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Theorem cdlemd1 30387
Description: Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 29-May-2012.)
Hypotheses
Ref Expression
cdlemd1.l  |-  .<_  =  ( le `  K )
cdlemd1.j  |-  .\/  =  ( join `  K )
cdlemd1.m  |-  ./\  =  ( meet `  K )
cdlemd1.a  |-  A  =  ( Atoms `  K )
cdlemd1.h  |-  H  =  ( LHyp `  K
)
Assertion
Ref Expression
cdlemd1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  P  =/= 
Q  /\  -.  R  .<_  ( P  .\/  Q
) ) ) )  ->  R  =  ( ( P  .\/  (
( P  .\/  R
)  ./\  W )
)  ./\  ( Q  .\/  ( ( Q  .\/  R )  ./\  W )
) ) )

Proof of Theorem cdlemd1
StepHypRef Expression
1 simpll 730 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  P  =/= 
Q  /\  -.  R  .<_  ( P  .\/  Q
) ) ) )  ->  K  e.  HL )
2 simpr1l 1012 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  P  =/= 
Q  /\  -.  R  .<_  ( P  .\/  Q
) ) ) )  ->  P  e.  A
)
3 simpr2l 1014 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  P  =/= 
Q  /\  -.  R  .<_  ( P  .\/  Q
) ) ) )  ->  Q  e.  A
)
4 simpr31 1045 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  P  =/= 
Q  /\  -.  R  .<_  ( P  .\/  Q
) ) ) )  ->  R  e.  A
)
5 simpr32 1046 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  P  =/= 
Q  /\  -.  R  .<_  ( P  .\/  Q
) ) ) )  ->  P  =/=  Q
)
6 simpr33 1047 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  P  =/= 
Q  /\  -.  R  .<_  ( P  .\/  Q
) ) ) )  ->  -.  R  .<_  ( P  .\/  Q ) )
7 cdlemd1.l . . . 4  |-  .<_  =  ( le `  K )
8 cdlemd1.j . . . 4  |-  .\/  =  ( join `  K )
9 cdlemd1.m . . . 4  |-  ./\  =  ( meet `  K )
10 cdlemd1.a . . . 4  |-  A  =  ( Atoms `  K )
117, 8, 9, 102llnma2 29978 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q
) ) )  -> 
( ( R  .\/  P )  ./\  ( R  .\/  Q ) )  =  R )
121, 2, 3, 4, 5, 6, 11syl132anc 1200 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  P  =/= 
Q  /\  -.  R  .<_  ( P  .\/  Q
) ) ) )  ->  ( ( R 
.\/  P )  ./\  ( R  .\/  Q ) )  =  R )
13 hllat 29553 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
1413ad2antrr 706 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  P  =/= 
Q  /\  -.  R  .<_  ( P  .\/  Q
) ) ) )  ->  K  e.  Lat )
15 eqid 2283 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
1615, 10atbase 29479 . . . . . 6  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
174, 16syl 15 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  P  =/= 
Q  /\  -.  R  .<_  ( P  .\/  Q
) ) ) )  ->  R  e.  (
Base `  K )
)
1815, 10atbase 29479 . . . . . 6  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
192, 18syl 15 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  P  =/= 
Q  /\  -.  R  .<_  ( P  .\/  Q
) ) ) )  ->  P  e.  (
Base `  K )
)
2015, 8latjcom 14165 . . . . 5  |-  ( ( K  e.  Lat  /\  R  e.  ( Base `  K )  /\  P  e.  ( Base `  K
) )  ->  ( R  .\/  P )  =  ( P  .\/  R
) )
2114, 17, 19, 20syl3anc 1182 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  P  =/= 
Q  /\  -.  R  .<_  ( P  .\/  Q
) ) ) )  ->  ( R  .\/  P )  =  ( P 
.\/  R ) )
22 simpl 443 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  P  =/= 
Q  /\  -.  R  .<_  ( P  .\/  Q
) ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
23 simpr1 961 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  P  =/= 
Q  /\  -.  R  .<_  ( P  .\/  Q
) ) ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
24 cdlemd1.h . . . . . 6  |-  H  =  ( LHyp `  K
)
2515, 7, 8, 9, 10, 24cdlemc1 30380 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  (
Base `  K )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( P  .\/  ( ( P  .\/  R )  ./\  W )
)  =  ( P 
.\/  R ) )
2622, 17, 23, 25syl3anc 1182 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  P  =/= 
Q  /\  -.  R  .<_  ( P  .\/  Q
) ) ) )  ->  ( P  .\/  ( ( P  .\/  R )  ./\  W )
)  =  ( P 
.\/  R ) )
2721, 26eqtr4d 2318 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  P  =/= 
Q  /\  -.  R  .<_  ( P  .\/  Q
) ) ) )  ->  ( R  .\/  P )  =  ( P 
.\/  ( ( P 
.\/  R )  ./\  W ) ) )
2815, 10atbase 29479 . . . . . 6  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
293, 28syl 15 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  P  =/= 
Q  /\  -.  R  .<_  ( P  .\/  Q
) ) ) )  ->  Q  e.  (
Base `  K )
)
3015, 8latjcom 14165 . . . . 5  |-  ( ( K  e.  Lat  /\  R  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  ->  ( R  .\/  Q )  =  ( Q  .\/  R
) )
3114, 17, 29, 30syl3anc 1182 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  P  =/= 
Q  /\  -.  R  .<_  ( P  .\/  Q
) ) ) )  ->  ( R  .\/  Q )  =  ( Q 
.\/  R ) )
32 simpr2 962 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  P  =/= 
Q  /\  -.  R  .<_  ( P  .\/  Q
) ) ) )  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
3315, 7, 8, 9, 10, 24cdlemc1 30380 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  (
Base `  K )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( Q  .\/  ( ( Q  .\/  R )  ./\  W )
)  =  ( Q 
.\/  R ) )
3422, 17, 32, 33syl3anc 1182 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  P  =/= 
Q  /\  -.  R  .<_  ( P  .\/  Q
) ) ) )  ->  ( Q  .\/  ( ( Q  .\/  R )  ./\  W )
)  =  ( Q 
.\/  R ) )
3531, 34eqtr4d 2318 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  P  =/= 
Q  /\  -.  R  .<_  ( P  .\/  Q
) ) ) )  ->  ( R  .\/  Q )  =  ( Q 
.\/  ( ( Q 
.\/  R )  ./\  W ) ) )
3627, 35oveq12d 5876 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  P  =/= 
Q  /\  -.  R  .<_  ( P  .\/  Q
) ) ) )  ->  ( ( R 
.\/  P )  ./\  ( R  .\/  Q ) )  =  ( ( P  .\/  ( ( P  .\/  R ) 
./\  W ) ) 
./\  ( Q  .\/  ( ( Q  .\/  R )  ./\  W )
) ) )
3712, 36eqtr3d 2317 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  P  =/= 
Q  /\  -.  R  .<_  ( P  .\/  Q
) ) ) )  ->  R  =  ( ( P  .\/  (
( P  .\/  R
)  ./\  W )
)  ./\  ( Q  .\/  ( ( Q  .\/  R )  ./\  W )
) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ 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   HLchlt 29540   LHypclh 30173
This theorem is referenced by:  cdlemd2  30388
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-iin 3908  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-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-psubsp 29692  df-pmap 29693  df-padd 29985  df-lhyp 30177
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