Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  exatleN Unicode version

Theorem exatleN 28282
Description: A condition for an atom to be less than or equal to a lattice element. Part of proof of Lemma A in [Crawley] p. 112. (Contributed by NM, 28-Apr-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
atomle.b  |-  B  =  ( Base `  K
)
atomle.l  |-  .<_  =  ( le `  K )
atomle.j  |-  .\/  =  ( join `  K )
atomle.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
exatleN  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  .<_  X  /\  -.  Q  .<_  X  /\  R  .<_  ( P  .\/  Q
) ) )  -> 
( R  .<_  X  <->  R  =  P ) )

Proof of Theorem exatleN
StepHypRef Expression
1 simpl32 1042 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  .<_  X  /\  -.  Q  .<_  X  /\  R  .<_  ( P  .\/  Q ) ) )  /\  R  =/=  P )  ->  -.  Q  .<_  X )
2 atomle.b . . . . . . 7  |-  B  =  ( Base `  K
)
3 atomle.l . . . . . . 7  |-  .<_  =  ( le `  K )
4 simp11l 1071 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  .<_  X  /\  -.  Q  .<_  X  /\  R  .<_  ( P  .\/  Q ) ) )  /\  R  =/=  P  /\  R  .<_  X )  ->  K  e.  HL )
5 hllat 28242 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  Lat )
64, 5syl 17 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  .<_  X  /\  -.  Q  .<_  X  /\  R  .<_  ( P  .\/  Q ) ) )  /\  R  =/=  P  /\  R  .<_  X )  ->  K  e.  Lat )
7 simp122 1093 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  .<_  X  /\  -.  Q  .<_  X  /\  R  .<_  ( P  .\/  Q ) ) )  /\  R  =/=  P  /\  R  .<_  X )  ->  Q  e.  A )
8 atomle.a . . . . . . . . 9  |-  A  =  ( Atoms `  K )
92, 8atbase 28168 . . . . . . . 8  |-  ( Q  e.  A  ->  Q  e.  B )
107, 9syl 17 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  .<_  X  /\  -.  Q  .<_  X  /\  R  .<_  ( P  .\/  Q ) ) )  /\  R  =/=  P  /\  R  .<_  X )  ->  Q  e.  B )
11 simp121 1092 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  .<_  X  /\  -.  Q  .<_  X  /\  R  .<_  ( P  .\/  Q ) ) )  /\  R  =/=  P  /\  R  .<_  X )  ->  P  e.  A )
122, 8atbase 28168 . . . . . . . . 9  |-  ( P  e.  A  ->  P  e.  B )
1311, 12syl 17 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  .<_  X  /\  -.  Q  .<_  X  /\  R  .<_  ( P  .\/  Q ) ) )  /\  R  =/=  P  /\  R  .<_  X )  ->  P  e.  B )
14 simp123 1094 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  .<_  X  /\  -.  Q  .<_  X  /\  R  .<_  ( P  .\/  Q ) ) )  /\  R  =/=  P  /\  R  .<_  X )  ->  R  e.  A )
152, 8atbase 28168 . . . . . . . . 9  |-  ( R  e.  A  ->  R  e.  B )
1614, 15syl 17 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  .<_  X  /\  -.  Q  .<_  X  /\  R  .<_  ( P  .\/  Q ) ) )  /\  R  =/=  P  /\  R  .<_  X )  ->  R  e.  B )
17 atomle.j . . . . . . . . 9  |-  .\/  =  ( join `  K )
182, 17latjcl 14000 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  R  e.  B )  ->  ( P  .\/  R
)  e.  B )
196, 13, 16, 18syl3anc 1187 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  .<_  X  /\  -.  Q  .<_  X  /\  R  .<_  ( P  .\/  Q ) ) )  /\  R  =/=  P  /\  R  .<_  X )  ->  ( P  .\/  R )  e.  B )
20 simp11r 1072 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  .<_  X  /\  -.  Q  .<_  X  /\  R  .<_  ( P  .\/  Q ) ) )  /\  R  =/=  P  /\  R  .<_  X )  ->  X  e.  B )
2114, 7, 113jca 1137 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  .<_  X  /\  -.  Q  .<_  X  /\  R  .<_  ( P  .\/  Q ) ) )  /\  R  =/=  P  /\  R  .<_  X )  ->  ( R  e.  A  /\  Q  e.  A  /\  P  e.  A )
)
22 simp2 961 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  .<_  X  /\  -.  Q  .<_  X  /\  R  .<_  ( P  .\/  Q ) ) )  /\  R  =/=  P  /\  R  .<_  X )  ->  R  =/=  P )
234, 21, 223jca 1137 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  .<_  X  /\  -.  Q  .<_  X  /\  R  .<_  ( P  .\/  Q ) ) )  /\  R  =/=  P  /\  R  .<_  X )  ->  ( K  e.  HL  /\  ( R  e.  A  /\  Q  e.  A  /\  P  e.  A )  /\  R  =/=  P
) )
24 simp133 1097 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  .<_  X  /\  -.  Q  .<_  X  /\  R  .<_  ( P  .\/  Q ) ) )  /\  R  =/=  P  /\  R  .<_  X )  ->  R  .<_  ( P  .\/  Q
) )
253, 17, 8hlatexch1 28273 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  Q  e.  A  /\  P  e.  A
)  /\  R  =/=  P )  ->  ( R  .<_  ( P  .\/  Q
)  ->  Q  .<_  ( P  .\/  R ) ) )
2623, 24, 25sylc 58 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  .<_  X  /\  -.  Q  .<_  X  /\  R  .<_  ( P  .\/  Q ) ) )  /\  R  =/=  P  /\  R  .<_  X )  ->  Q  .<_  ( P  .\/  R
) )
27 simp131 1095 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  .<_  X  /\  -.  Q  .<_  X  /\  R  .<_  ( P  .\/  Q ) ) )  /\  R  =/=  P  /\  R  .<_  X )  ->  P  .<_  X )
28 simp3 962 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  .<_  X  /\  -.  Q  .<_  X  /\  R  .<_  ( P  .\/  Q ) ) )  /\  R  =/=  P  /\  R  .<_  X )  ->  R  .<_  X )
292, 3, 17latjle12 14012 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  ( P  e.  B  /\  R  e.  B  /\  X  e.  B
) )  ->  (
( P  .<_  X  /\  R  .<_  X )  <->  ( P  .\/  R )  .<_  X ) )
306, 13, 16, 20, 29syl13anc 1189 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  .<_  X  /\  -.  Q  .<_  X  /\  R  .<_  ( P  .\/  Q ) ) )  /\  R  =/=  P  /\  R  .<_  X )  ->  (
( P  .<_  X  /\  R  .<_  X )  <->  ( P  .\/  R )  .<_  X ) )
3127, 28, 30mpbi2and 892 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  .<_  X  /\  -.  Q  .<_  X  /\  R  .<_  ( P  .\/  Q ) ) )  /\  R  =/=  P  /\  R  .<_  X )  ->  ( P  .\/  R )  .<_  X )
322, 3, 6, 10, 19, 20, 26, 31lattrd 14008 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  .<_  X  /\  -.  Q  .<_  X  /\  R  .<_  ( P  .\/  Q ) ) )  /\  R  =/=  P  /\  R  .<_  X )  ->  Q  .<_  X )
33323expia 1158 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  .<_  X  /\  -.  Q  .<_  X  /\  R  .<_  ( P  .\/  Q ) ) )  /\  R  =/=  P )  -> 
( R  .<_  X  ->  Q  .<_  X ) )
341, 33mtod 170 . . . 4  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  .<_  X  /\  -.  Q  .<_  X  /\  R  .<_  ( P  .\/  Q ) ) )  /\  R  =/=  P )  ->  -.  R  .<_  X )
3534ex 425 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  .<_  X  /\  -.  Q  .<_  X  /\  R  .<_  ( P  .\/  Q
) ) )  -> 
( R  =/=  P  ->  -.  R  .<_  X ) )
3635necon4ad 2473 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  .<_  X  /\  -.  Q  .<_  X  /\  R  .<_  ( P  .\/  Q
) ) )  -> 
( R  .<_  X  ->  R  =  P )
)
37 simp31 996 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  .<_  X  /\  -.  Q  .<_  X  /\  R  .<_  ( P  .\/  Q
) ) )  ->  P  .<_  X )
38 breq1 3923 . . 3  |-  ( R  =  P  ->  ( R  .<_  X  <->  P  .<_  X ) )
3937, 38syl5ibrcom 215 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  .<_  X  /\  -.  Q  .<_  X  /\  R  .<_  ( P  .\/  Q
) ) )  -> 
( R  =  P  ->  R  .<_  X ) )
4036, 39impbid 185 1  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  .<_  X  /\  -.  Q  .<_  X  /\  R  .<_  ( P  .\/  Q
) ) )  -> 
( R  .<_  X  <->  R  =  P ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2412   class class class wbr 3920   ` cfv 4592  (class class class)co 5710   Basecbs 13022   lecple 13089   joincjn 13922   Latclat 13995   Atomscatm 28142   HLchlt 28229
This theorem is referenced by:  cdlema2N  28670
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 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-undef 6182  df-riota 6190  df-poset 13924  df-plt 13936  df-lub 13952  df-join 13954  df-lat 13996  df-covers 28145  df-ats 28146  df-atl 28177  df-cvlat 28201  df-hlat 28230
  Copyright terms: Public domain W3C validator