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

Theorem exatleN 34218
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 1078 . . . . 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 1107 . . . . . . . 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 34178 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  Lat )
64, 5syl 16 . . . . . . 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 1129 . . . . . . . 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 34104 . . . . . . . 8  |-  ( Q  e.  A  ->  Q  e.  B )
107, 9syl 16 . . . . . . 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 1128 . . . . . . . . 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 34104 . . . . . . . . 9  |-  ( P  e.  A  ->  P  e.  B )
1311, 12syl 16 . . . . . . . 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 1130 . . . . . . . . 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 34104 . . . . . . . . 9  |-  ( R  e.  A  ->  R  e.  B )
1614, 15syl 16 . . . . . . . 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 15538 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  R  e.  B )  ->  ( P  .\/  R
)  e.  B )
196, 13, 16, 18syl3anc 1228 . . . . . . 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 1108 . . . . . . 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 1176 . . . . . . . . 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 997 . . . . . . . . 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 1176 . . . . . . . 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 1133 . . . . . . . 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 34209 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  Q  e.  A  /\  P  e.  A
)  /\  R  =/=  P )  ->  ( R  .<_  ( P  .\/  Q
)  ->  Q  .<_  ( P  .\/  R ) ) )
2623, 24, 25sylc 60 . . . . . . 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 1131 . . . . . . . 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 998 . . . . . . . 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 15549 . . . . . . . . 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 1230 . . . . . . . 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 919 . . . . . . 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 15545 . . . . . 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 1198 . . . . 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 177 . . . 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 434 . . 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 2687 . 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 1032 . . 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 4450 . . 3  |-  ( R  =  P  ->  ( R  .<_  X  <->  P  .<_  X ) )
3937, 38syl5ibrcom 222 . 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 191 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4447   ` cfv 5588  (class class class)co 6284   Basecbs 14490   lecple 14562   joincjn 15431   Latclat 15532   Atomscatm 34078   HLchlt 34165
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-poset 15433  df-plt 15445  df-lub 15461  df-glb 15462  df-join 15463  df-meet 15464  df-p0 15526  df-lat 15533  df-covers 34081  df-ats 34082  df-atl 34113  df-cvlat 34137  df-hlat 34166
This theorem is referenced by:  cdlema2N  34606
  Copyright terms: Public domain W3C validator