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Theorem cdleme19a 33322
Description: Part of proof of Lemma E in [Crawley] p. 113, 5th paragraph on p. 114, 1st line.  D represents s2. In their notation, we prove that if r  <_ s  \/ t, then s2=(s  \/ t)  /\ w. (Contributed by NM, 13-Nov-2012.)
Hypotheses
Ref Expression
cdleme19.l  |-  .<_  =  ( le `  K )
cdleme19.j  |-  .\/  =  ( join `  K )
cdleme19.m  |-  ./\  =  ( meet `  K )
cdleme19.a  |-  A  =  ( Atoms `  K )
cdleme19.h  |-  H  =  ( LHyp `  K
)
cdleme19.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
cdleme19.f  |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) )
cdleme19.g  |-  G  =  ( ( T  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  T )  ./\  W )
) )
cdleme19.d  |-  D  =  ( ( R  .\/  S )  ./\  W )
cdleme19.y  |-  Y  =  ( ( R  .\/  T )  ./\  W )
Assertion
Ref Expression
cdleme19a  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
)  /\  R  .<_  ( S  .\/  T ) ) )  ->  D  =  ( ( S 
.\/  T )  ./\  W ) )

Proof of Theorem cdleme19a
StepHypRef Expression
1 cdleme19.d . 2  |-  D  =  ( ( R  .\/  S )  ./\  W )
2 eqid 2402 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
3 cdleme19.l . . . 4  |-  .<_  =  ( le `  K )
4 hllat 32381 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
543ad2ant1 1018 . . . 4  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
)  /\  R  .<_  ( S  .\/  T ) ) )  ->  K  e.  Lat )
6 simp1 997 . . . . 5  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
)  /\  R  .<_  ( S  .\/  T ) ) )  ->  K  e.  HL )
7 simp21 1030 . . . . 5  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
)  /\  R  .<_  ( S  .\/  T ) ) )  ->  R  e.  A )
8 simp22 1031 . . . . 5  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
)  /\  R  .<_  ( S  .\/  T ) ) )  ->  S  e.  A )
9 cdleme19.j . . . . . 6  |-  .\/  =  ( join `  K )
10 cdleme19.a . . . . . 6  |-  A  =  ( Atoms `  K )
112, 9, 10hlatjcl 32384 . . . . 5  |-  ( ( K  e.  HL  /\  R  e.  A  /\  S  e.  A )  ->  ( R  .\/  S
)  e.  ( Base `  K ) )
126, 7, 8, 11syl3anc 1230 . . . 4  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
)  /\  R  .<_  ( S  .\/  T ) ) )  ->  ( R  .\/  S )  e.  ( Base `  K
) )
13 simp23 1032 . . . . 5  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
)  /\  R  .<_  ( S  .\/  T ) ) )  ->  T  e.  A )
142, 9, 10hlatjcl 32384 . . . . 5  |-  ( ( K  e.  HL  /\  S  e.  A  /\  T  e.  A )  ->  ( S  .\/  T
)  e.  ( Base `  K ) )
156, 8, 13, 14syl3anc 1230 . . . 4  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
)  /\  R  .<_  ( S  .\/  T ) ) )  ->  ( S  .\/  T )  e.  ( Base `  K
) )
16 simp33 1035 . . . . 5  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
)  /\  R  .<_  ( S  .\/  T ) ) )  ->  R  .<_  ( S  .\/  T
) )
173, 9, 10hlatlej1 32392 . . . . . 6  |-  ( ( K  e.  HL  /\  S  e.  A  /\  T  e.  A )  ->  S  .<_  ( S  .\/  T ) )
186, 8, 13, 17syl3anc 1230 . . . . 5  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
)  /\  R  .<_  ( S  .\/  T ) ) )  ->  S  .<_  ( S  .\/  T
) )
192, 10atbase 32307 . . . . . . 7  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
207, 19syl 17 . . . . . 6  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
)  /\  R  .<_  ( S  .\/  T ) ) )  ->  R  e.  ( Base `  K
) )
212, 10atbase 32307 . . . . . . 7  |-  ( S  e.  A  ->  S  e.  ( Base `  K
) )
228, 21syl 17 . . . . . 6  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
)  /\  R  .<_  ( S  .\/  T ) ) )  ->  S  e.  ( Base `  K
) )
232, 3, 9latjle12 16016 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( R  e.  ( Base `  K )  /\  S  e.  ( Base `  K )  /\  ( S  .\/  T )  e.  ( Base `  K
) ) )  -> 
( ( R  .<_  ( S  .\/  T )  /\  S  .<_  ( S 
.\/  T ) )  <-> 
( R  .\/  S
)  .<_  ( S  .\/  T ) ) )
245, 20, 22, 15, 23syl13anc 1232 . . . . 5  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
)  /\  R  .<_  ( S  .\/  T ) ) )  ->  (
( R  .<_  ( S 
.\/  T )  /\  S  .<_  ( S  .\/  T ) )  <->  ( R  .\/  S )  .<_  ( S 
.\/  T ) ) )
2516, 18, 24mpbi2and 922 . . . 4  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
)  /\  R  .<_  ( S  .\/  T ) ) )  ->  ( R  .\/  S )  .<_  ( S  .\/  T ) )
263, 9, 10hlatlej2 32393 . . . . . 6  |-  ( ( K  e.  HL  /\  R  e.  A  /\  S  e.  A )  ->  S  .<_  ( R  .\/  S ) )
276, 7, 8, 26syl3anc 1230 . . . . 5  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
)  /\  R  .<_  ( S  .\/  T ) ) )  ->  S  .<_  ( R  .\/  S
) )
28 hlcvl 32377 . . . . . . . . 9  |-  ( K  e.  HL  ->  K  e.  CvLat )
29283ad2ant1 1018 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
)  /\  R  .<_  ( S  .\/  T ) ) )  ->  K  e.  CvLat )
30 simp31 1033 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
)  /\  R  .<_  ( S  .\/  T ) ) )  ->  R  .<_  ( P  .\/  Q
) )
31 simp32 1034 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
)  /\  R  .<_  ( S  .\/  T ) ) )  ->  -.  S  .<_  ( P  .\/  Q ) )
32 nbrne2 4413 . . . . . . . . 9  |-  ( ( R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( P  .\/  Q ) )  ->  R  =/=  S )
3330, 31, 32syl2anc 659 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
)  /\  R  .<_  ( S  .\/  T ) ) )  ->  R  =/=  S )
343, 9, 10cvlatexch1 32354 . . . . . . . 8  |-  ( ( K  e.  CvLat  /\  ( R  e.  A  /\  T  e.  A  /\  S  e.  A )  /\  R  =/=  S
)  ->  ( R  .<_  ( S  .\/  T
)  ->  T  .<_  ( S  .\/  R ) ) )
3529, 7, 13, 8, 33, 34syl131anc 1243 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
)  /\  R  .<_  ( S  .\/  T ) ) )  ->  ( R  .<_  ( S  .\/  T )  ->  T  .<_  ( S  .\/  R ) ) )
3616, 35mpd 15 . . . . . 6  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
)  /\  R  .<_  ( S  .\/  T ) ) )  ->  T  .<_  ( S  .\/  R
) )
379, 10hlatjcom 32385 . . . . . . 7  |-  ( ( K  e.  HL  /\  R  e.  A  /\  S  e.  A )  ->  ( R  .\/  S
)  =  ( S 
.\/  R ) )
386, 7, 8, 37syl3anc 1230 . . . . . 6  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
)  /\  R  .<_  ( S  .\/  T ) ) )  ->  ( R  .\/  S )  =  ( S  .\/  R
) )
3936, 38breqtrrd 4421 . . . . 5  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
)  /\  R  .<_  ( S  .\/  T ) ) )  ->  T  .<_  ( R  .\/  S
) )
402, 10atbase 32307 . . . . . . 7  |-  ( T  e.  A  ->  T  e.  ( Base `  K
) )
4113, 40syl 17 . . . . . 6  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
)  /\  R  .<_  ( S  .\/  T ) ) )  ->  T  e.  ( Base `  K
) )
422, 3, 9latjle12 16016 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( S  e.  ( Base `  K )  /\  T  e.  ( Base `  K )  /\  ( R  .\/  S )  e.  ( Base `  K
) ) )  -> 
( ( S  .<_  ( R  .\/  S )  /\  T  .<_  ( R 
.\/  S ) )  <-> 
( S  .\/  T
)  .<_  ( R  .\/  S ) ) )
435, 22, 41, 12, 42syl13anc 1232 . . . . 5  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
)  /\  R  .<_  ( S  .\/  T ) ) )  ->  (
( S  .<_  ( R 
.\/  S )  /\  T  .<_  ( R  .\/  S ) )  <->  ( S  .\/  T )  .<_  ( R 
.\/  S ) ) )
4427, 39, 43mpbi2and 922 . . . 4  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
)  /\  R  .<_  ( S  .\/  T ) ) )  ->  ( S  .\/  T )  .<_  ( R  .\/  S ) )
452, 3, 5, 12, 15, 25, 44latasymd 16011 . . 3  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
)  /\  R  .<_  ( S  .\/  T ) ) )  ->  ( R  .\/  S )  =  ( S  .\/  T
) )
4645oveq1d 6293 . 2  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
)  /\  R  .<_  ( S  .\/  T ) ) )  ->  (
( R  .\/  S
)  ./\  W )  =  ( ( S 
.\/  T )  ./\  W ) )
471, 46syl5eq 2455 1  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
)  /\  R  .<_  ( S  .\/  T ) ) )  ->  D  =  ( ( S 
.\/  T )  ./\  W ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   class class class wbr 4395   ` cfv 5569  (class class class)co 6278   Basecbs 14841   lecple 14916   joincjn 15897   meetcmee 15898   Latclat 15999   Atomscatm 32281   CvLatclc 32283   HLchlt 32368   LHypclh 33001
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-preset 15881  df-poset 15899  df-plt 15912  df-lub 15928  df-glb 15929  df-join 15930  df-meet 15931  df-p0 15993  df-lat 16000  df-covers 32284  df-ats 32285  df-atl 32316  df-cvlat 32340  df-hlat 32369
This theorem is referenced by:  cdleme19b  33323
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