HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  chirredi Structured version   Unicode version

Theorem chirredi 25733
Description: The Hilbert lattice is irreducible: any element that commutes with all elements must be zero or one. Theorem 14.8.4 of [BeltramettiCassinelli] p. 166. (Contributed by NM, 15-Jun-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
chirred.1  |-  A  e. 
CH
chirred.2  |-  ( x  e.  CH  ->  A  C_H  x )
Assertion
Ref Expression
chirredi  |-  ( A  =  0H  \/  A  =  ~H )
Distinct variable group:    x, A

Proof of Theorem chirredi
Dummy variables  q  p  r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2441 . . 3  |-  0H  =  0H
2 ioran 487 . . . . 5  |-  ( -.  ( A  =  0H  \/  ( _|_ `  A
)  =  0H )  <-> 
( -.  A  =  0H  /\  -.  ( _|_ `  A )  =  0H ) )
3 df-ne 2606 . . . . . 6  |-  ( A  =/=  0H  <->  -.  A  =  0H )
4 df-ne 2606 . . . . . 6  |-  ( ( _|_ `  A )  =/=  0H  <->  -.  ( _|_ `  A )  =  0H )
53, 4anbi12i 692 . . . . 5  |-  ( ( A  =/=  0H  /\  ( _|_ `  A )  =/=  0H )  <->  ( -.  A  =  0H  /\  -.  ( _|_ `  A )  =  0H ) )
62, 5bitr4i 252 . . . 4  |-  ( -.  ( A  =  0H  \/  ( _|_ `  A
)  =  0H )  <-> 
( A  =/=  0H  /\  ( _|_ `  A
)  =/=  0H ) )
7 chirred.1 . . . . . . . 8  |-  A  e. 
CH
87hatomici 25698 . . . . . . 7  |-  ( A  =/=  0H  ->  E. p  e. HAtoms  p  C_  A )
97choccli 24645 . . . . . . . 8  |-  ( _|_ `  A )  e.  CH
109hatomici 25698 . . . . . . 7  |-  ( ( _|_ `  A )  =/=  0H  ->  E. q  e. HAtoms  q  C_  ( _|_ `  A ) )
118, 10anim12i 563 . . . . . 6  |-  ( ( A  =/=  0H  /\  ( _|_ `  A )  =/=  0H )  -> 
( E. p  e. HAtoms  p  C_  A  /\  E. q  e. HAtoms  q  C_  ( _|_ `  A ) ) )
12 reeanv 2886 . . . . . 6  |-  ( E. p  e. HAtoms  E. q  e. HAtoms  ( p  C_  A  /\  q  C_  ( _|_ `  A ) )  <->  ( E. p  e. HAtoms  p  C_  A  /\  E. q  e. HAtoms  q  C_  ( _|_ `  A
) ) )
1311, 12sylibr 212 . . . . 5  |-  ( ( A  =/=  0H  /\  ( _|_ `  A )  =/=  0H )  ->  E. p  e. HAtoms  E. q  e. HAtoms  ( p  C_  A  /\  q  C_  ( _|_ `  A ) ) )
14 simpll 748 . . . . . . . . . 10  |-  ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  (
q  e. HAtoms  /\  q  C_  ( _|_ `  A
) ) )  ->  p  e. HAtoms )
15 simprl 750 . . . . . . . . . 10  |-  ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  (
q  e. HAtoms  /\  q  C_  ( _|_ `  A
) ) )  -> 
q  e. HAtoms )
16 atelch 25683 . . . . . . . . . . . . . . . 16  |-  ( p  e. HAtoms  ->  p  e.  CH )
17 chsscon3 24838 . . . . . . . . . . . . . . . 16  |-  ( ( p  e.  CH  /\  A  e.  CH )  ->  ( p  C_  A  <->  ( _|_ `  A ) 
C_  ( _|_ `  p
) ) )
1816, 7, 17sylancl 657 . . . . . . . . . . . . . . 15  |-  ( p  e. HAtoms  ->  ( p  C_  A 
<->  ( _|_ `  A
)  C_  ( _|_ `  p ) ) )
1918biimpa 481 . . . . . . . . . . . . . 14  |-  ( ( p  e. HAtoms  /\  p  C_  A )  ->  ( _|_ `  A )  C_  ( _|_ `  p ) )
20 sstr 3361 . . . . . . . . . . . . . 14  |-  ( ( q  C_  ( _|_ `  A )  /\  ( _|_ `  A )  C_  ( _|_ `  p ) )  ->  q  C_  ( _|_ `  p ) )
2119, 20sylan2 471 . . . . . . . . . . . . 13  |-  ( ( q  C_  ( _|_ `  A )  /\  (
p  e. HAtoms  /\  p  C_  A ) )  -> 
q  C_  ( _|_ `  p ) )
2221ancoms 450 . . . . . . . . . . . 12  |-  ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  q  C_  ( _|_ `  A
) )  ->  q  C_  ( _|_ `  p
) )
23 atne0 25684 . . . . . . . . . . . . . . 15  |-  ( p  e. HAtoms  ->  p  =/=  0H )
2423adantr 462 . . . . . . . . . . . . . 14  |-  ( ( p  e. HAtoms  /\  q  C_  ( _|_ `  p
) )  ->  p  =/=  0H )
25 sseq1 3374 . . . . . . . . . . . . . . . . . . . 20  |-  ( p  =  q  ->  (
p  C_  ( _|_ `  p )  <->  q  C_  ( _|_ `  p ) ) )
2625bicomd 201 . . . . . . . . . . . . . . . . . . 19  |-  ( p  =  q  ->  (
q  C_  ( _|_ `  p )  <->  p  C_  ( _|_ `  p ) ) )
27 chssoc 24834 . . . . . . . . . . . . . . . . . . . 20  |-  ( p  e.  CH  ->  (
p  C_  ( _|_ `  p )  <->  p  =  0H ) )
2816, 27syl 16 . . . . . . . . . . . . . . . . . . 19  |-  ( p  e. HAtoms  ->  ( p  C_  ( _|_ `  p )  <-> 
p  =  0H ) )
2926, 28sylan9bbr 695 . . . . . . . . . . . . . . . . . 18  |-  ( ( p  e. HAtoms  /\  p  =  q )  -> 
( q  C_  ( _|_ `  p )  <->  p  =  0H ) )
3029biimpa 481 . . . . . . . . . . . . . . . . 17  |-  ( ( ( p  e. HAtoms  /\  p  =  q )  /\  q  C_  ( _|_ `  p
) )  ->  p  =  0H )
3130an32s 797 . . . . . . . . . . . . . . . 16  |-  ( ( ( p  e. HAtoms  /\  q  C_  ( _|_ `  p
) )  /\  p  =  q )  ->  p  =  0H )
3231ex 434 . . . . . . . . . . . . . . 15  |-  ( ( p  e. HAtoms  /\  q  C_  ( _|_ `  p
) )  ->  (
p  =  q  ->  p  =  0H )
)
3332necon3d 2644 . . . . . . . . . . . . . 14  |-  ( ( p  e. HAtoms  /\  q  C_  ( _|_ `  p
) )  ->  (
p  =/=  0H  ->  p  =/=  q ) )
3424, 33mpd 15 . . . . . . . . . . . . 13  |-  ( ( p  e. HAtoms  /\  q  C_  ( _|_ `  p
) )  ->  p  =/=  q )
3534adantlr 709 . . . . . . . . . . . 12  |-  ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  q  C_  ( _|_ `  p
) )  ->  p  =/=  q )
3622, 35syldan 467 . . . . . . . . . . 11  |-  ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  q  C_  ( _|_ `  A
) )  ->  p  =/=  q )
3736adantrl 710 . . . . . . . . . 10  |-  ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  (
q  e. HAtoms  /\  q  C_  ( _|_ `  A
) ) )  ->  p  =/=  q )
38 superpos 25693 . . . . . . . . . 10  |-  ( ( p  e. HAtoms  /\  q  e. HAtoms  /\  p  =/=  q
)  ->  E. r  e. HAtoms  ( r  =/=  p  /\  r  =/=  q  /\  r  C_  ( p  vH  q ) ) )
3914, 15, 37, 38syl3anc 1213 . . . . . . . . 9  |-  ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  (
q  e. HAtoms  /\  q  C_  ( _|_ `  A
) ) )  ->  E. r  e. HAtoms  ( r  =/=  p  /\  r  =/=  q  /\  r  C_  ( p  vH  q
) ) )
40 df-3an 962 . . . . . . . . . . . 12  |-  ( ( r  =/=  p  /\  r  =/=  q  /\  r  C_  ( p  vH  q
) )  <->  ( (
r  =/=  p  /\  r  =/=  q )  /\  r  C_  ( p  vH  q ) ) )
41 neanior 2695 . . . . . . . . . . . . 13  |-  ( ( r  =/=  p  /\  r  =/=  q )  <->  -.  (
r  =  p  \/  r  =  q ) )
4241anbi1i 690 . . . . . . . . . . . 12  |-  ( ( ( r  =/=  p  /\  r  =/=  q
)  /\  r  C_  ( p  vH  q
) )  <->  ( -.  ( r  =  p  \/  r  =  q )  /\  r  C_  ( p  vH  q
) ) )
4340, 42bitri 249 . . . . . . . . . . 11  |-  ( ( r  =/=  p  /\  r  =/=  q  /\  r  C_  ( p  vH  q
) )  <->  ( -.  ( r  =  p  \/  r  =  q )  /\  r  C_  ( p  vH  q
) ) )
44 chirred.2 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  CH  ->  A  C_H  x )
457, 44chirredlem4 25732 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  ( q  e. HAtoms  /\  q  C_  ( _|_ `  A ) ) )  /\  ( r  e. HAtoms  /\  r  C_  ( p  vH  q ) ) )  ->  ( r  =  p  \/  r  =  q ) )
4645anassrs 643 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( p  e. HAtoms  /\  p  C_  A
)  /\  ( q  e. HAtoms  /\  q  C_  ( _|_ `  A ) ) )  /\  r  e. HAtoms
)  /\  r  C_  ( p  vH  q
) )  ->  (
r  =  p  \/  r  =  q ) )
4746pm2.24d 143 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( p  e. HAtoms  /\  p  C_  A
)  /\  ( q  e. HAtoms  /\  q  C_  ( _|_ `  A ) ) )  /\  r  e. HAtoms
)  /\  r  C_  ( p  vH  q
) )  ->  ( -.  ( r  =  p  \/  r  =  q )  ->  -.  0H  =  0H ) )
4847ex 434 . . . . . . . . . . . . 13  |-  ( ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  ( q  e. HAtoms  /\  q  C_  ( _|_ `  A ) ) )  /\  r  e. HAtoms )  ->  ( r  C_  (
p  vH  q )  ->  ( -.  ( r  =  p  \/  r  =  q )  ->  -.  0H  =  0H ) ) )
4948com23 78 . . . . . . . . . . . 12  |-  ( ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  ( q  e. HAtoms  /\  q  C_  ( _|_ `  A ) ) )  /\  r  e. HAtoms )  ->  ( -.  ( r  =  p  \/  r  =  q )  -> 
( r  C_  (
p  vH  q )  ->  -.  0H  =  0H ) ) )
5049imp3a 431 . . . . . . . . . . 11  |-  ( ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  ( q  e. HAtoms  /\  q  C_  ( _|_ `  A ) ) )  /\  r  e. HAtoms )  ->  ( ( -.  (
r  =  p  \/  r  =  q )  /\  r  C_  (
p  vH  q )
)  ->  -.  0H  =  0H ) )
5143, 50syl5bi 217 . . . . . . . . . 10  |-  ( ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  ( q  e. HAtoms  /\  q  C_  ( _|_ `  A ) ) )  /\  r  e. HAtoms )  ->  ( ( r  =/=  p  /\  r  =/=  q  /\  r  C_  ( p  vH  q
) )  ->  -.  0H  =  0H )
)
5251rexlimdva 2839 . . . . . . . . 9  |-  ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  (
q  e. HAtoms  /\  q  C_  ( _|_ `  A
) ) )  -> 
( E. r  e. HAtoms  ( r  =/=  p  /\  r  =/=  q  /\  r  C_  ( p  vH  q ) )  ->  -.  0H  =  0H ) )
5339, 52mpd 15 . . . . . . . 8  |-  ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  (
q  e. HAtoms  /\  q  C_  ( _|_ `  A
) ) )  ->  -.  0H  =  0H )
5453an4s 817 . . . . . . 7  |-  ( ( ( p  e. HAtoms  /\  q  e. HAtoms )  /\  ( p 
C_  A  /\  q  C_  ( _|_ `  A
) ) )  ->  -.  0H  =  0H )
5554ex 434 . . . . . 6  |-  ( ( p  e. HAtoms  /\  q  e. HAtoms )  ->  ( (
p  C_  A  /\  q  C_  ( _|_ `  A
) )  ->  -.  0H  =  0H )
)
5655rexlimivv 2844 . . . . 5  |-  ( E. p  e. HAtoms  E. q  e. HAtoms  ( p  C_  A  /\  q  C_  ( _|_ `  A ) )  ->  -.  0H  =  0H )
5713, 56syl 16 . . . 4  |-  ( ( A  =/=  0H  /\  ( _|_ `  A )  =/=  0H )  ->  -.  0H  =  0H )
586, 57sylbi 195 . . 3  |-  ( -.  ( A  =  0H  \/  ( _|_ `  A
)  =  0H )  ->  -.  0H  =  0H )
591, 58mt4 137 . 2  |-  ( A  =  0H  \/  ( _|_ `  A )  =  0H )
60 fveq2 5688 . . . 4  |-  ( ( _|_ `  A )  =  0H  ->  ( _|_ `  ( _|_ `  A
) )  =  ( _|_ `  0H ) )
617ococi 24743 . . . 4  |-  ( _|_ `  ( _|_ `  A
) )  =  A
62 choc0 24664 . . . 4  |-  ( _|_ `  0H )  =  ~H
6360, 61, 623eqtr3g 2496 . . 3  |-  ( ( _|_ `  A )  =  0H  ->  A  =  ~H )
6463orim2i 515 . 2  |-  ( ( A  =  0H  \/  ( _|_ `  A )  =  0H )  -> 
( A  =  0H  \/  A  =  ~H ) )
6559, 64ax-mp 5 1  |-  ( A  =  0H  \/  A  =  ~H )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761    =/= wne 2604   E.wrex 2714    C_ wss 3325   class class class wbr 4289   ` cfv 5415  (class class class)co 6090   ~Hchil 24256   CHcch 24266   _|_cort 24267    vH chj 24270   0Hc0h 24272    C_H ccm 24273  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-shs 24646  df-span 24647  df-chj 24648  df-chsup 24649  df-pjh 24733  df-cm 24921  df-cv 25618  df-at 25677
This theorem is referenced by:  chirred  25734
  Copyright terms: Public domain W3C validator