax-i2m1 9614  ax-1ne0 9615  ax-1rid 9616  ax-rnegex 9617  ax-rrecex 9618  ax-cnre 9619  ax-pre-lttri 9620  ax-pre-lttrn 9621  ax-pre-ltadd 9622  ax-pre-mulgt0 9623  ax-pre-sup 9624  ax-addf 9625  ax-mulf 9626  ax-hilex 26650  ax-hfvadd 26651  ax-hvcom 26652  ax-hvass 26653  ax-hv0cl 26654  ax-hvaddid 26655  ax-hfvmul 26656  ax-hvmulid 26657  ax-hvmulass 26658  ax-hvdistr1 26659  ax-hvdistr2 26660  ax-hvmul0 26661  ax-hfi 26730  ax-his1 26733  ax-his2 26734  ax-his3 26735  ax-his4 26736  ax-hcompl 26853
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-int 4256  df-iun 4301  df-iin 4302  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-se 4813  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-of 6545  df-om 6707  df-1st 6807  df-2nd 6808  df-supp 6926  df-wrecs 7039  df-recs 7101  df-rdg 7139  df-1o 7193  df-2o 7194  df-oadd 7197  df-omul 7198  df-er 7374  df-map 7485  df-pm 7486  df-ixp 7534  df-en 7581  df-dom 7582  df-sdom 7583  df-fin 7584  df-fsupp 7893  df-fi 7934  df-sup 7965  df-inf 7966  df-oi 8034  df-card 8381  df-acn 8384  df-cda 8605  df-pnf 9684  df-mnf 9685  df-xr 9686  df-ltxr 9687  df-le 9688  df-sub 9869  df-neg 9870  df-div 10277  df-nn 10617  df-2 10675  df-3 10676  df-4 10677  df-5 10678  df-6 10679  df-7 10680  df-8 10681  df-9 10682  df-10 10683  df-n0 10877  df-z 10945  df-dec 11059  df-uz 11167  df-q 11272  df-rp 11310  df-xneg 11416  df-xadd 11417  df-xmul 11418  df-ioo 11646  df-ico 11648  df-icc 11649  df-fz 11792  df-fzo 11923  df-fl 12034  df-seq 12220  df-exp 12279  df-hash 12522  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-clim 13551  df-rlim 13552  df-sum 13752  df-struct 15122  df-ndx 15123  df-slot 15124  df-base 15125  df-sets 15126  df-ress 15127  df-plusg 15202  df-mulr 15203  df-starv 15204  df-sca 15205  df-vsca 15206  df-ip 15207  df-tset 15208  df-ple 15209  df-ds 15211  df-unif 15212  df-hom 15213  df-cco 15214  df-rest 15320  df-topn 15321  df-0g 15339  df-gsum 15340  df-topgen 15341  df-pt 15342  df-prds 15345  df-xrs 15399  df-qtop 15405  df-imas 15406  df-xps 15409  df-mre 15491  df-mrc 15492  df-acs 15494  df-mgm 16487  df-sgrp 16526  df-mnd 16536  df-submnd 16582  df-mulg 16675  df-cntz 16970  df-cmn 17431  df-psmet 18961  df-xmet 18962  df-met 18963  df-bl 18964  df-mopn 18965  df-fbas 18966  df-fg 18967  df-cnfld 18970  df-top 19919  df-bases 19920  df-topon 19921  df-topsp 19922  df-cld 20032  df-ntr 20033  df-cls 20034  df-nei 20112  df-cn 20241  df-cnp 20242  df-lm 20243  df-haus 20329  df-tx 20575  df-hmeo 20768  df-fil 20859  df-fm 20951  df-flim 20952  df-flf 20953  df-xms 21333  df-ms 21334  df-tms 21335  df-cfil 22223  df-cau 22224  df-cmet 22225  df-grpo 25917  df-gid 25918  df-ginv 25919  df-gdiv 25920  df-ablo 26008  df-subgo 26028  df-vc 26163  df-nv 26209  df-va 26212  df-ba 26213  df-sm 26214  df-0v 26215  df-vs 26216  df-nmcv 26217  df-ims 26218  df-dip 26335  df-ssp 26359  df-ph 26452  df-cbn 26503  df-hnorm 26619  df-hba 26620  df-hvsub 26622  df-hlim 26623  df-hcau 26624  df-sh 26858  df-ch 26872  df-oc 26903  df-ch0 26904  df-span 26960  df-cv 27930  df-at 27989
This theorem is referenced by:  superpos  28005  chcv1  28006  chjatom  28008
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