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Theorem chirredi 22974
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 2283 . . 3  |-  0H  =  0H
2 ioran 476 . . . . 5  |-  ( -.  ( A  =  0H  \/  ( _|_ `  A
)  =  0H )  <-> 
( -.  A  =  0H  /\  -.  ( _|_ `  A )  =  0H ) )
3 df-ne 2448 . . . . . 6  |-  ( A  =/=  0H  <->  -.  A  =  0H )
4 df-ne 2448 . . . . . 6  |-  ( ( _|_ `  A )  =/=  0H  <->  -.  ( _|_ `  A )  =  0H )
53, 4anbi12i 678 . . . . 5  |-  ( ( A  =/=  0H  /\  ( _|_ `  A )  =/=  0H )  <->  ( -.  A  =  0H  /\  -.  ( _|_ `  A )  =  0H ) )
62, 5bitr4i 243 . . . 4  |-  ( -.  ( A  =  0H  \/  ( _|_ `  A
)  =  0H )  <-> 
( A  =/=  0H  /\  ( _|_ `  A
)  =/=  0H ) )
7 chirred.1 . . . . . . . 8  |-  A  e. 
CH
87hatomici 22939 . . . . . . 7  |-  ( A  =/=  0H  ->  E. p  e. HAtoms  p  C_  A )
97choccli 21886 . . . . . . . 8  |-  ( _|_ `  A )  e.  CH
109hatomici 22939 . . . . . . 7  |-  ( ( _|_ `  A )  =/=  0H  ->  E. q  e. HAtoms  q  C_  ( _|_ `  A ) )
118, 10anim12i 549 . . . . . 6  |-  ( ( A  =/=  0H  /\  ( _|_ `  A )  =/=  0H )  -> 
( E. p  e. HAtoms  p  C_  A  /\  E. q  e. HAtoms  q  C_  ( _|_ `  A ) ) )
12 reeanv 2707 . . . . . 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 203 . . . . 5  |-  ( ( A  =/=  0H  /\  ( _|_ `  A )  =/=  0H )  ->  E. p  e. HAtoms  E. q  e. HAtoms  ( p  C_  A  /\  q  C_  ( _|_ `  A ) ) )
14 simpll 730 . . . . . . . . . 10  |-  ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  (
q  e. HAtoms  /\  q  C_  ( _|_ `  A
) ) )  ->  p  e. HAtoms )
15 simprl 732 . . . . . . . . . 10  |-  ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  (
q  e. HAtoms  /\  q  C_  ( _|_ `  A
) ) )  -> 
q  e. HAtoms )
16 atelch 22924 . . . . . . . . . . . . . . . 16  |-  ( p  e. HAtoms  ->  p  e.  CH )
17 chsscon3 22079 . . . . . . . . . . . . . . . 16  |-  ( ( p  e.  CH  /\  A  e.  CH )  ->  ( p  C_  A  <->  ( _|_ `  A ) 
C_  ( _|_ `  p
) ) )
1816, 7, 17sylancl 643 . . . . . . . . . . . . . . 15  |-  ( p  e. HAtoms  ->  ( p  C_  A 
<->  ( _|_ `  A
)  C_  ( _|_ `  p ) ) )
1918biimpa 470 . . . . . . . . . . . . . 14  |-  ( ( p  e. HAtoms  /\  p  C_  A )  ->  ( _|_ `  A )  C_  ( _|_ `  p ) )
20 sstr 3187 . . . . . . . . . . . . . 14  |-  ( ( q  C_  ( _|_ `  A )  /\  ( _|_ `  A )  C_  ( _|_ `  p ) )  ->  q  C_  ( _|_ `  p ) )
2119, 20sylan2 460 . . . . . . . . . . . . 13  |-  ( ( q  C_  ( _|_ `  A )  /\  (
p  e. HAtoms  /\  p  C_  A ) )  -> 
q  C_  ( _|_ `  p ) )
2221ancoms 439 . . . . . . . . . . . 12  |-  ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  q  C_  ( _|_ `  A
) )  ->  q  C_  ( _|_ `  p
) )
23 atne0 22925 . . . . . . . . . . . . . . 15  |-  ( p  e. HAtoms  ->  p  =/=  0H )
2423adantr 451 . . . . . . . . . . . . . 14  |-  ( ( p  e. HAtoms  /\  q  C_  ( _|_ `  p
) )  ->  p  =/=  0H )
25 sseq1 3199 . . . . . . . . . . . . . . . . . . . 20  |-  ( p  =  q  ->  (
p  C_  ( _|_ `  p )  <->  q  C_  ( _|_ `  p ) ) )
2625bicomd 192 . . . . . . . . . . . . . . . . . . 19  |-  ( p  =  q  ->  (
q  C_  ( _|_ `  p )  <->  p  C_  ( _|_ `  p ) ) )
27 chssoc 22075 . . . . . . . . . . . . . . . . . . . 20  |-  ( p  e.  CH  ->  (
p  C_  ( _|_ `  p )  <->  p  =  0H ) )
2816, 27syl 15 . . . . . . . . . . . . . . . . . . 19  |-  ( p  e. HAtoms  ->  ( p  C_  ( _|_ `  p )  <-> 
p  =  0H ) )
2926, 28sylan9bbr 681 . . . . . . . . . . . . . . . . . 18  |-  ( ( p  e. HAtoms  /\  p  =  q )  -> 
( q  C_  ( _|_ `  p )  <->  p  =  0H ) )
3029biimpa 470 . . . . . . . . . . . . . . . . 17  |-  ( ( ( p  e. HAtoms  /\  p  =  q )  /\  q  C_  ( _|_ `  p
) )  ->  p  =  0H )
3130an32s 779 . . . . . . . . . . . . . . . 16  |-  ( ( ( p  e. HAtoms  /\  q  C_  ( _|_ `  p
) )  /\  p  =  q )  ->  p  =  0H )
3231ex 423 . . . . . . . . . . . . . . 15  |-  ( ( p  e. HAtoms  /\  q  C_  ( _|_ `  p
) )  ->  (
p  =  q  ->  p  =  0H )
)
3332necon3d 2484 . . . . . . . . . . . . . 14  |-  ( ( p  e. HAtoms  /\  q  C_  ( _|_ `  p
) )  ->  (
p  =/=  0H  ->  p  =/=  q ) )
3424, 33mpd 14 . . . . . . . . . . . . 13  |-  ( ( p  e. HAtoms  /\  q  C_  ( _|_ `  p
) )  ->  p  =/=  q )
3534adantlr 695 . . . . . . . . . . . 12  |-  ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  q  C_  ( _|_ `  p
) )  ->  p  =/=  q )
3622, 35syldan 456 . . . . . . . . . . 11  |-  ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  q  C_  ( _|_ `  A
) )  ->  p  =/=  q )
3736adantrl 696 . . . . . . . . . 10  |-  ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  (
q  e. HAtoms  /\  q  C_  ( _|_ `  A
) ) )  ->  p  =/=  q )
38 superpos 22934 . . . . . . . . . 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 1182 . . . . . . . . 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 936 . . . . . . . . . . . 12  |-  ( ( r  =/=  p  /\  r  =/=  q  /\  r  C_  ( p  vH  q
) )  <->  ( (
r  =/=  p  /\  r  =/=  q )  /\  r  C_  ( p  vH  q ) ) )
41 neanior 2531 . . . . . . . . . . . . 13  |-  ( ( r  =/=  p  /\  r  =/=  q )  <->  -.  (
r  =  p  \/  r  =  q ) )
4241anbi1i 676 . . . . . . . . . . . 12  |-  ( ( ( r  =/=  p  /\  r  =/=  q
)  /\  r  C_  ( p  vH  q
) )  <->  ( -.  ( r  =  p  \/  r  =  q )  /\  r  C_  ( p  vH  q
) ) )
4340, 42bitri 240 . . . . . . . . . . 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 22973 . . . . . . . . . . . . . . . 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 629 . . . . . . . . . . . . . . 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 135 . . . . . . . . . . . . . 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 423 . . . . . . . . . . . . 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 72 . . . . . . . . . . . 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 420 . . . . . . . . . . 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 208 . . . . . . . . . 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 2667 . . . . . . . . 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 14 . . . . . . . 8  |-  ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  (
q  e. HAtoms  /\  q  C_  ( _|_ `  A
) ) )  ->  -.  0H  =  0H )
5453an4s 799 . . . . . . 7  |-  ( ( ( p  e. HAtoms  /\  q  e. HAtoms )  /\  ( p 
C_  A  /\  q  C_  ( _|_ `  A
) ) )  ->  -.  0H  =  0H )
5554ex 423 . . . . . 6  |-  ( ( p  e. HAtoms  /\  q  e. HAtoms )  ->  ( (
p  C_  A  /\  q  C_  ( _|_ `  A
) )  ->  -.  0H  =  0H )
)
5655rexlimivv 2672 . . . . 5  |-  ( E. p  e. HAtoms  E. q  e. HAtoms  ( p  C_  A  /\  q  C_  ( _|_ `  A ) )  ->  -.  0H  =  0H )
5713, 56syl 15 . . . 4  |-  ( ( A  =/=  0H  /\  ( _|_ `  A )  =/=  0H )  ->  -.  0H  =  0H )
586, 57sylbi 187 . . 3  |-  ( -.  ( A  =  0H  \/  ( _|_ `  A
)  =  0H )  ->  -.  0H  =  0H )
591, 58mt4 129 . 2  |-  ( A  =  0H  \/  ( _|_ `  A )  =  0H )
60 fveq2 5525 . . . 4  |-  ( ( _|_ `  A )  =  0H  ->  ( _|_ `  ( _|_ `  A
) )  =  ( _|_ `  0H ) )
617ococi 21984 . . . 4  |-  ( _|_ `  ( _|_ `  A
) )  =  A
62 choc0 21905 . . . 4  |-  ( _|_ `  0H )  =  ~H
6360, 61, 623eqtr3g 2338 . . 3  |-  ( ( _|_ `  A )  =  0H  ->  A  =  ~H )
6463orim2i 504 . 2  |-  ( ( A  =  0H  \/  ( _|_ `  A )  =  0H )  -> 
( A  =  0H  \/  A  =  ~H ) )
6559, 64ax-mp 8 1  |-  ( A  =  0H  \/  A  =  ~H )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544    C_ wss 3152   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   ~Hchil 21499   CHcch 21509   _|_cort 21510    vH chj 21513   0Hc0h 21515    C_H ccm 21516  HAtomscat 21545
This theorem is referenced by:  chirred  22975
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cc 8061  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817  ax-hilex 21579  ax-hfvadd 21580  ax-hvcom 21581  ax-hvass 21582  ax-hv0cl 21583  ax-hvaddid 21584  ax-hfvmul 21585  ax-hvmulid 21586  ax-hvmulass 21587  ax-hvdistr1 21588  ax-hvdistr2 21589  ax-hvmul0 21590  ax-hfi 21658  ax-his1 21661  ax-his2 21662  ax-his3 21663  ax-his4 21664  ax-hcompl 21781
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-omul 6484  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-acn 7575  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-rlim 11963  df-sum 12159  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-cn 16957  df-cnp 16958  df-lm 16959  df-haus 17043  df-tx 17257  df-hmeo 17446  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-tms 17887  df-cfil 18681  df-cau 18682  df-cmet 18683  df-grpo 20858  df-gid 20859  df-ginv 20860  df-gdiv 20861  df-ablo 20949  df-subgo 20969  df-vc 21102  df-nv 21148  df-va 21151  df-ba 21152  df-sm 21153  df-0v 21154  df-vs 21155  df-nmcv 21156  df-ims 21157  df-dip 21274  df-ssp 21298  df-ph 21391  df-cbn 21442  df-hnorm 21548  df-hba 21549  df-hvsub 21551  df-hlim 21552  df-hcau 21553  df-sh 21786  df-ch 21801  df-oc 21831  df-ch0 21832  df-shs 21887  df-span 21888  df-chj 21889  df-chsup 21890  df-pjh 21974  df-cm 22162  df-cv 22859  df-at 22918
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