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Theorem cdleme20zN 30563
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 29626 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
213ad2ant1 976 . . 3  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  ->  K  e.  Lat )
3 simp1 955 . . . 4  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  ->  K  e.  HL )
4 simp22 989 . . . 4  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  ->  S  e.  A )
5 simp21 988 . . . 4  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  ->  R  e.  A )
6 eqid 2285 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
7 cdleme20z.j . . . . 5  |-  .\/  =  ( join `  K )
8 cdleme20z.a . . . . 5  |-  A  =  ( Atoms `  K )
96, 7, 8hlatjcl 29629 . . . 4  |-  ( ( K  e.  HL  /\  S  e.  A  /\  R  e.  A )  ->  ( S  .\/  R
)  e.  ( Base `  K ) )
103, 4, 5, 9syl3anc 1182 . . 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 990 . . . 4  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  ->  T  e.  A )
126, 8atbase 29552 . . . 4  |-  ( T  e.  A  ->  T  e.  ( Base `  K
) )
1311, 12syl 15 . . 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 14183 . . 3  |-  ( ( K  e.  Lat  /\  ( S  .\/  R )  e.  ( Base `  K
)  /\  T  e.  ( Base `  K )
)  ->  ( ( S  .\/  R )  ./\  T )  =  ( T 
./\  ( S  .\/  R ) ) )
162, 10, 13, 15syl3anc 1182 . 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 984 . . . 4  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  ->  -.  R  .<_  ( S 
.\/  T ) )
18 hlcvl 29622 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  CvLat )
19183ad2ant1 976 . . . . 5  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  ->  K  e.  CvLat )
20 simp3l 983 . . . . . 6  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  ->  S  =/=  T )
2120necomd 2531 . . . . 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 29599 . . . . 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 1195 . . . 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 168 . . 3  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  ->  -.  T  .<_  ( S 
.\/  R ) )
26 hlatl 29623 . . . . 5  |-  ( K  e.  HL  ->  K  e.  AtLat )
27263ad2ant1 976 . . . 4  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  ->  K  e.  AtLat )
28 eqid 2285 . . . . 5  |-  ( 0.
`  K )  =  ( 0. `  K
)
296, 22, 14, 28, 8atnle 29580 . . . 4  |-  ( ( K  e.  AtLat  /\  T  e.  A  /\  ( S  .\/  R )  e.  ( Base `  K
) )  ->  ( -.  T  .<_  ( S 
.\/  R )  <->  ( T  ./\  ( S  .\/  R
) )  =  ( 0. `  K ) ) )
3027, 11, 10, 29syl3anc 1182 . . 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 201 . 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 2317 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 176    /\ wa 358    /\ w3a 934    = wceq 1625    e. wcel 1686    =/= wne 2448   class class class wbr 4025   ` cfv 5257  (class class class)co 5860   Basecbs 13150   lecple 13217   joincjn 14080   meetcmee 14081   0.cp0 14145   Latclat 14153   Atomscatm 29526   AtLatcal 29527   CvLatclc 29528   HLchlt 29613
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-1st 6124  df-2nd 6125  df-undef 6300  df-riota 6306  df-poset 14082  df-plt 14094  df-lub 14110  df-glb 14111  df-join 14112  df-meet 14113  df-p0 14147  df-lat 14154  df-covers 29529  df-ats 29530  df-atl 29561  df-cvlat 29585  df-hlat 29614
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