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Theorem chirredi 22899
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
StepHypRef Expression
1 eqid 2256 . . 3  |-  0H  =  0H
2 ioran 478 . . . . 5  |-  ( -.  ( A  =  0H  \/  ( _|_ `  A
)  =  0H )  <-> 
( -.  A  =  0H  /\  -.  ( _|_ `  A )  =  0H ) )
3 df-ne 2421 . . . . . 6  |-  ( A  =/=  0H  <->  -.  A  =  0H )
4 df-ne 2421 . . . . . 6  |-  ( ( _|_ `  A )  =/=  0H  <->  -.  ( _|_ `  A )  =  0H )
53, 4anbi12i 681 . . . . 5  |-  ( ( A  =/=  0H  /\  ( _|_ `  A )  =/=  0H )  <->  ( -.  A  =  0H  /\  -.  ( _|_ `  A )  =  0H ) )
62, 5bitr4i 245 . . . 4  |-  ( -.  ( A  =  0H  \/  ( _|_ `  A
)  =  0H )  <-> 
( A  =/=  0H  /\  ( _|_ `  A
)  =/=  0H ) )
7 chirred.1 . . . . . . . 8  |-  A  e. 
CH
87hatomici 22864 . . . . . . 7  |-  ( A  =/=  0H  ->  E. p  e. HAtoms  p  C_  A )
97choccli 21811 . . . . . . . 8  |-  ( _|_ `  A )  e.  CH
109hatomici 22864 . . . . . . 7  |-  ( ( _|_ `  A )  =/=  0H  ->  E. q  e. HAtoms  q  C_  ( _|_ `  A ) )
118, 10anim12i 551 . . . . . 6  |-  ( ( A  =/=  0H  /\  ( _|_ `  A )  =/=  0H )  -> 
( E. p  e. HAtoms  p  C_  A  /\  E. q  e. HAtoms  q  C_  ( _|_ `  A ) ) )
12 reeanv 2678 . . . . . 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 205 . . . . 5  |-  ( ( A  =/=  0H  /\  ( _|_ `  A )  =/=  0H )  ->  E. p  e. HAtoms  E. q  e. HAtoms  ( p  C_  A  /\  q  C_  ( _|_ `  A ) ) )
14 simpll 733 . . . . . . . . . 10  |-  ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  (
q  e. HAtoms  /\  q  C_  ( _|_ `  A
) ) )  ->  p  e. HAtoms )
15 simprl 735 . . . . . . . . . 10  |-  ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  (
q  e. HAtoms  /\  q  C_  ( _|_ `  A
) ) )  -> 
q  e. HAtoms )
16 atelch 22849 . . . . . . . . . . . . . . . 16  |-  ( p  e. HAtoms  ->  p  e.  CH )
17 chsscon3 22004 . . . . . . . . . . . . . . . 16  |-  ( ( p  e.  CH  /\  A  e.  CH )  ->  ( p  C_  A  <->  ( _|_ `  A ) 
C_  ( _|_ `  p
) ) )
1816, 7, 17sylancl 646 . . . . . . . . . . . . . . 15  |-  ( p  e. HAtoms  ->  ( p  C_  A 
<->  ( _|_ `  A
)  C_  ( _|_ `  p ) ) )
1918biimpa 472 . . . . . . . . . . . . . 14  |-  ( ( p  e. HAtoms  /\  p  C_  A )  ->  ( _|_ `  A )  C_  ( _|_ `  p ) )
20 sstr 3129 . . . . . . . . . . . . . 14  |-  ( ( q  C_  ( _|_ `  A )  /\  ( _|_ `  A )  C_  ( _|_ `  p ) )  ->  q  C_  ( _|_ `  p ) )
2119, 20sylan2 462 . . . . . . . . . . . . 13  |-  ( ( q  C_  ( _|_ `  A )  /\  (
p  e. HAtoms  /\  p  C_  A ) )  -> 
q  C_  ( _|_ `  p ) )
2221ancoms 441 . . . . . . . . . . . 12  |-  ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  q  C_  ( _|_ `  A
) )  ->  q  C_  ( _|_ `  p
) )
23 atne0 22850 . . . . . . . . . . . . . . 15  |-  ( p  e. HAtoms  ->  p  =/=  0H )
2423adantr 453 . . . . . . . . . . . . . 14  |-  ( ( p  e. HAtoms  /\  q  C_  ( _|_ `  p
) )  ->  p  =/=  0H )
25 sseq1 3141 . . . . . . . . . . . . . . . . . . . 20  |-  ( p  =  q  ->  (
p  C_  ( _|_ `  p )  <->  q  C_  ( _|_ `  p ) ) )
2625bicomd 194 . . . . . . . . . . . . . . . . . . 19  |-  ( p  =  q  ->  (
q  C_  ( _|_ `  p )  <->  p  C_  ( _|_ `  p ) ) )
27 chssoc 22000 . . . . . . . . . . . . . . . . . . . 20  |-  ( p  e.  CH  ->  (
p  C_  ( _|_ `  p )  <->  p  =  0H ) )
2816, 27syl 17 . . . . . . . . . . . . . . . . . . 19  |-  ( p  e. HAtoms  ->  ( p  C_  ( _|_ `  p )  <-> 
p  =  0H ) )
2926, 28sylan9bbr 684 . . . . . . . . . . . . . . . . . 18  |-  ( ( p  e. HAtoms  /\  p  =  q )  -> 
( q  C_  ( _|_ `  p )  <->  p  =  0H ) )
3029biimpa 472 . . . . . . . . . . . . . . . . 17  |-  ( ( ( p  e. HAtoms  /\  p  =  q )  /\  q  C_  ( _|_ `  p
) )  ->  p  =  0H )
3130an32s 782 . . . . . . . . . . . . . . . 16  |-  ( ( ( p  e. HAtoms  /\  q  C_  ( _|_ `  p
) )  /\  p  =  q )  ->  p  =  0H )
3231ex 425 . . . . . . . . . . . . . . 15  |-  ( ( p  e. HAtoms  /\  q  C_  ( _|_ `  p
) )  ->  (
p  =  q  ->  p  =  0H )
)
3332necon3d 2457 . . . . . . . . . . . . . 14  |-  ( ( p  e. HAtoms  /\  q  C_  ( _|_ `  p
) )  ->  (
p  =/=  0H  ->  p  =/=  q ) )
3424, 33mpd 16 . . . . . . . . . . . . 13  |-  ( ( p  e. HAtoms  /\  q  C_  ( _|_ `  p
) )  ->  p  =/=  q )
3534adantlr 698 . . . . . . . . . . . 12  |-  ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  q  C_  ( _|_ `  p
) )  ->  p  =/=  q )
3622, 35syldan 458 . . . . . . . . . . 11  |-  ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  q  C_  ( _|_ `  A
) )  ->  p  =/=  q )
3736adantrl 699 . . . . . . . . . 10  |-  ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  (
q  e. HAtoms  /\  q  C_  ( _|_ `  A
) ) )  ->  p  =/=  q )
38 superpos 22859 . . . . . . . . . 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 1187 . . . . . . . . 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 941 . . . . . . . . . . . 12  |-  ( ( r  =/=  p  /\  r  =/=  q  /\  r  C_  ( p  vH  q
) )  <->  ( (
r  =/=  p  /\  r  =/=  q )  /\  r  C_  ( p  vH  q ) ) )
41 neanior 2504 . . . . . . . . . . . . 13  |-  ( ( r  =/=  p  /\  r  =/=  q )  <->  -.  (
r  =  p  \/  r  =  q ) )
4241anbi1i 679 . . . . . . . . . . . 12  |-  ( ( ( r  =/=  p  /\  r  =/=  q
)  /\  r  C_  ( p  vH  q
) )  <->  ( -.  ( r  =  p  \/  r  =  q )  /\  r  C_  ( p  vH  q
) ) )
4340, 42bitri 242 . . . . . . . . . . 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 22898 . . . . . . . . . . . . . . . 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 632 . . . . . . . . . . . . . . 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 425 . . . . . . . . . . . . 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 74 . . . . . . . . . . . 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 422 . . . . . . . . . . 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 210 . . . . . . . . . 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 2638 . . . . . . . . 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 16 . . . . . . . 8  |-  ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  (
q  e. HAtoms  /\  q  C_  ( _|_ `  A
) ) )  ->  -.  0H  =  0H )
5453an4s 802 . . . . . . 7  |-  ( ( ( p  e. HAtoms  /\  q  e. HAtoms )  /\  ( p 
C_  A  /\  q  C_  ( _|_ `  A
) ) )  ->  -.  0H  =  0H )
5554ex 425 . . . . . 6  |-  ( ( p  e. HAtoms  /\  q  e. HAtoms )  ->  ( (
p  C_  A  /\  q  C_  ( _|_ `  A
) )  ->  -.  0H  =  0H )
)
5655rexlimivv 2643 . . . . 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 189 . . 3  |-  ( -.  ( A  =  0H  \/  ( _|_ `  A
)  =  0H )  ->  -.  0H  =  0H )
591, 58mt4 131 . 2  |-  ( A  =  0H  \/  ( _|_ `  A )  =  0H )
60 fveq2 5423 . . . 4  |-  ( ( _|_ `  A )  =  0H  ->  ( _|_ `  ( _|_ `  A
) )  =  ( _|_ `  0H ) )
617ococi 21909 . . . 4  |-  ( _|_ `  ( _|_ `  A
) )  =  A
62 choc0 21830 . . . 4  |-  ( _|_ `  0H )  =  ~H
6360, 61, 623eqtr3g 2311 . . 3  |-  ( ( _|_ `  A )  =  0H  ->  A  =  ~H )
6463orim2i 506 . 2  |-  ( ( A  =  0H  \/  ( _|_ `  A )  =  0H )  -> 
( A  =  0H  \/  A  =  ~H ) )
6559, 64ax-mp 10 1  |-  ( A  =  0H  \/  A  =  ~H )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    \/ wo 359    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2419   E.wrex 2517    C_ wss 3094   class class class wbr 3963   ` cfv 4638  (class class class)co 5757   ~Hchil 21424   CHcch 21434   _|_cort 21435    vH chj 21438   0Hc0h 21440    C_H ccm 21441  HAtomscat 21470
This theorem is referenced by:  chirred  22900
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-inf2 7275  ax-cc 7994  ax-cnex 8726  ax-resscn 8727  ax-1cn 8728  ax-icn 8729  ax-addcl 8730  ax-addrcl 8731  ax-mulcl 8732  ax-mulrcl 8733  ax-mulcom 8734  ax-addass 8735  ax-mulass 8736  ax-distr 8737  ax-i2m1 8738  ax-1ne0 8739  ax-1rid 8740  ax-rnegex 8741  ax-rrecex 8742  ax-cnre 8743  ax-pre-lttri 8744  ax-pre-lttrn 8745  ax-pre-ltadd 8746  ax-pre-mulgt0 8747  ax-pre-sup 8748  ax-addf 8749  ax-mulf 8750  ax-hilex 21504  ax-hfvadd 21505  ax-hvcom 21506  ax-hvass 21507  ax-hv0cl 21508  ax-hvaddid 21509  ax-hfvmul 21510  ax-hvmulid 21511  ax-hvmulass 21512  ax-hvdistr1 21513  ax-hvdistr2 21514  ax-hvmul0 21515  ax-hfi 21583  ax-his1 21586  ax-his2 21587  ax-his3 21588  ax-his4 21589  ax-hcompl 21706
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-int 3804  df-iun 3848  df-iin 3849  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-se 4290  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-isom 4655  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-of 5977  df-1st 6021  df-2nd 6022  df-iota 6190  df-riota 6237  df-recs 6321  df-rdg 6356  df-1o 6412  df-2o 6413  df-oadd 6416  df-omul 6417  df-er 6593  df-map 6707  df-pm 6708  df-ixp 6751  df-en 6797  df-dom 6798  df-sdom 6799  df-fin 6800  df-fi 7098  df-sup 7127  df-oi 7158  df-card 7505  df-acn 7508  df-cda 7727  df-pnf 8802  df-mnf 8803  df-xr 8804  df-ltxr 8805  df-le 8806  df-sub 8972  df-neg 8973  df-div 9357  df-n 9680  df-2 9737  df-3 9738  df-4 9739  df-5 9740  df-6 9741  df-7 9742  df-8 9743  df-9 9744  df-10 9745  df-n0 9898  df-z 9957  df-dec 10057  df-uz 10163  df-q 10249  df-rp 10287  df-xneg 10384  df-xadd 10385  df-xmul 10386  df-ioo 10591  df-ico 10593  df-icc 10594  df-fz 10714  df-fzo 10802  df-fl 10856  df-seq 10978  df-exp 11036  df-hash 11269  df-cj 11514  df-re 11515  df-im 11516  df-sqr 11650  df-abs 11651  df-clim 11892  df-rlim 11893  df-sum 12089  df-struct 13077  df-ndx 13078  df-slot 13079  df-base 13080  df-sets 13081  df-ress 13082  df-plusg 13148  df-mulr 13149  df-starv 13150  df-sca 13151  df-vsca 13152  df-tset 13154  df-ple 13155  df-ds 13157  df-hom 13159  df-cco 13160  df-rest 13254  df-topn 13255  df-topgen 13271  df-pt 13272  df-prds 13275  df-xrs 13330  df-0g 13331  df-gsum 13332  df-qtop 13337  df-imas 13338  df-xps 13340  df-mre 13415  df-mrc 13416  df-acs 13418  df-mnd 14294  df-submnd 14343  df-mulg 14419  df-cntz 14720  df-cmn 15018  df-xmet 16300  df-met 16301  df-bl 16302  df-mopn 16303  df-cnfld 16305  df-top 16563  df-bases 16565  df-topon 16566  df-topsp 16567  df-cld 16683  df-ntr 16684  df-cls 16685  df-nei 16762  df-cn 16884  df-cnp 16885  df-lm 16886  df-haus 16970  df-tx 17184  df-hmeo 17373  df-fbas 17447  df-fg 17448  df-fil 17468  df-fm 17560  df-flim 17561  df-flf 17562  df-xms 17812  df-ms 17813  df-tms 17814  df-cfil 18608  df-cau 18609  df-cmet 18610  df-grpo 20783  df-gid 20784  df-ginv 20785  df-gdiv 20786  df-ablo 20874  df-subgo 20894  df-vc 21027  df-nv 21073  df-va 21076  df-ba 21077  df-sm 21078  df-0v 21079  df-vs 21080  df-nmcv 21081  df-ims 21082  df-dip 21199  df-ssp 21223  df-ph 21316  df-cbn 21367  df-hnorm 21473  df-hba 21474  df-hvsub 21476  df-hlim 21477  df-hcau 21478  df-sh 21711  df-ch 21726  df-oc 21756  df-ch0 21757  df-shs 21812  df-span 21813  df-chj 21814  df-chsup 21815  df-pjh 21899  df-cm 22105  df-cv 22784  df-at 22843
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