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Theorem chirredi 27439
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 2457 . . 3  |-  0H  =  0H
2 ioran 490 . . . . 5  |-  ( -.  ( A  =  0H  \/  ( _|_ `  A
)  =  0H )  <-> 
( -.  A  =  0H  /\  -.  ( _|_ `  A )  =  0H ) )
3 df-ne 2654 . . . . . 6  |-  ( A  =/=  0H  <->  -.  A  =  0H )
4 df-ne 2654 . . . . . 6  |-  ( ( _|_ `  A )  =/=  0H  <->  -.  ( _|_ `  A )  =  0H )
53, 4anbi12i 697 . . . . 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 27404 . . . . . . 7  |-  ( A  =/=  0H  ->  E. p  e. HAtoms  p  C_  A )
97choccli 26351 . . . . . . . 8  |-  ( _|_ `  A )  e.  CH
109hatomici 27404 . . . . . . 7  |-  ( ( _|_ `  A )  =/=  0H  ->  E. q  e. HAtoms  q  C_  ( _|_ `  A ) )
118, 10anim12i 566 . . . . . 6  |-  ( ( A  =/=  0H  /\  ( _|_ `  A )  =/=  0H )  -> 
( E. p  e. HAtoms  p  C_  A  /\  E. q  e. HAtoms  q  C_  ( _|_ `  A ) ) )
12 reeanv 3025 . . . . . 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 753 . . . . . . . . . 10  |-  ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  (
q  e. HAtoms  /\  q  C_  ( _|_ `  A
) ) )  ->  p  e. HAtoms )
15 simprl 756 . . . . . . . . . 10  |-  ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  (
q  e. HAtoms  /\  q  C_  ( _|_ `  A
) ) )  -> 
q  e. HAtoms )
16 atelch 27389 . . . . . . . . . . . . . . . 16  |-  ( p  e. HAtoms  ->  p  e.  CH )
17 chsscon3 26544 . . . . . . . . . . . . . . . 16  |-  ( ( p  e.  CH  /\  A  e.  CH )  ->  ( p  C_  A  <->  ( _|_ `  A ) 
C_  ( _|_ `  p
) ) )
1816, 7, 17sylancl 662 . . . . . . . . . . . . . . 15  |-  ( p  e. HAtoms  ->  ( p  C_  A 
<->  ( _|_ `  A
)  C_  ( _|_ `  p ) ) )
1918biimpa 484 . . . . . . . . . . . . . 14  |-  ( ( p  e. HAtoms  /\  p  C_  A )  ->  ( _|_ `  A )  C_  ( _|_ `  p ) )
20 sstr 3507 . . . . . . . . . . . . . 14  |-  ( ( q  C_  ( _|_ `  A )  /\  ( _|_ `  A )  C_  ( _|_ `  p ) )  ->  q  C_  ( _|_ `  p ) )
2119, 20sylan2 474 . . . . . . . . . . . . 13  |-  ( ( q  C_  ( _|_ `  A )  /\  (
p  e. HAtoms  /\  p  C_  A ) )  -> 
q  C_  ( _|_ `  p ) )
2221ancoms 453 . . . . . . . . . . . 12  |-  ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  q  C_  ( _|_ `  A
) )  ->  q  C_  ( _|_ `  p
) )
23 atne0 27390 . . . . . . . . . . . . . . 15  |-  ( p  e. HAtoms  ->  p  =/=  0H )
2423adantr 465 . . . . . . . . . . . . . 14  |-  ( ( p  e. HAtoms  /\  q  C_  ( _|_ `  p
) )  ->  p  =/=  0H )
25 sseq1 3520 . . . . . . . . . . . . . . . . . . . 20  |-  ( p  =  q  ->  (
p  C_  ( _|_ `  p )  <->  q  C_  ( _|_ `  p ) ) )
2625bicomd 201 . . . . . . . . . . . . . . . . . . 19  |-  ( p  =  q  ->  (
q  C_  ( _|_ `  p )  <->  p  C_  ( _|_ `  p ) ) )
27 chssoc 26540 . . . . . . . . . . . . . . . . . . . 20  |-  ( p  e.  CH  ->  (
p  C_  ( _|_ `  p )  <->  p  =  0H ) )
2816, 27syl 16 . . . . . . . . . . . . . . . . . . 19  |-  ( p  e. HAtoms  ->  ( p  C_  ( _|_ `  p )  <-> 
p  =  0H ) )
2926, 28sylan9bbr 700 . . . . . . . . . . . . . . . . . 18  |-  ( ( p  e. HAtoms  /\  p  =  q )  -> 
( q  C_  ( _|_ `  p )  <->  p  =  0H ) )
3029biimpa 484 . . . . . . . . . . . . . . . . 17  |-  ( ( ( p  e. HAtoms  /\  p  =  q )  /\  q  C_  ( _|_ `  p
) )  ->  p  =  0H )
3130an32s 804 . . . . . . . . . . . . . . . 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 2681 . . . . . . . . . . . . . 14  |-  ( ( p  e. HAtoms  /\  q  C_  ( _|_ `  p
) )  ->  (
p  =/=  0H  ->  p  =/=  q ) )
3424, 33mpd 15 . . . . . . . . . . . . 13  |-  ( ( p  e. HAtoms  /\  q  C_  ( _|_ `  p
) )  ->  p  =/=  q )
3534adantlr 714 . . . . . . . . . . . 12  |-  ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  q  C_  ( _|_ `  p
) )  ->  p  =/=  q )
3622, 35syldan 470 . . . . . . . . . . 11  |-  ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  q  C_  ( _|_ `  A
) )  ->  p  =/=  q )
3736adantrl 715 . . . . . . . . . 10  |-  ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  (
q  e. HAtoms  /\  q  C_  ( _|_ `  A
) ) )  ->  p  =/=  q )
38 superpos 27399 . . . . . . . . . 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 1228 . . . . . . . . 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 975 . . . . . . . . . . . 12  |-  ( ( r  =/=  p  /\  r  =/=  q  /\  r  C_  ( p  vH  q
) )  <->  ( (
r  =/=  p  /\  r  =/=  q )  /\  r  C_  ( p  vH  q ) ) )
41 neanior 2782 . . . . . . . . . . . . 13  |-  ( ( r  =/=  p  /\  r  =/=  q )  <->  -.  (
r  =  p  \/  r  =  q ) )
4241anbi1i 695 . . . . . . . . . . . 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 27438 . . . . . . . . . . . . . . . 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 648 . . . . . . . . . . . . . . 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 ) ) )
5049impd 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 2949 . . . . . . . . 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 826 . . . . . . 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 2954 . . . . 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 5872 . . . 4  |-  ( ( _|_ `  A )  =  0H  ->  ( _|_ `  ( _|_ `  A
) )  =  ( _|_ `  0H ) )
617ococi 26449 . . . 4  |-  ( _|_ `  ( _|_ `  A
) )  =  A
62 choc0 26370 . . . 4  |-  ( _|_ `  0H )  =  ~H
6360, 61, 623eqtr3g 2521 . . 3  |-  ( ( _|_ `  A )  =  0H  ->  A  =  ~H )
6463orim2i 518 . 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 973    = wceq 1395    e. wcel 1819    =/= wne 2652   E.wrex 2808    C_ wss 3471   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   ~Hchil 25962   CHcch 25972   _|_cort 25973    vH chj 25976   0Hc0h 25978    C_H ccm 25979  HAtomscat 26008
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cc 8832  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587  ax-addf 9588  ax-mulf 9589  ax-hilex 26042  ax-hfvadd 26043  ax-hvcom 26044  ax-hvass 26045  ax-hv0cl 26046  ax-hvaddid 26047  ax-hfvmul 26048  ax-hvmulid 26049  ax-hvmulass 26050  ax-hvdistr1 26051  ax-hvdistr2 26052  ax-hvmul0 26053  ax-hfi 26122  ax-his1 26125  ax-his2 26126  ax-his3 26127  ax-his4 26128  ax-hcompl 26245
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-omul 7153  df-er 7329  df-map 7440  df-pm 7441  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-fi 7889  df-sup 7919  df-oi 7953  df-card 8337  df-acn 8340  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-q 11208  df-rp 11246  df-xneg 11343  df-xadd 11344  df-xmul 11345  df-ioo 11558  df-ico 11560  df-icc 11561  df-fz 11698  df-fzo 11821  df-fl 11931  df-seq 12110  df-exp 12169  df-hash 12408  df-cj 12943  df-re 12944  df-im 12945  df-sqrt 13079  df-abs 13080  df-clim 13322  df-rlim 13323  df-sum 13520  df-struct 14645  df-ndx 14646  df-slot 14647  df-base 14648  df-sets 14649  df-ress 14650  df-plusg 14724  df-mulr 14725  df-starv 14726  df-sca 14727  df-vsca 14728  df-ip 14729  df-tset 14730  df-ple 14731  df-ds 14733  df-unif 14734  df-hom 14735  df-cco 14736  df-rest 14839  df-topn 14840  df-0g 14858  df-gsum 14859  df-topgen 14860  df-pt 14861  df-prds 14864  df-xrs 14918  df-qtop 14923  df-imas 14924  df-xps 14926  df-mre 15002  df-mrc 15003  df-acs 15005  df-mgm 15998  df-sgrp 16037  df-mnd 16047  df-submnd 16093  df-mulg 16186  df-cntz 16481  df-cmn 16926  df-psmet 18537  df-xmet 18538  df-met 18539  df-bl 18540  df-mopn 18541  df-fbas 18542  df-fg 18543  df-cnfld 18547  df-top 19525  df-bases 19527  df-topon 19528  df-topsp 19529  df-cld 19646  df-ntr 19647  df-cls 19648  df-nei 19725  df-cn 19854  df-cnp 19855  df-lm 19856  df-haus 19942  df-tx 20188  df-hmeo 20381  df-fil 20472  df-fm 20564  df-flim 20565  df-flf 20566  df-xms 20948  df-ms 20949  df-tms 20950  df-cfil 21819  df-cau 21820  df-cmet 21821  df-grpo 25319  df-gid 25320  df-ginv 25321  df-gdiv 25322  df-ablo 25410  df-subgo 25430  df-vc 25565  df-nv 25611  df-va 25614  df-ba 25615  df-sm 25616  df-0v 25617  df-vs 25618  df-nmcv 25619  df-ims 25620  df-dip 25737  df-ssp 25761  df-ph 25854  df-cbn 25905  df-hnorm 26011  df-hba 26012  df-hvsub 26014  df-hlim 26015  df-hcau 26016  df-sh 26250  df-ch 26265  df-oc 26296  df-ch0 26297  df-shs 26352  df-span 26353  df-chj 26354  df-chsup 26355  df-pjh 26439  df-cm 26627  df-cv 27324  df-at 27383
This theorem is referenced by:  chirred  27440
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