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

Theorem chirredi 28032
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 2422 . . 3  |-  0H  =  0H
2 ioran 492 . . . . 5  |-  ( -.  ( A  =  0H  \/  ( _|_ `  A
)  =  0H )  <-> 
( -.  A  =  0H  /\  -.  ( _|_ `  A )  =  0H ) )
3 df-ne 2620 . . . . . 6  |-  ( A  =/=  0H  <->  -.  A  =  0H )
4 df-ne 2620 . . . . . 6  |-  ( ( _|_ `  A )  =/=  0H  <->  -.  ( _|_ `  A )  =  0H )
53, 4anbi12i 701 . . . . 5  |-  ( ( A  =/=  0H  /\  ( _|_ `  A )  =/=  0H )  <->  ( -.  A  =  0H  /\  -.  ( _|_ `  A )  =  0H ) )
62, 5bitr4i 255 . . . 4  |-  ( -.  ( A  =  0H  \/  ( _|_ `  A
)  =  0H )  <-> 
( A  =/=  0H  /\  ( _|_ `  A
)  =/=  0H ) )
7 chirred.1 . . . . . . . 8  |-  A  e. 
CH
87hatomici 27997 . . . . . . 7  |-  ( A  =/=  0H  ->  E. p  e. HAtoms  p  C_  A )
97choccli 26945 . . . . . . . 8  |-  ( _|_ `  A )  e.  CH
109hatomici 27997 . . . . . . 7  |-  ( ( _|_ `  A )  =/=  0H  ->  E. q  e. HAtoms  q  C_  ( _|_ `  A ) )
118, 10anim12i 568 . . . . . 6  |-  ( ( A  =/=  0H  /\  ( _|_ `  A )  =/=  0H )  -> 
( E. p  e. HAtoms  p  C_  A  /\  E. q  e. HAtoms  q  C_  ( _|_ `  A ) ) )
12 reeanv 2996 . . . . . 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 215 . . . . 5  |-  ( ( A  =/=  0H  /\  ( _|_ `  A )  =/=  0H )  ->  E. p  e. HAtoms  E. q  e. HAtoms  ( p  C_  A  /\  q  C_  ( _|_ `  A ) ) )
14 simpll 758 . . . . . . . . . 10  |-  ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  (
q  e. HAtoms  /\  q  C_  ( _|_ `  A
) ) )  ->  p  e. HAtoms )
15 simprl 762 . . . . . . . . . 10  |-  ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  (
q  e. HAtoms  /\  q  C_  ( _|_ `  A
) ) )  -> 
q  e. HAtoms )
16 atelch 27982 . . . . . . . . . . . . . . . 16  |-  ( p  e. HAtoms  ->  p  e.  CH )
17 chsscon3 27138 . . . . . . . . . . . . . . . 16  |-  ( ( p  e.  CH  /\  A  e.  CH )  ->  ( p  C_  A  <->  ( _|_ `  A ) 
C_  ( _|_ `  p
) ) )
1816, 7, 17sylancl 666 . . . . . . . . . . . . . . 15  |-  ( p  e. HAtoms  ->  ( p  C_  A 
<->  ( _|_ `  A
)  C_  ( _|_ `  p ) ) )
1918biimpa 486 . . . . . . . . . . . . . 14  |-  ( ( p  e. HAtoms  /\  p  C_  A )  ->  ( _|_ `  A )  C_  ( _|_ `  p ) )
20 sstr 3472 . . . . . . . . . . . . . 14  |-  ( ( q  C_  ( _|_ `  A )  /\  ( _|_ `  A )  C_  ( _|_ `  p ) )  ->  q  C_  ( _|_ `  p ) )
2119, 20sylan2 476 . . . . . . . . . . . . 13  |-  ( ( q  C_  ( _|_ `  A )  /\  (
p  e. HAtoms  /\  p  C_  A ) )  -> 
q  C_  ( _|_ `  p ) )
2221ancoms 454 . . . . . . . . . . . 12  |-  ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  q  C_  ( _|_ `  A
) )  ->  q  C_  ( _|_ `  p
) )
23 atne0 27983 . . . . . . . . . . . . . . 15  |-  ( p  e. HAtoms  ->  p  =/=  0H )
2423adantr 466 . . . . . . . . . . . . . 14  |-  ( ( p  e. HAtoms  /\  q  C_  ( _|_ `  p
) )  ->  p  =/=  0H )
25 sseq1 3485 . . . . . . . . . . . . . . . . . . . 20  |-  ( p  =  q  ->  (
p  C_  ( _|_ `  p )  <->  q  C_  ( _|_ `  p ) ) )
2625bicomd 204 . . . . . . . . . . . . . . . . . . 19  |-  ( p  =  q  ->  (
q  C_  ( _|_ `  p )  <->  p  C_  ( _|_ `  p ) ) )
27 chssoc 27134 . . . . . . . . . . . . . . . . . . . 20  |-  ( p  e.  CH  ->  (
p  C_  ( _|_ `  p )  <->  p  =  0H ) )
2816, 27syl 17 . . . . . . . . . . . . . . . . . . 19  |-  ( p  e. HAtoms  ->  ( p  C_  ( _|_ `  p )  <-> 
p  =  0H ) )
2926, 28sylan9bbr 705 . . . . . . . . . . . . . . . . . 18  |-  ( ( p  e. HAtoms  /\  p  =  q )  -> 
( q  C_  ( _|_ `  p )  <->  p  =  0H ) )
3029biimpa 486 . . . . . . . . . . . . . . . . 17  |-  ( ( ( p  e. HAtoms  /\  p  =  q )  /\  q  C_  ( _|_ `  p
) )  ->  p  =  0H )
3130an32s 811 . . . . . . . . . . . . . . . 16  |-  ( ( ( p  e. HAtoms  /\  q  C_  ( _|_ `  p
) )  /\  p  =  q )  ->  p  =  0H )
3231ex 435 . . . . . . . . . . . . . . 15  |-  ( ( p  e. HAtoms  /\  q  C_  ( _|_ `  p
) )  ->  (
p  =  q  ->  p  =  0H )
)
3332necon3d 2648 . . . . . . . . . . . . . 14  |-  ( ( p  e. HAtoms  /\  q  C_  ( _|_ `  p
) )  ->  (
p  =/=  0H  ->  p  =/=  q ) )
3424, 33mpd 15 . . . . . . . . . . . . 13  |-  ( ( p  e. HAtoms  /\  q  C_  ( _|_ `  p
) )  ->  p  =/=  q )
3534adantlr 719 . . . . . . . . . . . 12  |-  ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  q  C_  ( _|_ `  p
) )  ->  p  =/=  q )
3622, 35syldan 472 . . . . . . . . . . 11  |-  ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  q  C_  ( _|_ `  A
) )  ->  p  =/=  q )
3736adantrl 720 . . . . . . . . . 10  |-  ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  (
q  e. HAtoms  /\  q  C_  ( _|_ `  A
) ) )  ->  p  =/=  q )
38 superpos 27992 . . . . . . . . . 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 1264 . . . . . . . . 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 984 . . . . . . . . . . . 12  |-  ( ( r  =/=  p  /\  r  =/=  q  /\  r  C_  ( p  vH  q
) )  <->  ( (
r  =/=  p  /\  r  =/=  q )  /\  r  C_  ( p  vH  q ) ) )
41 neanior 2749 . . . . . . . . . . . . 13  |-  ( ( r  =/=  p  /\  r  =/=  q )  <->  -.  (
r  =  p  \/  r  =  q ) )
4241anbi1i 699 . . . . . . . . . . . 12  |-  ( ( ( r  =/=  p  /\  r  =/=  q
)  /\  r  C_  ( p  vH  q
) )  <->  ( -.  ( r  =  p  \/  r  =  q )  /\  r  C_  ( p  vH  q
) ) )
4340, 42bitri 252 . . . . . . . . . . 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 28031 . . . . . . . . . . . . . . . 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 652 . . . . . . . . . . . . . . 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 137 . . . . . . . . . . . . . 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 435 . . . . . . . . . . . . 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 81 . . . . . . . . . . . 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 ) ) )
5049impd 432 . . . . . . . . . . 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 220 . . . . . . . . . 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 2917 . . . . . . . . 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 833 . . . . . . 7  |-  ( ( ( p  e. HAtoms  /\  q  e. HAtoms )  /\  ( p 
C_  A  /\  q  C_  ( _|_ `  A
) ) )  ->  -.  0H  =  0H )
5554ex 435 . . . . . 6  |-  ( ( p  e. HAtoms  /\  q  e. HAtoms )  ->  ( (
p  C_  A  /\  q  C_  ( _|_ `  A
) )  ->  -.  0H  =  0H )
)
5655rexlimivv 2922 . . . . 5  |-  ( E. p  e. HAtoms  E. q  e. HAtoms  ( p  C_  A  /\  q  C_  ( _|_ `  A ) )  ->  -.  0H  =  0H )
5713, 56syl 17 . . . 4  |-  ( ( A  =/=  0H  /\  ( _|_ `  A )  =/=  0H )  ->  -.  0H  =  0H )
586, 57sylbi 198 . . 3  |-  ( -.  ( A  =  0H  \/  ( _|_ `  A
)  =  0H )  ->  -.  0H  =  0H )
591, 58mt4 142 . 2  |-  ( A  =  0H  \/  ( _|_ `  A )  =  0H )
60 fveq2 5877 . . . 4  |-  ( ( _|_ `  A )  =  0H  ->  ( _|_ `  ( _|_ `  A
) )  =  ( _|_ `  0H ) )
617ococi 27043 . . . 4  |-  ( _|_ `  ( _|_ `  A
) )  =  A
62 choc0 26964 . . . 4  |-  ( _|_ `  0H )  =  ~H
6360, 61, 623eqtr3g 2486 . . 3  |-  ( ( _|_ `  A )  =  0H  ->  A  =  ~H )
6463orim2i 520 . 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 187    \/ wo 369    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1868    =/= wne 2618   E.wrex 2776    C_ wss 3436   class class class wbr 4420   ` cfv 5597  (class class class)co 6301   ~Hchil 26557   CHcch 26567   _|_cort 26568    vH chj 26571   0Hc0h 26573    C_H ccm 26574  HAtomscat 26603
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593  ax-inf2 8148  ax-cc 8865  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617  ax-addf 9618  ax-mulf 9619  ax-hilex 26637  ax-hfvadd 26638  ax-hvcom 26639  ax-hvass 26640  ax-hv0cl 26641  ax-hvaddid 26642  ax-hfvmul 26643  ax-hvmulid 26644  ax-hvmulass 26645  ax-hvdistr1 26646  ax-hvdistr2 26647  ax-hvmul0 26648  ax-hfi 26717  ax-his1 26720  ax-his2 26721  ax-his3 26722  ax-his4 26723  ax-hcompl 26840
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 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  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 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-iin 4299  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4760  df-id 4764  df-po 4770  df-so 4771  df-fr 4808  df-se 4809  df-we 4810  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-pred 5395  df-ord 5441  df-on 5442  df-lim 5443  df-suc 5444  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-isom 5606  df-riota 6263  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-of 6541  df-om 6703  df-1st 6803  df-2nd 6804  df-supp 6922  df-wrecs 7032  df-recs 7094  df-rdg 7132  df-1o 7186  df-2o 7187  df-oadd 7190  df-omul 7191  df-er 7367  df-map 7478  df-pm 7479  df-ixp 7527  df-en 7574  df-dom 7575  df-sdom 7576  df-fin 7577  df-fsupp 7886  df-fi 7927  df-sup 7958  df-inf 7959  df-oi 8027  df-card 8374  df-acn 8377  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ioo 11639  df-ico 11641  df-icc 11642  df-fz 11785  df-fzo 11916  df-fl 12027  df-seq 12213  df-exp 12272  df-hash 12515  df-cj 13150  df-re 13151  df-im 13152  df-sqrt 13286  df-abs 13287  df-clim 13539  df-rlim 13540  df-sum 13740  df-struct 15110  df-ndx 15111  df-slot 15112  df-base 15113  df-sets 15114  df-ress 15115  df-plusg 15190  df-mulr 15191  df-starv 15192  df-sca 15193  df-vsca 15194  df-ip 15195  df-tset 15196  df-ple 15197  df-ds 15199  df-unif 15200  df-hom 15201  df-cco 15202  df-rest 15308  df-topn 15309  df-0g 15327  df-gsum 15328  df-topgen 15329  df-pt 15330  df-prds 15333  df-xrs 15387  df-qtop 15393  df-imas 15394  df-xps 15397  df-mre 15479  df-mrc 15480  df-acs 15482  df-mgm 16475  df-sgrp 16514  df-mnd 16524  df-submnd 16570  df-mulg 16663  df-cntz 16958  df-cmn 17419  df-psmet 18949  df-xmet 18950  df-met 18951  df-bl 18952  df-mopn 18953  df-fbas 18954  df-fg 18955  df-cnfld 18958  df-top 19907  df-bases 19908  df-topon 19909  df-topsp 19910  df-cld 20020  df-ntr 20021  df-cls 20022  df-nei 20100  df-cn 20229  df-cnp 20230  df-lm 20231  df-haus 20317  df-tx 20563  df-hmeo 20756  df-fil 20847  df-fm 20939  df-flim 20940  df-flf 20941  df-xms 21321  df-ms 21322  df-tms 21323  df-cfil 22211  df-cau 22212  df-cmet 22213  df-grpo 25904  df-gid 25905  df-ginv 25906  df-gdiv 25907  df-ablo 25995  df-subgo 26015  df-vc 26150  df-nv 26196  df-va 26199  df-ba 26200  df-sm 26201  df-0v 26202  df-vs 26203  df-nmcv 26204  df-ims 26205  df-dip 26322  df-ssp 26346  df-ph 26439  df-cbn 26490  df-hnorm 26606  df-hba 26607  df-hvsub 26609  df-hlim 26610  df-hcau 26611  df-sh 26845  df-ch 26859  df-oc 26890  df-ch0 26891  df-shs 26946  df-span 26947  df-chj 26948  df-chsup 26949  df-pjh 27033  df-cm 27221  df-cv 27917  df-at 27976
This theorem is referenced by:  chirred  28033
  Copyright terms: Public domain W3C validator