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Theorem cdlemf2 30019
Description: Part of Lemma F in [Crawley] p. 116. (Contributed by NM, 12-Apr-2013.)
Hypotheses
Ref Expression
cdlemf1.l  |-  .<_  =  ( le `  K )
cdlemf1.j  |-  .\/  =  ( join `  K )
cdlemf1.a  |-  A  =  ( Atoms `  K )
cdlemf1.h  |-  H  =  ( LHyp `  K
)
cdlemf2.m  |-  ./\  =  ( meet `  K )
Assertion
Ref Expression
cdlemf2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W ) )  ->  E. p  e.  A  E. q  e.  A  ( ( -.  p  .<_  W  /\  -.  q  .<_  W )  /\  U  =  ( ( p  .\/  q
)  ./\  W )
) )
Distinct variable groups:    q, p, A    H, p, q    K, p, q    .<_ , p, q    U, p, q    W, p, q
Allowed substitution hints:    .\/ ( q, p)    ./\ ( q, p)

Proof of Theorem cdlemf2
StepHypRef Expression
1 cdlemf1.l . . . 4  |-  .<_  =  ( le `  K )
2 cdlemf1.a . . . 4  |-  A  =  ( Atoms `  K )
3 cdlemf1.h . . . 4  |-  H  =  ( LHyp `  K
)
41, 2, 3lhpexnle 29463 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  -.  p  .<_  W )
54adantr 453 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W ) )  ->  E. p  e.  A  -.  p  .<_  W )
6 cdlemf1.j . . . . . . 7  |-  .\/  =  ( join `  K )
71, 6, 2, 3cdlemf1 30018 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( p  e.  A  /\  -.  p  .<_  W ) )  ->  E. q  e.  A  ( p  =/=  q  /\  -.  q  .<_  W  /\  U  .<_  ( p  .\/  q ) ) )
8 simpr1r 1015 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W ) )  /\  ( ( p  e.  A  /\  -.  p  .<_  W )  /\  q  e.  A  /\  ( p  =/=  q  /\  -.  q  .<_  W  /\  U  .<_  ( p  .\/  q ) ) ) )  ->  -.  p  .<_  W )
9 simpr32 1048 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W ) )  /\  ( ( p  e.  A  /\  -.  p  .<_  W )  /\  q  e.  A  /\  ( p  =/=  q  /\  -.  q  .<_  W  /\  U  .<_  ( p  .\/  q ) ) ) )  ->  -.  q  .<_  W )
10 simpr33 1049 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W ) )  /\  ( ( p  e.  A  /\  -.  p  .<_  W )  /\  q  e.  A  /\  ( p  =/=  q  /\  -.  q  .<_  W  /\  U  .<_  ( p  .\/  q ) ) ) )  ->  U  .<_  ( p  .\/  q ) )
11 simplrr 739 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W ) )  /\  ( ( p  e.  A  /\  -.  p  .<_  W )  /\  q  e.  A  /\  ( p  =/=  q  /\  -.  q  .<_  W  /\  U  .<_  ( p  .\/  q ) ) ) )  ->  U  .<_  W )
12 hllat 28821 . . . . . . . . . . . . . 14  |-  ( K  e.  HL  ->  K  e.  Lat )
1312ad3antrrr 712 . . . . . . . . . . . . 13  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W ) )  /\  ( ( p  e.  A  /\  -.  p  .<_  W )  /\  q  e.  A  /\  ( p  =/=  q  /\  -.  q  .<_  W  /\  U  .<_  ( p  .\/  q ) ) ) )  ->  K  e.  Lat )
14 simplrl 738 . . . . . . . . . . . . . 14  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W ) )  /\  ( ( p  e.  A  /\  -.  p  .<_  W )  /\  q  e.  A  /\  ( p  =/=  q  /\  -.  q  .<_  W  /\  U  .<_  ( p  .\/  q ) ) ) )  ->  U  e.  A )
15 eqid 2285 . . . . . . . . . . . . . . 15  |-  ( Base `  K )  =  (
Base `  K )
1615, 2atbase 28747 . . . . . . . . . . . . . 14  |-  ( U  e.  A  ->  U  e.  ( Base `  K
) )
1714, 16syl 17 . . . . . . . . . . . . 13  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W ) )  /\  ( ( p  e.  A  /\  -.  p  .<_  W )  /\  q  e.  A  /\  ( p  =/=  q  /\  -.  q  .<_  W  /\  U  .<_  ( p  .\/  q ) ) ) )  ->  U  e.  ( Base `  K )
)
18 simplll 736 . . . . . . . . . . . . . 14  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W ) )  /\  ( ( p  e.  A  /\  -.  p  .<_  W )  /\  q  e.  A  /\  ( p  =/=  q  /\  -.  q  .<_  W  /\  U  .<_  ( p  .\/  q ) ) ) )  ->  K  e.  HL )
19 simpr1l 1014 . . . . . . . . . . . . . 14  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W ) )  /\  ( ( p  e.  A  /\  -.  p  .<_  W )  /\  q  e.  A  /\  ( p  =/=  q  /\  -.  q  .<_  W  /\  U  .<_  ( p  .\/  q ) ) ) )  ->  p  e.  A )
20 simpr2 964 . . . . . . . . . . . . . 14  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W ) )  /\  ( ( p  e.  A  /\  -.  p  .<_  W )  /\  q  e.  A  /\  ( p  =/=  q  /\  -.  q  .<_  W  /\  U  .<_  ( p  .\/  q ) ) ) )  ->  q  e.  A )
2115, 6, 2hlatjcl 28824 . . . . . . . . . . . . . 14  |-  ( ( K  e.  HL  /\  p  e.  A  /\  q  e.  A )  ->  ( p  .\/  q
)  e.  ( Base `  K ) )
2218, 19, 20, 21syl3anc 1184 . . . . . . . . . . . . 13  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W ) )  /\  ( ( p  e.  A  /\  -.  p  .<_  W )  /\  q  e.  A  /\  ( p  =/=  q  /\  -.  q  .<_  W  /\  U  .<_  ( p  .\/  q ) ) ) )  ->  ( p  .\/  q )  e.  (
Base `  K )
)
23 simpllr 737 . . . . . . . . . . . . . 14  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W ) )  /\  ( ( p  e.  A  /\  -.  p  .<_  W )  /\  q  e.  A  /\  ( p  =/=  q  /\  -.  q  .<_  W  /\  U  .<_  ( p  .\/  q ) ) ) )  ->  W  e.  H )
2415, 3lhpbase 29455 . . . . . . . . . . . . . 14  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
2523, 24syl 17 . . . . . . . . . . . . 13  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W ) )  /\  ( ( p  e.  A  /\  -.  p  .<_  W )  /\  q  e.  A  /\  ( p  =/=  q  /\  -.  q  .<_  W  /\  U  .<_  ( p  .\/  q ) ) ) )  ->  W  e.  ( Base `  K )
)
26 cdlemf2.m . . . . . . . . . . . . . 14  |-  ./\  =  ( meet `  K )
2715, 1, 26latlem12 14179 . . . . . . . . . . . . 13  |-  ( ( K  e.  Lat  /\  ( U  e.  ( Base `  K )  /\  ( p  .\/  q )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
) )  ->  (
( U  .<_  ( p 
.\/  q )  /\  U  .<_  W )  <->  U  .<_  ( ( p  .\/  q
)  ./\  W )
) )
2813, 17, 22, 25, 27syl13anc 1186 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W ) )  /\  ( ( p  e.  A  /\  -.  p  .<_  W )  /\  q  e.  A  /\  ( p  =/=  q  /\  -.  q  .<_  W  /\  U  .<_  ( p  .\/  q ) ) ) )  ->  ( ( U  .<_  ( p  .\/  q )  /\  U  .<_  W )  <->  U  .<_  ( ( p  .\/  q
)  ./\  W )
) )
2910, 11, 28mpbi2and 889 . . . . . . . . . . 11  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W ) )  /\  ( ( p  e.  A  /\  -.  p  .<_  W )  /\  q  e.  A  /\  ( p  =/=  q  /\  -.  q  .<_  W  /\  U  .<_  ( p  .\/  q ) ) ) )  ->  U  .<_  ( ( p  .\/  q
)  ./\  W )
)
30 hlatl 28818 . . . . . . . . . . . . 13  |-  ( K  e.  HL  ->  K  e.  AtLat )
3130ad3antrrr 712 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W ) )  /\  ( ( p  e.  A  /\  -.  p  .<_  W )  /\  q  e.  A  /\  ( p  =/=  q  /\  -.  q  .<_  W  /\  U  .<_  ( p  .\/  q ) ) ) )  ->  K  e.  AtLat
)
32 simpll 732 . . . . . . . . . . . . 13  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W ) )  /\  ( ( p  e.  A  /\  -.  p  .<_  W )  /\  q  e.  A  /\  ( p  =/=  q  /\  -.  q  .<_  W  /\  U  .<_  ( p  .\/  q ) ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
33 simpr31 1047 . . . . . . . . . . . . 13  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W ) )  /\  ( ( p  e.  A  /\  -.  p  .<_  W )  /\  q  e.  A  /\  ( p  =/=  q  /\  -.  q  .<_  W  /\  U  .<_  ( p  .\/  q ) ) ) )  ->  p  =/=  q )
341, 6, 26, 2, 3lhpat 29500 . . . . . . . . . . . . 13  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( p  e.  A  /\  -.  p  .<_  W )  /\  (
q  e.  A  /\  p  =/=  q ) )  ->  ( ( p 
.\/  q )  ./\  W )  e.  A )
3532, 19, 8, 20, 33, 34syl122anc 1193 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W ) )  /\  ( ( p  e.  A  /\  -.  p  .<_  W )  /\  q  e.  A  /\  ( p  =/=  q  /\  -.  q  .<_  W  /\  U  .<_  ( p  .\/  q ) ) ) )  ->  ( (
p  .\/  q )  ./\  W )  e.  A
)
361, 2atcmp 28769 . . . . . . . . . . . 12  |-  ( ( K  e.  AtLat  /\  U  e.  A  /\  (
( p  .\/  q
)  ./\  W )  e.  A )  ->  ( U  .<_  ( ( p 
.\/  q )  ./\  W )  <->  U  =  (
( p  .\/  q
)  ./\  W )
) )
3731, 14, 35, 36syl3anc 1184 . . . . . . . . . . 11  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W ) )  /\  ( ( p  e.  A  /\  -.  p  .<_  W )  /\  q  e.  A  /\  ( p  =/=  q  /\  -.  q  .<_  W  /\  U  .<_  ( p  .\/  q ) ) ) )  ->  ( U  .<_  ( ( p  .\/  q )  ./\  W
)  <->  U  =  (
( p  .\/  q
)  ./\  W )
) )
3829, 37mpbid 203 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W ) )  /\  ( ( p  e.  A  /\  -.  p  .<_  W )  /\  q  e.  A  /\  ( p  =/=  q  /\  -.  q  .<_  W  /\  U  .<_  ( p  .\/  q ) ) ) )  ->  U  =  ( ( p  .\/  q )  ./\  W
) )
398, 9, 38jca31 522 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W ) )  /\  ( ( p  e.  A  /\  -.  p  .<_  W )  /\  q  e.  A  /\  ( p  =/=  q  /\  -.  q  .<_  W  /\  U  .<_  ( p  .\/  q ) ) ) )  ->  ( ( -.  p  .<_  W  /\  -.  q  .<_  W )  /\  U  =  ( ( p  .\/  q
)  ./\  W )
) )
40393exp2 1171 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W ) )  ->  (
( p  e.  A  /\  -.  p  .<_  W )  ->  ( q  e.  A  ->  ( (
p  =/=  q  /\  -.  q  .<_  W  /\  U  .<_  ( p  .\/  q ) )  -> 
( ( -.  p  .<_  W  /\  -.  q  .<_  W )  /\  U  =  ( ( p 
.\/  q )  ./\  W ) ) ) ) ) )
41403impia 1150 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( p  e.  A  /\  -.  p  .<_  W ) )  ->  ( q  e.  A  ->  ( (
p  =/=  q  /\  -.  q  .<_  W  /\  U  .<_  ( p  .\/  q ) )  -> 
( ( -.  p  .<_  W  /\  -.  q  .<_  W )  /\  U  =  ( ( p 
.\/  q )  ./\  W ) ) ) ) )
4241reximdvai 2655 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( p  e.  A  /\  -.  p  .<_  W ) )  ->  ( E. q  e.  A  ( p  =/=  q  /\  -.  q  .<_  W  /\  U  .<_  ( p  .\/  q ) )  ->  E. q  e.  A  ( ( -.  p  .<_  W  /\  -.  q  .<_  W )  /\  U  =  ( ( p  .\/  q
)  ./\  W )
) ) )
437, 42mpd 16 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( p  e.  A  /\  -.  p  .<_  W ) )  ->  E. q  e.  A  ( ( -.  p  .<_  W  /\  -.  q  .<_  W )  /\  U  =  ( ( p 
.\/  q )  ./\  W ) ) )
44433expia 1155 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W ) )  ->  (
( p  e.  A  /\  -.  p  .<_  W )  ->  E. q  e.  A  ( ( -.  p  .<_  W  /\  -.  q  .<_  W )  /\  U  =  ( ( p 
.\/  q )  ./\  W ) ) ) )
4544exp3a 427 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W ) )  ->  (
p  e.  A  -> 
( -.  p  .<_  W  ->  E. q  e.  A  ( ( -.  p  .<_  W  /\  -.  q  .<_  W )  /\  U  =  ( ( p 
.\/  q )  ./\  W ) ) ) ) )
4645reximdvai 2655 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W ) )  ->  ( E. p  e.  A  -.  p  .<_  W  ->  E. p  e.  A  E. q  e.  A  ( ( -.  p  .<_  W  /\  -.  q  .<_  W )  /\  U  =  ( ( p 
.\/  q )  ./\  W ) ) ) )
475, 46mpd 16 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W ) )  ->  E. p  e.  A  E. q  e.  A  ( ( -.  p  .<_  W  /\  -.  q  .<_  W )  /\  U  =  ( ( p  .\/  q
)  ./\  W )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 936    = wceq 1624    e. wcel 1685    =/= wne 2448   E.wrex 2546   class class class wbr 4025   ` cfv 5222  (class class class)co 5820   Basecbs 13143   lecple 13210   joincjn 14073   meetcmee 14074   Latclat 14146   Atomscatm 28721   AtLatcal 28722   HLchlt 28808   LHypclh 29441
This theorem is referenced by:  cdlemf  30020
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 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4188  ax-pr 4214  ax-un 4512
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 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  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-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-iota 6253  df-undef 6292  df-riota 6300  df-poset 14075  df-plt 14087  df-lub 14103  df-glb 14104  df-join 14105  df-meet 14106  df-p0 14140  df-p1 14141  df-lat 14147  df-clat 14209  df-oposet 28634  df-ol 28636  df-oml 28637  df-covers 28724  df-ats 28725  df-atl 28756  df-cvlat 28780  df-hlat 28809  df-lhyp 29445
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