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Theorem cdlemg5 30863
Description: TODO: Is there a simpler more direct proof, that could be placed earlier e.g. near lhpexle 30263? 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 30263 . . 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 30819 . . . . . 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 2726 . . . . 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 2742 . 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 1642    e. wcel 1710    =/= wne 2521   E.wrex 2620   class class class wbr 4104   ` cfv 5337  (class class class)co 5945   lecple 13312   joincjn 14177   Atomscatm 29522   HLchlt 29609   LHypclh 30242
This theorem is referenced by:  cdlemb3  30864
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-undef 6385  df-riota 6391  df-poset 14179  df-plt 14191  df-lub 14207  df-glb 14208  df-join 14209  df-meet 14210  df-p0 14244  df-p1 14245  df-lat 14251  df-clat 14313  df-oposet 29435  df-ol 29437  df-oml 29438  df-covers 29525  df-ats 29526  df-atl 29557  df-cvlat 29581  df-hlat 29610  df-lhyp 30246
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