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Theorem cdlemf2 31090
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 30534 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  -.  p  .<_  W )
54adantr 452 . 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 31089 . . . . . 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 738 . . . . . . . . . . . 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 29892 . . . . . . . . . . . . . 14  |-  ( K  e.  HL  ->  K  e.  Lat )
1312ad3antrrr 711 . . . . . . . . . . . . 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 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 ) ) ) )  ->  U  e.  A )
15 eqid 2430 . . . . . . . . . . . . . . 15  |-  ( Base `  K )  =  (
Base `  K )
1615, 2atbase 29818 . . . . . . . . . . . . . 14  |-  ( U  e.  A  ->  U  e.  ( Base `  K
) )
1714, 16syl 16 . . . . . . . . . . . . 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 735 . . . . . . . . . . . . . 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 29895 . . . . . . . . . . . . . 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 )
)
2315, 3lhpbase 30526 . . . . . . . . . . . . . 14  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
2423ad3antlr 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 ) ) ) )  ->  W  e.  ( Base `  K )
)
25 cdlemf2.m . . . . . . . . . . . . . 14  |-  ./\  =  ( meet `  K )
2615, 1, 25latlem12 14490 . . . . . . . . . . . . 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 )
) )
2713, 17, 22, 24, 26syl13anc 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 )
) )
2810, 11, 27mpbi2and 888 . . . . . . . . . . 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 )
)
29 hlatl 29889 . . . . . . . . . . . . 13  |-  ( K  e.  HL  ->  K  e.  AtLat )
3029ad3antrrr 711 . . . . . . . . . . . 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
)
31 simpll 731 . . . . . . . . . . . . 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 ) )
32 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 )
331, 6, 25, 2, 3lhpat 30571 . . . . . . . . . . . . 13  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( p  e.  A  /\  -.  p  .<_  W )  /\  (
q  e.  A  /\  p  =/=  q ) )  ->  ( ( p 
.\/  q )  ./\  W )  e.  A )
3431, 19, 8, 20, 32, 33syl122anc 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
)
351, 2atcmp 29840 . . . . . . . . . . . 12  |-  ( ( K  e.  AtLat  /\  U  e.  A  /\  (
( p  .\/  q
)  ./\  W )  e.  A )  ->  ( U  .<_  ( ( p 
.\/  q )  ./\  W )  <->  U  =  (
( p  .\/  q
)  ./\  W )
) )
3630, 14, 34, 35syl3anc 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 )
) )
3728, 36mpbid 202 . . . . . . . . . 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
) )
388, 9, 37jca31 521 . . . . . . . . 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 )
) )
39383exp2 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 ) ) ) ) ) )
40393impia 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 ) ) ) ) )
4140reximdvai 2803 . . . . . 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 )
) ) )
427, 41mpd 15 . . . . 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 ) ) )
43423expia 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 ) ) ) )
4443exp3a 426 . . 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 ) ) ) ) )
4544reximdvai 2803 . 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 ) ) ) )
465, 45mpd 15 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 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2593   E.wrex 2693   class class class wbr 4199   ` cfv 5440  (class class class)co 6067   Basecbs 13452   lecple 13519   joincjn 14384   meetcmee 14385   Latclat 14457   Atomscatm 29792   AtLatcal 29793   HLchlt 29879   LHypclh 30512
This theorem is referenced by:  cdlemf  31091
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411  ax-rep 4307  ax-sep 4317  ax-nul 4325  ax-pow 4364  ax-pr 4390  ax-un 4687
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-nel 2596  df-ral 2697  df-rex 2698  df-reu 2699  df-rab 2701  df-v 2945  df-sbc 3149  df-csb 3239  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-nul 3616  df-if 3727  df-pw 3788  df-sn 3807  df-pr 3808  df-op 3810  df-uni 4003  df-iun 4082  df-br 4200  df-opab 4254  df-mpt 4255  df-id 4485  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-dm 4874  df-rn 4875  df-res 4876  df-ima 4877  df-iota 5404  df-fun 5442  df-fn 5443  df-f 5444  df-f1 5445  df-fo 5446  df-f1o 5447  df-fv 5448  df-ov 6070  df-oprab 6071  df-mpt2 6072  df-1st 6335  df-2nd 6336  df-undef 6529  df-riota 6535  df-poset 14386  df-plt 14398  df-lub 14414  df-glb 14415  df-join 14416  df-meet 14417  df-p0 14451  df-p1 14452  df-lat 14458  df-clat 14520  df-oposet 29705  df-ol 29707  df-oml 29708  df-covers 29795  df-ats 29796  df-atl 29827  df-cvlat 29851  df-hlat 29880  df-lhyp 30516
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