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Theorem pexmidlem2N 34644
Description: Lemma for pexmidN 34642. (Contributed by NM, 2-Feb-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
pexmidlem.l  |-  .<_  =  ( le `  K )
pexmidlem.j  |-  .\/  =  ( join `  K )
pexmidlem.a  |-  A  =  ( Atoms `  K )
pexmidlem.p  |-  .+  =  ( +P `  K
)
pexmidlem.o  |-  ._|_  =  ( _|_P `  K
)
pexmidlem.m  |-  M  =  ( X  .+  {
p } )
Assertion
Ref Expression
pexmidlem2N  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X )  /\  p  .<_  ( r  .\/  q ) ) )  ->  p  e.  ( X  .+  (  ._|_  `  X ) ) )

Proof of Theorem pexmidlem2N
StepHypRef Expression
1 simpl1 994 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X )  /\  p  .<_  ( r  .\/  q ) ) )  ->  K  e.  HL )
2 hllat 34037 . . 3  |-  ( K  e.  HL  ->  K  e.  Lat )
31, 2syl 16 . 2  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X )  /\  p  .<_  ( r  .\/  q ) ) )  ->  K  e.  Lat )
4 simpl2 995 . 2  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X )  /\  p  .<_  ( r  .\/  q ) ) )  ->  X  C_  A
)
5 pexmidlem.a . . . 4  |-  A  =  ( Atoms `  K )
6 pexmidlem.o . . . 4  |-  ._|_  =  ( _|_P `  K
)
75, 6polssatN 34581 . . 3  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
(  ._|_  `  X )  C_  A )
81, 4, 7syl2anc 661 . 2  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X )  /\  p  .<_  ( r  .\/  q ) ) )  ->  (  ._|_  `  X
)  C_  A )
9 simpr1 997 . 2  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X )  /\  p  .<_  ( r  .\/  q ) ) )  ->  r  e.  X
)
10 simpr2 998 . 2  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X )  /\  p  .<_  ( r  .\/  q ) ) )  ->  q  e.  ( 
._|_  `  X ) )
11 simpl3 996 . 2  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X )  /\  p  .<_  ( r  .\/  q ) ) )  ->  p  e.  A
)
12 simpr3 999 . 2  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X )  /\  p  .<_  ( r  .\/  q ) ) )  ->  p  .<_  ( r 
.\/  q ) )
13 pexmidlem.l . . 3  |-  .<_  =  ( le `  K )
14 pexmidlem.j . . 3  |-  .\/  =  ( join `  K )
15 pexmidlem.p . . 3  |-  .+  =  ( +P `  K
)
1613, 14, 5, 15elpaddri 34475 . 2  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  (  ._|_  `  X )  C_  A )  /\  (
r  e.  X  /\  q  e.  (  ._|_  `  X ) )  /\  ( p  e.  A  /\  p  .<_  ( r 
.\/  q ) ) )  ->  p  e.  ( X  .+  (  ._|_  `  X ) ) )
173, 4, 8, 9, 10, 11, 12, 16syl322anc 1251 1  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X )  /\  p  .<_  ( r  .\/  q ) ) )  ->  p  e.  ( X  .+  (  ._|_  `  X ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762    C_ wss 3471   {csn 4022   class class class wbr 4442   ` cfv 5581  (class class class)co 6277   lecple 14553   joincjn 15422   Latclat 15523   Atomscatm 33937   HLchlt 34024   +Pcpadd 34468   _|_PcpolN 34575
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-riotaBAD 33633
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-iin 4323  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6776  df-2nd 6777  df-undef 6994  df-poset 15424  df-lub 15452  df-glb 15453  df-join 15454  df-meet 15455  df-p1 15518  df-lat 15524  df-clat 15586  df-oposet 33850  df-ol 33852  df-oml 33853  df-ats 33941  df-atl 33972  df-cvlat 33996  df-hlat 34025  df-psubsp 34176  df-pmap 34177  df-padd 34469  df-polarityN 34576
This theorem is referenced by:  pexmidlem3N  34645
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