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Theorem cdlemg5 35802
Description: TODO: Is there a simpler more direct proof, that could be placed earlier e.g. near lhpexle 35202? 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 35202 . . 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 35758 . . . 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 1228 . . 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 993 . . . 4  |-  ( ( P  =/=  q  /\  -.  q  .<_  W  /\  r  .<_  ( P  .\/  q ) )  -> 
( P  =/=  q  /\  -.  q  .<_  W ) )
1312reximi 2935 . . 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 2963 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   E.wrex 2818   class class class wbr 4453   ` cfv 5594  (class class class)co 6295   lecple 14579   joincjn 15448   Atomscatm 34461   HLchlt 34548   LHypclh 35181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-poset 15450  df-plt 15462  df-lub 15478  df-glb 15479  df-join 15480  df-meet 15481  df-p0 15543  df-p1 15544  df-lat 15550  df-clat 15612  df-oposet 34374  df-ol 34376  df-oml 34377  df-covers 34464  df-ats 34465  df-atl 34496  df-cvlat 34520  df-hlat 34549  df-lhyp 35185
This theorem is referenced by:  cdlemb3  35803
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