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Theorem cdlemg5 34249
Description: TODO: Is there a simpler more direct proof, that could be placed earlier e.g. near lhpexle 33649? 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 33649 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. r  e.  A  r  .<_  W )
54adantr 465 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  E. r  e.  A  r  .<_  W )
6 simpll 753 . . . 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 461 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( r  e.  A  /\  r  .<_  W ) )  -> 
( r  e.  A  /\  r  .<_  W ) )
8 simplr 754 . . . 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 34205 . . . 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 1218 . . 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 985 . . . 4  |-  ( ( P  =/=  q  /\  -.  q  .<_  W  /\  r  .<_  ( P  .\/  q ) )  -> 
( P  =/=  q  /\  -.  q  .<_  W ) )
1312reximi 2823 . . 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 2845 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  E. q  e.  A  ( P  =/=  q  /\  -.  q  .<_  W ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2606   E.wrex 2716   class class class wbr 4292   ` cfv 5418  (class class class)co 6091   lecple 14245   joincjn 15114   Atomscatm 32908   HLchlt 32995   LHypclh 33628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-poset 15116  df-plt 15128  df-lub 15144  df-glb 15145  df-join 15146  df-meet 15147  df-p0 15209  df-p1 15210  df-lat 15216  df-clat 15278  df-oposet 32821  df-ol 32823  df-oml 32824  df-covers 32911  df-ats 32912  df-atl 32943  df-cvlat 32967  df-hlat 32996  df-lhyp 33632
This theorem is referenced by:  cdlemb3  34250
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