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Theorem cdleme20y 29758
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 986 . . . . . 6  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  ->  -.  R  .<_  ( S 
.\/  T ) )
2 simp1 957 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  ->  K  e.  HL )
3 simp22 991 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  ->  S  e.  A )
4 simp23 992 . . . . . . . 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 28824 . . . . . . . 8  |-  ( ( K  e.  HL  /\  S  e.  A  /\  T  e.  A )  ->  ( S  .\/  T
)  =  ( T 
.\/  S ) )
82, 3, 4, 7syl3anc 1184 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  -> 
( S  .\/  T
)  =  ( T 
.\/  S ) )
98breq2d 4036 . . . . . 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 28816 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  CvLat )
12113ad2ant1 978 . . . . . 6  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  ->  K  e.  CvLat )
13 simp21 990 . . . . . 6  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  ->  R  e.  A )
14 simp3l 985 . . . . . 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 28793 . . . . . 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 1197 . . . . 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 28817 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  AtLat )
20193ad2ant1 978 . . . . 5  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  ->  K  e.  AtLat )
21 eqid 2284 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
2221, 5, 6hlatjcl 28823 . . . . . 6  |-  ( ( K  e.  HL  /\  T  e.  A  /\  R  e.  A )  ->  ( T  .\/  R
)  e.  ( Base `  K ) )
232, 4, 13, 22syl3anc 1184 . . . . 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 2284 . . . . . 6  |-  ( 0.
`  K )  =  ( 0. `  K
)
2621, 15, 24, 25, 6atnle 28774 . . . . 5  |-  ( ( K  e.  AtLat  /\  S  e.  A  /\  ( T  .\/  R )  e.  ( Base `  K
) )  ->  ( -.  S  .<_  ( T 
.\/  R )  <->  ( S  ./\  ( T  .\/  R
) )  =  ( 0. `  K ) ) )
2720, 3, 23, 26syl3anc 1184 . . . 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 5834 . 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 28746 . . . 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 28832 . . . 4  |-  ( ( K  e.  HL  /\  T  e.  A  /\  R  e.  A )  ->  R  .<_  ( T  .\/  R ) )
332, 4, 13, 32syl3anc 1184 . . 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 29323 . . 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 1197 . 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 28818 . . . 4  |-  ( K  e.  HL  ->  K  e.  OL )
37363ad2ant1 978 . . 3  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( S  =/=  T  /\  -.  R  .<_  ( S  .\/  T
) ) )  ->  K  e.  OL )
3821, 5, 25olj02 28683 . . 3  |-  ( ( K  e.  OL  /\  R  e.  ( Base `  K ) )  -> 
( ( 0. `  K )  .\/  R
)  =  R )
3937, 31, 38syl2anc 644 . 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 2324 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 936    = wceq 1624    e. wcel 1685    =/= wne 2447   class class class wbr 4024   ` cfv 5221  (class class class)co 5819   Basecbs 13142   lecple 13209   joincjn 14072   meetcmee 14073   0.cp0 14137   OLcol 28631   Atomscatm 28720   AtLatcal 28721   CvLatclc 28722   HLchlt 28807
This theorem is referenced by:  cdleme20h  29772
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-iin 3909  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-iota 6252  df-undef 6291  df-riota 6299  df-poset 14074  df-plt 14086  df-lub 14102  df-glb 14103  df-join 14104  df-meet 14105  df-p0 14139  df-lat 14146  df-clat 14208  df-oposet 28633  df-ol 28635  df-oml 28636  df-covers 28723  df-ats 28724  df-atl 28755  df-cvlat 28779  df-hlat 28808  df-psubsp 28959  df-pmap 28960  df-padd 29252
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