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Theorem cdleme20y 29642
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 989 . . . . . 6  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  ->  -.  R  .<_  ( S 
.\/  T ) )
2 simp1 960 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  ->  K  e.  HL )
3 simp22 994 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  ->  S  e.  A )
4 simp23 995 . . . . . . . 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 28708 . . . . . . . 8  |-  ( ( K  e.  HL  /\  S  e.  A  /\  T  e.  A )  ->  ( S  .\/  T
)  =  ( T 
.\/  S ) )
82, 3, 4, 7syl3anc 1187 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  -> 
( S  .\/  T
)  =  ( T 
.\/  S ) )
98breq2d 3995 . . . . . 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 293 . . . . 5  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  ->  -.  R  .<_  ( T 
.\/  S ) )
11 hlcvl 28700 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  CvLat )
12113ad2ant1 981 . . . . . 6  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  ->  K  e.  CvLat )
13 simp21 993 . . . . . 6  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  ->  R  e.  A )
14 simp3l 988 . . . . . 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 28677 . . . . . 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 1200 . . . . 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 170 . . . 4  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  ->  -.  S  .<_  ( T 
.\/  R ) )
19 hlatl 28701 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  AtLat )
20193ad2ant1 981 . . . . 5  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  ->  K  e.  AtLat )
21 eqid 2256 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
2221, 5, 6hlatjcl 28707 . . . . . 6  |-  ( ( K  e.  HL  /\  T  e.  A  /\  R  e.  A )  ->  ( T  .\/  R
)  e.  ( Base `  K ) )
232, 4, 13, 22syl3anc 1187 . . . . 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 2256 . . . . . 6  |-  ( 0.
`  K )  =  ( 0. `  K
)
2621, 15, 24, 25, 6atnle 28658 . . . . 5  |-  ( ( K  e.  AtLat  /\  S  e.  A  /\  ( T  .\/  R )  e.  ( Base `  K
) )  ->  ( -.  S  .<_  ( T 
.\/  R )  <->  ( S  ./\  ( T  .\/  R
) )  =  ( 0. `  K ) ) )
2720, 3, 23, 26syl3anc 1187 . . . 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 203 . . 3  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  -> 
( S  ./\  ( T  .\/  R ) )  =  ( 0. `  K ) )
2928oveq1d 5793 . 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 28630 . . . 4  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
3113, 30syl 17 . . 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 28716 . . . 4  |-  ( ( K  e.  HL  /\  T  e.  A  /\  R  e.  A )  ->  R  .<_  ( T  .\/  R ) )
332, 4, 13, 32syl3anc 1187 . . 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 29207 . . 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 1200 . 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 28702 . . . 4  |-  ( K  e.  HL  ->  K  e.  OL )
37363ad2ant1 981 . . 3  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  ->  K  e.  OL )
3821, 5, 25olj02 28567 . . 3  |-  ( ( K  e.  OL  /\  R  e.  ( Base `  K ) )  -> 
( ( 0. `  K )  .\/  R
)  =  R )
3937, 31, 38syl2anc 645 . 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 2296 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 set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2419   class class class wbr 3983   ` cfv 4659  (class class class)co 5778   Basecbs 13096   lecple 13163   joincjn 14026   meetcmee 14027   0.cp0 14091   OLcol 28515   Atomscatm 28604   AtLatcal 28605   CvLatclc 28606   HLchlt 28691
This theorem is referenced by:  cdleme20h  29656
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4091  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2521  df-rex 2522  df-reu 2523  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-op 3609  df-uni 3788  df-iun 3867  df-iin 3868  df-br 3984  df-opab 4038  df-mpt 4039  df-id 4267  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-1st 6042  df-2nd 6043  df-iota 6211  df-undef 6250  df-riota 6258  df-poset 14028  df-plt 14040  df-lub 14056  df-glb 14057  df-join 14058  df-meet 14059  df-p0 14093  df-lat 14100  df-clat 14162  df-oposet 28517  df-ol 28519  df-oml 28520  df-covers 28607  df-ats 28608  df-atl 28639  df-cvlat 28663  df-hlat 28692  df-psubsp 28843  df-pmap 28844  df-padd 29136
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