HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  elat2 Unicode version

Theorem elat2 22922
Description: Expanded membership relation for the set of atoms, i.e. the predicate "is an atom (of the Hilbert lattice)." An atom is a nonzero element of a lattice such that anything less than it is zero, i.e. it is the smallest nonzero element of the lattice. (Contributed by NM, 9-Jun-2004.) (New usage is discouraged.)
Assertion
Ref Expression
elat2  |-  ( A  e. HAtoms 
<->  ( A  e.  CH  /\  ( A  =/=  0H  /\ 
A. x  e.  CH  ( x  C_  A  -> 
( x  =  A  \/  x  =  0H ) ) ) ) )
Distinct variable group:    x, A

Proof of Theorem elat2
StepHypRef Expression
1 ela 22921 . 2  |-  ( A  e. HAtoms 
<->  ( A  e.  CH  /\  0H  <oH  A ) )
2 h0elch 21836 . . . . 5  |-  0H  e.  CH
3 cvbr2 22865 . . . . 5  |-  ( ( 0H  e.  CH  /\  A  e.  CH )  ->  ( 0H  <oH  A  <->  ( 0H  C.  A  /\  A. x  e.  CH  ( ( 0H 
C.  x  /\  x  C_  A )  ->  x  =  A ) ) ) )
42, 3mpan 651 . . . 4  |-  ( A  e.  CH  ->  ( 0H  <oH  A  <->  ( 0H  C.  A  /\  A. x  e.  CH  ( ( 0H 
C.  x  /\  x  C_  A )  ->  x  =  A ) ) ) )
5 ch0pss 22026 . . . . 5  |-  ( A  e.  CH  ->  ( 0H  C.  A  <->  A  =/=  0H ) )
6 ch0pss 22026 . . . . . . . . . 10  |-  ( x  e.  CH  ->  ( 0H  C.  x  <->  x  =/=  0H ) )
76imbi1d 308 . . . . . . . . 9  |-  ( x  e.  CH  ->  (
( 0H  C.  x  ->  x  =  A )  <-> 
( x  =/=  0H  ->  x  =  A ) ) )
87imbi2d 307 . . . . . . . 8  |-  ( x  e.  CH  ->  (
( x  C_  A  ->  ( 0H  C.  x  ->  x  =  A ) )  <->  ( x  C_  A  ->  ( x  =/= 
0H  ->  x  =  A ) ) ) )
9 impexp 433 . . . . . . . . 9  |-  ( ( ( 0H  C.  x  /\  x  C_  A )  ->  x  =  A )  <->  ( 0H  C.  x  ->  ( x  C_  A  ->  x  =  A ) ) )
10 bi2.04 350 . . . . . . . . 9  |-  ( ( 0H  C.  x  -> 
( x  C_  A  ->  x  =  A ) )  <->  ( x  C_  A  ->  ( 0H  C.  x  ->  x  =  A ) ) )
119, 10bitri 240 . . . . . . . 8  |-  ( ( ( 0H  C.  x  /\  x  C_  A )  ->  x  =  A )  <->  ( x  C_  A  ->  ( 0H  C.  x  ->  x  =  A ) ) )
12 orcom 376 . . . . . . . . . 10  |-  ( ( x  =  A  \/  x  =  0H )  <->  ( x  =  0H  \/  x  =  A )
)
13 neor 2532 . . . . . . . . . 10  |-  ( ( x  =  0H  \/  x  =  A )  <->  ( x  =/=  0H  ->  x  =  A ) )
1412, 13bitri 240 . . . . . . . . 9  |-  ( ( x  =  A  \/  x  =  0H )  <->  ( x  =/=  0H  ->  x  =  A ) )
1514imbi2i 303 . . . . . . . 8  |-  ( ( x  C_  A  ->  ( x  =  A  \/  x  =  0H )
)  <->  ( x  C_  A  ->  ( x  =/= 
0H  ->  x  =  A ) ) )
168, 11, 153bitr4g 279 . . . . . . 7  |-  ( x  e.  CH  ->  (
( ( 0H  C.  x  /\  x  C_  A
)  ->  x  =  A )  <->  ( x  C_  A  ->  ( x  =  A  \/  x  =  0H ) ) ) )
1716ralbiia 2577 . . . . . 6  |-  ( A. x  e.  CH  ( ( 0H  C.  x  /\  x  C_  A )  ->  x  =  A )  <->  A. x  e.  CH  (
x  C_  A  ->  ( x  =  A  \/  x  =  0H )
) )
1817a1i 10 . . . . 5  |-  ( A  e.  CH  ->  ( A. x  e.  CH  (
( 0H  C.  x  /\  x  C_  A )  ->  x  =  A )  <->  A. x  e.  CH  ( x  C_  A  -> 
( x  =  A  \/  x  =  0H ) ) ) )
195, 18anbi12d 691 . . . 4  |-  ( A  e.  CH  ->  (
( 0H  C.  A  /\  A. x  e.  CH  ( ( 0H  C.  x  /\  x  C_  A
)  ->  x  =  A ) )  <->  ( A  =/=  0H  /\  A. x  e.  CH  ( x  C_  A  ->  ( x  =  A  \/  x  =  0H ) ) ) ) )
204, 19bitr2d 245 . . 3  |-  ( A  e.  CH  ->  (
( A  =/=  0H  /\ 
A. x  e.  CH  ( x  C_  A  -> 
( x  =  A  \/  x  =  0H ) ) )  <->  0H  <oH  A ) )
2120pm5.32i 618 . 2  |-  ( ( A  e.  CH  /\  ( A  =/=  0H  /\ 
A. x  e.  CH  ( x  C_  A  -> 
( x  =  A  \/  x  =  0H ) ) ) )  <-> 
( A  e.  CH  /\  0H  <oH  A ) )
221, 21bitr4i 243 1  |-  ( A  e. HAtoms 
<->  ( A  e.  CH  /\  ( A  =/=  0H  /\ 
A. x  e.  CH  ( x  C_  A  -> 
( x  =  A  \/  x  =  0H ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1625    e. wcel 1686    =/= wne 2448   A.wral 2545    C_ wss 3154    C. wpss 3155   class class class wbr 4025   CHcch 21511   0Hc0h 21517    <oH ccv 21546  HAtomscat 21547
This theorem is referenced by:  atne0  22927  atss  22928  h1da  22931  atom1d  22935
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816  ax-pre-sup 8817  ax-addf 8818  ax-mulf 8819  ax-hilex 21581  ax-hfvadd 21582  ax-hvcom 21583  ax-hvass 21584  ax-hv0cl 21585  ax-hvaddid 21586  ax-hfvmul 21587  ax-hvmulid 21588  ax-hvmulass 21589  ax-hvdistr1 21590  ax-hvdistr2 21591  ax-hvmul0 21592  ax-hfi 21660  ax-his1 21663  ax-his2 21664  ax-his3 21665  ax-his4 21666
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-1st 6124  df-2nd 6125  df-riota 6306  df-recs 6390  df-rdg 6425  df-er 6662  df-map 6776  df-pm 6777  df-en 6866  df-dom 6867  df-sdom 6868  df-sup 7196  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-div 9426  df-nn 9749  df-2 9806  df-3 9807  df-4 9808  df-n0 9968  df-z 10027  df-uz 10233  df-q 10319  df-rp 10357  df-xneg 10454  df-xadd 10455  df-xmul 10456  df-icc 10665  df-seq 11049  df-exp 11107  df-cj 11586  df-re 11587  df-im 11588  df-sqr 11722  df-abs 11723  df-topgen 13346  df-xmet 16375  df-met 16376  df-bl 16377  df-mopn 16378  df-top 16638  df-bases 16640  df-topon 16641  df-lm 16961  df-haus 17045  df-grpo 20860  df-gid 20861  df-ginv 20862  df-gdiv 20863  df-ablo 20951  df-vc 21104  df-nv 21150  df-va 21153  df-ba 21154  df-sm 21155  df-0v 21156  df-vs 21157  df-nmcv 21158  df-ims 21159  df-hnorm 21550  df-hvsub 21553  df-hlim 21554  df-sh 21788  df-ch 21803  df-ch0 21834  df-cv 22861  df-at 22920
  Copyright terms: Public domain W3C validator