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Theorem cdlemg5 31402
Description: TODO: Is there a simpler more direct proof, that could be placed earlier e.g. near lhpexle 30802? TODO: The  .\/ hypothesis is unused. FIX COMMENT (Contributed by NM, 26-Apr-2013.)
Hypotheses
Ref Expression
cdlemg5.l  |-  .<_  =  ( le `  K )
cdlemg5.j  |-  .\/  =  ( join `  K )
cdlemg5.a  |-  A  =  ( Atoms `  K )
cdlemg5.h  |-  H  =  ( LHyp `  K
)
Assertion
Ref Expression
cdlemg5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  E. q  e.  A  ( P  =/=  q  /\  -.  q  .<_  W ) )
Distinct variable groups:    A, q    H, q    K, q    .<_ , q    P, q    W, q
Allowed substitution hint:    .\/ ( q)

Proof of Theorem cdlemg5
Dummy variable  r is distinct from all other variables.
StepHypRef Expression
1 cdlemg5.l . . . 4  |-  .<_  =  ( le `  K )
2 cdlemg5.a . . . 4  |-  A  =  ( Atoms `  K )
3 cdlemg5.h . . . 4  |-  H  =  ( LHyp `  K
)
41, 2, 3lhpexle 30802 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. r  e.  A  r  .<_  W )
54adantr 452 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  E. r  e.  A  r  .<_  W )
6 simpll 731 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( r  e.  A  /\  r  .<_  W ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
7 simpr 448 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( r  e.  A  /\  r  .<_  W ) )  -> 
( r  e.  A  /\  r  .<_  W ) )
8 simplr 732 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( r  e.  A  /\  r  .<_  W ) )  -> 
( P  e.  A  /\  -.  P  .<_  W ) )
9 cdlemg5.j . . . . 5  |-  .\/  =  ( join `  K )
101, 9, 2, 3cdlemf1 31358 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( r  e.  A  /\  r  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  E. q  e.  A  ( P  =/=  q  /\  -.  q  .<_  W  /\  r  .<_  ( P  .\/  q ) ) )
116, 7, 8, 10syl3anc 1184 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( r  e.  A  /\  r  .<_  W ) )  ->  E. q  e.  A  ( P  =/=  q  /\  -.  q  .<_  W  /\  r  .<_  ( P  .\/  q ) ) )
12 3simpa 954 . . . 4  |-  ( ( P  =/=  q  /\  -.  q  .<_  W  /\  r  .<_  ( P  .\/  q ) )  -> 
( P  =/=  q  /\  -.  q  .<_  W ) )
1312reximi 2813 . . 3  |-  ( E. q  e.  A  ( P  =/=  q  /\  -.  q  .<_  W  /\  r  .<_  ( P  .\/  q ) )  ->  E. q  e.  A  ( P  =/=  q  /\  -.  q  .<_  W ) )
1411, 13syl 16 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( r  e.  A  /\  r  .<_  W ) )  ->  E. q  e.  A  ( P  =/=  q  /\  -.  q  .<_  W ) )
155, 14rexlimddv 2834 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  E. q  e.  A  ( P  =/=  q  /\  -.  q  .<_  W ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   E.wrex 2706   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   lecple 13536   joincjn 14401   Atomscatm 30061   HLchlt 30148   LHypclh 30781
This theorem is referenced by:  cdlemb3  31403
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-undef 6543  df-riota 6549  df-poset 14403  df-plt 14415  df-lub 14431  df-glb 14432  df-join 14433  df-meet 14434  df-p0 14468  df-p1 14469  df-lat 14475  df-clat 14537  df-oposet 29974  df-ol 29976  df-oml 29977  df-covers 30064  df-ats 30065  df-atl 30096  df-cvlat 30120  df-hlat 30149  df-lhyp 30785
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