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Theorem cdleme19a 30492
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 2283 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
3 cdleme19.l . . . 4  |-  .<_  =  ( le `  K )
4 hllat 29553 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
543ad2ant1 976 . . . 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 955 . . . . 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 988 . . . . 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 989 . . . . 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 29556 . . . . 5  |-  ( ( K  e.  HL  /\  R  e.  A  /\  S  e.  A )  ->  ( R  .\/  S
)  e.  ( Base `  K ) )
126, 7, 8, 11syl3anc 1182 . . . 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 990 . . . . 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 29556 . . . . 5  |-  ( ( K  e.  HL  /\  S  e.  A  /\  T  e.  A )  ->  ( S  .\/  T
)  e.  ( Base `  K ) )
156, 8, 13, 14syl3anc 1182 . . . 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 993 . . . . 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 29564 . . . . . 6  |-  ( ( K  e.  HL  /\  S  e.  A  /\  T  e.  A )  ->  S  .<_  ( S  .\/  T ) )
186, 8, 13, 17syl3anc 1182 . . . . 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 29479 . . . . . . 7  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
207, 19syl 15 . . . . . 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 29479 . . . . . . 7  |-  ( S  e.  A  ->  S  e.  ( Base `  K
) )
228, 21syl 15 . . . . . 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 14168 . . . . . 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 1184 . . . . 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 887 . . . 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 29565 . . . . . 6  |-  ( ( K  e.  HL  /\  R  e.  A  /\  S  e.  A )  ->  S  .<_  ( R  .\/  S ) )
276, 7, 8, 26syl3anc 1182 . . . . 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 29549 . . . . . . . . 9  |-  ( K  e.  HL  ->  K  e.  CvLat )
29283ad2ant1 976 . . . . . . . 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 991 . . . . . . . . 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 992 . . . . . . . . 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 4041 . . . . . . . . 9  |-  ( ( R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( P  .\/  Q ) )  ->  R  =/=  S )
3330, 31, 32syl2anc 642 . . . . . . . 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 29526 . . . . . . . 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 1195 . . . . . . 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 14 . . . . . 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 29557 . . . . . . 7  |-  ( ( K  e.  HL  /\  R  e.  A  /\  S  e.  A )  ->  ( R  .\/  S
)  =  ( S 
.\/  R ) )
386, 7, 8, 37syl3anc 1182 . . . . . 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 4049 . . . . 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 29479 . . . . . . 7  |-  ( T  e.  A  ->  T  e.  ( Base `  K
) )
4113, 40syl 15 . . . . . 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 14168 . . . . . 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 1184 . . . . 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 887 . . . 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 14163 . . 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 5873 . 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 2327 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 set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   joincjn 14078   meetcmee 14079   Latclat 14151   Atomscatm 29453   CvLatclc 29455   HLchlt 29540   LHypclh 30173
This theorem is referenced by:  cdleme19b  30493
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-plt 14092  df-lub 14108  df-join 14110  df-lat 14152  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541
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