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Theorem cdlemg5 30794
Description: TODO: Is there a simpler more direct proof, that could be placed earlier e.g. near lhpexle 30194? 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 30194 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. r  e.  A  r  .<_  W )
54adantr 451 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  E. r  e.  A  r  .<_  W )
6 simpll 730 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( r  e.  A  /\  r  .<_  W ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
7 simpr 447 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( r  e.  A  /\  r  .<_  W ) )  -> 
( r  e.  A  /\  r  .<_  W ) )
8 simplr 731 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( r  e.  A  /\  r  .<_  W ) )  -> 
( P  e.  A  /\  -.  P  .<_  W ) )
9 cdlemg5.j . . . . . . 7  |-  .\/  =  ( join `  K )
101, 9, 2, 3cdlemf1 30750 . . . . . 6  |-  ( ( ( 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 1182 . . . . 5  |-  ( ( ( ( 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 952 . . . . . 6  |-  ( ( P  =/=  q  /\  -.  q  .<_  W  /\  r  .<_  ( P  .\/  q ) )  -> 
( P  =/=  q  /\  -.  q  .<_  W ) )
1312reximi 2650 . . . . 5  |-  ( E. q  e.  A  ( P  =/=  q  /\  -.  q  .<_  W  /\  r  .<_  ( P  .\/  q ) )  ->  E. q  e.  A  ( P  =/=  q  /\  -.  q  .<_  W ) )
1411, 13syl 15 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( r  e.  A  /\  r  .<_  W ) )  ->  E. q  e.  A  ( P  =/=  q  /\  -.  q  .<_  W ) )
1514exp32 588 . . 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 ) ) ) )
1615rexlimdv 2666 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  -> 
( E. r  e.  A  r  .<_  W  ->  E. q  e.  A  ( P  =/=  q  /\  -.  q  .<_  W ) ) )
175, 16mpd 14 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   lecple 13215   joincjn 14078   Atomscatm 29453   HLchlt 29540   LHypclh 30173
This theorem is referenced by:  cdlemb3  30795
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-lhyp 30177
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