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Theorem cdleme20zN 30466
Description: Part of proof of Lemma E in [Crawley] p. 113. Utility lemma. (Contributed by NM, 17-Nov-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdleme20z.l  |-  .<_  =  ( le `  K )
cdleme20z.j  |-  .\/  =  ( join `  K )
cdleme20z.m  |-  ./\  =  ( meet `  K )
cdleme20z.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
cdleme20zN  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  -> 
( ( S  .\/  R )  ./\  T )  =  ( 0. `  K ) )

Proof of Theorem cdleme20zN
StepHypRef Expression
1 hllat 29529 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
213ad2ant1 978 . . 3  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  ->  K  e.  Lat )
3 simp1 957 . . . 4  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  ->  K  e.  HL )
4 simp22 991 . . . 4  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  ->  S  e.  A )
5 simp21 990 . . . 4  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  ->  R  e.  A )
6 eqid 2380 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
7 cdleme20z.j . . . . 5  |-  .\/  =  ( join `  K )
8 cdleme20z.a . . . . 5  |-  A  =  ( Atoms `  K )
96, 7, 8hlatjcl 29532 . . . 4  |-  ( ( K  e.  HL  /\  S  e.  A  /\  R  e.  A )  ->  ( S  .\/  R
)  e.  ( Base `  K ) )
103, 4, 5, 9syl3anc 1184 . . 3  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  -> 
( S  .\/  R
)  e.  ( Base `  K ) )
11 simp23 992 . . . 4  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  ->  T  e.  A )
126, 8atbase 29455 . . . 4  |-  ( T  e.  A  ->  T  e.  ( Base `  K
) )
1311, 12syl 16 . . 3  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  ->  T  e.  ( Base `  K ) )
14 cdleme20z.m . . . 4  |-  ./\  =  ( meet `  K )
156, 14latmcom 14424 . . 3  |-  ( ( K  e.  Lat  /\  ( S  .\/  R )  e.  ( Base `  K
)  /\  T  e.  ( Base `  K )
)  ->  ( ( S  .\/  R )  ./\  T )  =  ( T 
./\  ( S  .\/  R ) ) )
162, 10, 13, 15syl3anc 1184 . 2  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  -> 
( ( S  .\/  R )  ./\  T )  =  ( T  ./\  ( S  .\/  R ) ) )
17 simp3r 986 . . . 4  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  ->  -.  R  .<_  ( S 
.\/  T ) )
18 hlcvl 29525 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  CvLat )
19183ad2ant1 978 . . . . 5  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  ->  K  e.  CvLat )
20 simp3l 985 . . . . . 6  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  ->  S  =/=  T )
2120necomd 2626 . . . . 5  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  ->  T  =/=  S )
22 cdleme20z.l . . . . . 6  |-  .<_  =  ( le `  K )
2322, 7, 8cvlatexch1 29502 . . . . 5  |-  ( ( K  e.  CvLat  /\  ( T  e.  A  /\  R  e.  A  /\  S  e.  A )  /\  T  =/=  S
)  ->  ( T  .<_  ( S  .\/  R
)  ->  R  .<_  ( S  .\/  T ) ) )
2419, 11, 5, 4, 21, 23syl131anc 1197 . . . 4  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  -> 
( T  .<_  ( S 
.\/  R )  ->  R  .<_  ( S  .\/  T ) ) )
2517, 24mtod 170 . . 3  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  ->  -.  T  .<_  ( S 
.\/  R ) )
26 hlatl 29526 . . . . 5  |-  ( K  e.  HL  ->  K  e.  AtLat )
27263ad2ant1 978 . . . 4  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  ->  K  e.  AtLat )
28 eqid 2380 . . . . 5  |-  ( 0.
`  K )  =  ( 0. `  K
)
296, 22, 14, 28, 8atnle 29483 . . . 4  |-  ( ( K  e.  AtLat  /\  T  e.  A  /\  ( S  .\/  R )  e.  ( Base `  K
) )  ->  ( -.  T  .<_  ( S 
.\/  R )  <->  ( T  ./\  ( S  .\/  R
) )  =  ( 0. `  K ) ) )
3027, 11, 10, 29syl3anc 1184 . . 3  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  -> 
( -.  T  .<_  ( S  .\/  R )  <-> 
( T  ./\  ( S  .\/  R ) )  =  ( 0. `  K ) ) )
3125, 30mpbid 202 . 2  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  -> 
( T  ./\  ( S  .\/  R ) )  =  ( 0. `  K ) )
3216, 31eqtrd 2412 1  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  -> 
( ( S  .\/  R )  ./\  T )  =  ( 0. `  K ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2543   class class class wbr 4146   ` cfv 5387  (class class class)co 6013   Basecbs 13389   lecple 13456   joincjn 14321   meetcmee 14322   0.cp0 14386   Latclat 14394   Atomscatm 29429   AtLatcal 29430   CvLatclc 29431   HLchlt 29516
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-undef 6472  df-riota 6478  df-poset 14323  df-plt 14335  df-lub 14351  df-glb 14352  df-join 14353  df-meet 14354  df-p0 14388  df-lat 14395  df-covers 29432  df-ats 29433  df-atl 29464  df-cvlat 29488  df-hlat 29517
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