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Theorem exatleN 30040
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 1039 . . . . 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 1068 . . . . . . . 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 30000 . . . . . . . 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 1090 . . . . . . . 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 29926 . . . . . . . 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 1089 . . . . . . . . 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 29926 . . . . . . . . 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 1091 . . . . . . . . 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 29926 . . . . . . . . 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 14467 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  R  e.  B )  ->  ( P  .\/  R
)  e.  B )
196, 13, 16, 18syl3anc 1184 . . . . . . 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 1069 . . . . . . 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 1134 . . . . . . . . 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 958 . . . . . . . . 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 1134 . . . . . . . 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 1094 . . . . . . . 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 30031 . . . . . . . 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 1092 . . . . . . . 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 959 . . . . . . . 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 14479 . . . . . . . . 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 1186 . . . . . . . 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 888 . . . . . . 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 14475 . . . . . 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 1155 . . . . 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 424 . . 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 2659 . 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 993 . . 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 4207 . . 3  |-  ( R  =  P  ->  ( R  .<_  X  <->  P  .<_  X ) )
3937, 38syl5ibrcom 214 . 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 184 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 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   class class class wbr 4204   ` cfv 5445  (class class class)co 6072   Basecbs 13457   lecple 13524   joincjn 14389   Latclat 14462   Atomscatm 29900   HLchlt 29987
This theorem is referenced by:  cdlema2N  30428
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-1st 6340  df-2nd 6341  df-undef 6534  df-riota 6540  df-poset 14391  df-plt 14403  df-lub 14419  df-join 14421  df-lat 14463  df-covers 29903  df-ats 29904  df-atl 29935  df-cvlat 29959  df-hlat 29988
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