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Theorem cdleme20y 33946
Description: Part of proof of Lemma E in [Crawley] p. 113. Utility lemma. (Contributed by NM, 17-Nov-2012.)
Hypotheses
Ref Expression
cdleme20z.l  |-  .<_  =  ( le `  K )
cdleme20z.j  |-  .\/  =  ( join `  K )
cdleme20z.m  |-  ./\  =  ( meet `  K )
cdleme20z.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
cdleme20y  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  -> 
( ( S  .\/  R )  ./\  ( T  .\/  R ) )  =  R )

Proof of Theorem cdleme20y
StepHypRef Expression
1 simp3r 1017 . . . . . 6  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  ->  -.  R  .<_  ( S 
.\/  T ) )
2 simp1 988 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  ->  K  e.  HL )
3 simp22 1022 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  ->  S  e.  A )
4 simp23 1023 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  ->  T  e.  A )
5 cdleme20z.j . . . . . . . . 9  |-  .\/  =  ( join `  K )
6 cdleme20z.a . . . . . . . . 9  |-  A  =  ( Atoms `  K )
75, 6hlatjcom 33012 . . . . . . . 8  |-  ( ( K  e.  HL  /\  S  e.  A  /\  T  e.  A )  ->  ( S  .\/  T
)  =  ( T 
.\/  S ) )
82, 3, 4, 7syl3anc 1218 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  -> 
( S  .\/  T
)  =  ( T 
.\/  S ) )
98breq2d 4304 . . . . . 6  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  -> 
( R  .<_  ( S 
.\/  T )  <->  R  .<_  ( T  .\/  S ) ) )
101, 9mtbid 300 . . . . 5  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  ->  -.  R  .<_  ( T 
.\/  S ) )
11 hlcvl 33004 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  CvLat )
12113ad2ant1 1009 . . . . . 6  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  ->  K  e.  CvLat )
13 simp21 1021 . . . . . 6  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  ->  R  e.  A )
14 simp3l 1016 . . . . . 6  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  ->  S  =/=  T )
15 cdleme20z.l . . . . . . 7  |-  .<_  =  ( le `  K )
1615, 5, 6cvlatexch1 32981 . . . . . 6  |-  ( ( K  e.  CvLat  /\  ( S  e.  A  /\  R  e.  A  /\  T  e.  A )  /\  S  =/=  T
)  ->  ( S  .<_  ( T  .\/  R
)  ->  R  .<_  ( T  .\/  S ) ) )
1712, 3, 13, 4, 14, 16syl131anc 1231 . . . . 5  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  -> 
( S  .<_  ( T 
.\/  R )  ->  R  .<_  ( T  .\/  S ) ) )
1810, 17mtod 177 . . . 4  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  ->  -.  S  .<_  ( T 
.\/  R ) )
19 hlatl 33005 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  AtLat )
20193ad2ant1 1009 . . . . 5  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  ->  K  e.  AtLat )
21 eqid 2443 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
2221, 5, 6hlatjcl 33011 . . . . . 6  |-  ( ( K  e.  HL  /\  T  e.  A  /\  R  e.  A )  ->  ( T  .\/  R
)  e.  ( Base `  K ) )
232, 4, 13, 22syl3anc 1218 . . . . 5  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  -> 
( T  .\/  R
)  e.  ( Base `  K ) )
24 cdleme20z.m . . . . . 6  |-  ./\  =  ( meet `  K )
25 eqid 2443 . . . . . 6  |-  ( 0.
`  K )  =  ( 0. `  K
)
2621, 15, 24, 25, 6atnle 32962 . . . . 5  |-  ( ( K  e.  AtLat  /\  S  e.  A  /\  ( T  .\/  R )  e.  ( Base `  K
) )  ->  ( -.  S  .<_  ( T 
.\/  R )  <->  ( S  ./\  ( T  .\/  R
) )  =  ( 0. `  K ) ) )
2720, 3, 23, 26syl3anc 1218 . . . 4  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  -> 
( -.  S  .<_  ( T  .\/  R )  <-> 
( S  ./\  ( T  .\/  R ) )  =  ( 0. `  K ) ) )
2818, 27mpbid 210 . . 3  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  -> 
( S  ./\  ( T  .\/  R ) )  =  ( 0. `  K ) )
2928oveq1d 6106 . 2  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  -> 
( ( S  ./\  ( T  .\/  R ) )  .\/  R )  =  ( ( 0.
`  K )  .\/  R ) )
3021, 6atbase 32934 . . . 4  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
3113, 30syl 16 . . 3  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  ->  R  e.  ( Base `  K ) )
3215, 5, 6hlatlej2 33020 . . . 4  |-  ( ( K  e.  HL  /\  T  e.  A  /\  R  e.  A )  ->  R  .<_  ( T  .\/  R ) )
332, 4, 13, 32syl3anc 1218 . . 3  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  ->  R  .<_  ( T  .\/  R ) )
3421, 15, 5, 24, 6atmod4i2 33511 . . 3  |-  ( ( K  e.  HL  /\  ( S  e.  A  /\  R  e.  ( Base `  K )  /\  ( T  .\/  R )  e.  ( Base `  K
) )  /\  R  .<_  ( T  .\/  R
) )  ->  (
( S  ./\  ( T  .\/  R ) ) 
.\/  R )  =  ( ( S  .\/  R )  ./\  ( T  .\/  R ) ) )
352, 3, 31, 23, 33, 34syl131anc 1231 . 2  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  -> 
( ( S  ./\  ( T  .\/  R ) )  .\/  R )  =  ( ( S 
.\/  R )  ./\  ( T  .\/  R ) ) )
36 hlol 33006 . . . 4  |-  ( K  e.  HL  ->  K  e.  OL )
37363ad2ant1 1009 . . 3  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  ->  K  e.  OL )
3821, 5, 25olj02 32871 . . 3  |-  ( ( K  e.  OL  /\  R  e.  ( Base `  K ) )  -> 
( ( 0. `  K )  .\/  R
)  =  R )
3937, 31, 38syl2anc 661 . 2  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  -> 
( ( 0. `  K )  .\/  R
)  =  R )
4029, 35, 393eqtr3d 2483 1  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  -> 
( ( S  .\/  R )  ./\  ( T  .\/  R ) )  =  R )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2606   class class class wbr 4292   ` cfv 5418  (class class class)co 6091   Basecbs 14174   lecple 14245   joincjn 15114   meetcmee 15115   0.cp0 15207   OLcol 32819   Atomscatm 32908   AtLatcal 32909   CvLatclc 32910   HLchlt 32995
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-iin 4174  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-1st 6577  df-2nd 6578  df-poset 15116  df-plt 15128  df-lub 15144  df-glb 15145  df-join 15146  df-meet 15147  df-p0 15209  df-lat 15216  df-clat 15278  df-oposet 32821  df-ol 32823  df-oml 32824  df-covers 32911  df-ats 32912  df-atl 32943  df-cvlat 32967  df-hlat 32996  df-psubsp 33147  df-pmap 33148  df-padd 33440
This theorem is referenced by:  cdleme20h  33960
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