Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdleme20zN Structured version   Unicode version

Theorem cdleme20zN 33283
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 32345 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
213ad2ant1 1016 . . 3  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  ->  K  e.  Lat )
3 simp1 995 . . . 4  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  ->  K  e.  HL )
4 simp22 1029 . . . 4  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  ->  S  e.  A )
5 simp21 1028 . . . 4  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  ->  R  e.  A )
6 eqid 2400 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
7 cdleme20z.j . . . . 5  |-  .\/  =  ( join `  K )
8 cdleme20z.a . . . . 5  |-  A  =  ( Atoms `  K )
96, 7, 8hlatjcl 32348 . . . 4  |-  ( ( K  e.  HL  /\  S  e.  A  /\  R  e.  A )  ->  ( S  .\/  R
)  e.  ( Base `  K ) )
103, 4, 5, 9syl3anc 1228 . . 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 1030 . . . 4  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  ->  T  e.  A )
126, 8atbase 32271 . . . 4  |-  ( T  e.  A  ->  T  e.  ( Base `  K
) )
1311, 12syl 17 . . 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 15919 . . 3  |-  ( ( K  e.  Lat  /\  ( S  .\/  R )  e.  ( Base `  K
)  /\  T  e.  ( Base `  K )
)  ->  ( ( S  .\/  R )  ./\  T )  =  ( T 
./\  ( S  .\/  R ) ) )
162, 10, 13, 15syl3anc 1228 . 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 1024 . . . 4  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  ->  -.  R  .<_  ( S 
.\/  T ) )
18 hlcvl 32341 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  CvLat )
19183ad2ant1 1016 . . . . 5  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  ->  K  e.  CvLat )
20 simp3l 1023 . . . . . 6  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  ->  S  =/=  T )
2120necomd 2672 . . . . 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 32318 . . . . 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 1241 . . . 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 177 . . 3  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  ->  -.  T  .<_  ( S 
.\/  R ) )
26 hlatl 32342 . . . . 5  |-  ( K  e.  HL  ->  K  e.  AtLat )
27263ad2ant1 1016 . . . 4  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  ->  K  e.  AtLat )
28 eqid 2400 . . . . 5  |-  ( 0.
`  K )  =  ( 0. `  K
)
296, 22, 14, 28, 8atnle 32299 . . . 4  |-  ( ( K  e.  AtLat  /\  T  e.  A  /\  ( S  .\/  R )  e.  ( Base `  K
) )  ->  ( -.  T  .<_  ( S 
.\/  R )  <->  ( T  ./\  ( S  .\/  R
) )  =  ( 0. `  K ) ) )
3027, 11, 10, 29syl3anc 1228 . . 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 210 . 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 2441 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 972    = wceq 1403    e. wcel 1840    =/= wne 2596   class class class wbr 4392   ` cfv 5523  (class class class)co 6232   Basecbs 14731   lecple 14806   joincjn 15787   meetcmee 15788   0.cp0 15881   Latclat 15889   Atomscatm 32245   AtLatcal 32246   CvLatclc 32247   HLchlt 32332
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-reu 2758  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-riota 6194  df-ov 6235  df-oprab 6236  df-preset 15771  df-poset 15789  df-plt 15802  df-lub 15818  df-glb 15819  df-join 15820  df-meet 15821  df-p0 15883  df-lat 15890  df-covers 32248  df-ats 32249  df-atl 32280  df-cvlat 32304  df-hlat 32333
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator