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Theorem cdlemf2 29918
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 29362 . . 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 29917 . . . . . 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 1018 . . . . . . . . . 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 1051 . . . . . . . . . 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 1052 . . . . . . . . . . . 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 740 . . . . . . . . . . . 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 28720 . . . . . . . . . . . . . 14  |-  ( K  e.  HL  ->  K  e.  Lat )
1312ad3antrrr 713 . . . . . . . . . . . . 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 739 . . . . . . . . . . . . . 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 2258 . . . . . . . . . . . . . . 15  |-  ( Base `  K )  =  (
Base `  K )
1615, 2atbase 28646 . . . . . . . . . . . . . 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 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 ) ) ) )  ->  K  e.  HL )
19 simpr1l 1017 . . . . . . . . . . . . . 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 967 . . . . . . . . . . . . . 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 28723 . . . . . . . . . . . . . 14  |-  ( ( K  e.  HL  /\  p  e.  A  /\  q  e.  A )  ->  ( p  .\/  q
)  e.  ( Base `  K ) )
2218, 19, 20, 21syl3anc 1187 . . . . . . . . . . . . 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 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 ) ) ) )  ->  W  e.  H )
2415, 3lhpbase 29354 . . . . . . . . . . . . . 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 14146 . . . . . . . . . . . . 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 1189 . . . . . . . . . . . 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 892 . . . . . . . . . . 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 28717 . . . . . . . . . . . . 13  |-  ( K  e.  HL  ->  K  e.  AtLat )
3130ad3antrrr 713 . . . . . . . . . . . 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 733 . . . . . . . . . . . . 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 1050 . . . . . . . . . . . . 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 29399 . . . . . . . . . . . . 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 1196 . . . . . . . . . . . 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 28668 . . . . . . . . . . . 12  |-  ( ( K  e.  AtLat  /\  U  e.  A  /\  (
( p  .\/  q
)  ./\  W )  e.  A )  ->  ( U  .<_  ( ( p 
.\/  q )  ./\  W )  <->  U  =  (
( p  .\/  q
)  ./\  W )
) )
3731, 14, 35, 36syl3anc 1187 . . . . . . . . . . 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 1174 . . . . . . . 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 1153 . . . . . . 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 2628 . . . . . 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 1158 . . . 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 2628 . 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 939    = wceq 1619    e. wcel 1621    =/= wne 2421   E.wrex 2519   class class class wbr 3997   ` cfv 4673  (class class class)co 5792   Basecbs 13110   lecple 13177   joincjn 14040   meetcmee 14041   Latclat 14113   Atomscatm 28620   AtLatcal 28621   HLchlt 28707   LHypclh 29340
This theorem is referenced by:  cdlemf  29919
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-iota 6225  df-undef 6264  df-riota 6272  df-poset 14042  df-plt 14054  df-lub 14070  df-glb 14071  df-join 14072  df-meet 14073  df-p0 14107  df-p1 14108  df-lat 14114  df-clat 14176  df-oposet 28533  df-ol 28535  df-oml 28536  df-covers 28623  df-ats 28624  df-atl 28655  df-cvlat 28679  df-hlat 28708  df-lhyp 29344
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