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Theorem exatleN 28860
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 28820 . . . . . . . 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 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 28746 . . . . . . . 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 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 28746 . . . . . . . . 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 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 28746 . . . . . . . . 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 14150 . . . . . . . 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 28851 . . . . . . . 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 14162 . . . . . . . . 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 889 . . . . . . 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 14158 . . . . . 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 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 2508 . 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 4027 . . 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 936    = wceq 1624    e. wcel 1685    =/= wne 2447   class class class wbr 4024   ` cfv 5221  (class class class)co 5819   Basecbs 13142   lecple 13209   joincjn 14072   Latclat 14145   Atomscatm 28720   HLchlt 28807
This theorem is referenced by:  cdlema2N  29248
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-iota 6252  df-undef 6291  df-riota 6299  df-poset 14074  df-plt 14086  df-lub 14102  df-join 14104  df-lat 14146  df-covers 28723  df-ats 28724  df-atl 28755  df-cvlat 28779  df-hlat 28808
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