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Theorem pexmidlem3N 34768
Description: Lemma for pexmidN 34765. Use atom exchange hlatexch1 34191 to swap  p and  q. (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
pexmidlem3N  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X ) )  /\  q  .<_  ( r 
.\/  p ) )  ->  p  e.  ( X  .+  (  ._|_  `  X ) ) )

Proof of Theorem pexmidlem3N
StepHypRef Expression
1 simp1 996 . 2  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X ) )  /\  q  .<_  ( r 
.\/  p ) )  ->  ( K  e.  HL  /\  X  C_  A  /\  p  e.  A
) )
2 simp2l 1022 . 2  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X ) )  /\  q  .<_  ( r 
.\/  p ) )  ->  r  e.  X
)
3 simp2r 1023 . 2  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X ) )  /\  q  .<_  ( r 
.\/  p ) )  ->  q  e.  ( 
._|_  `  X ) )
4 simpl1 999 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X ) ) )  ->  K  e.  HL )
5 simpl2 1000 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X ) ) )  ->  X  C_  A
)
6 pexmidlem.a . . . . . . 7  |-  A  =  ( Atoms `  K )
7 pexmidlem.o . . . . . . 7  |-  ._|_  =  ( _|_P `  K
)
86, 7polssatN 34704 . . . . . 6  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
(  ._|_  `  X )  C_  A )
94, 5, 8syl2anc 661 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X ) ) )  ->  (  ._|_  `  X )  C_  A
)
10 simprr 756 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X ) ) )  ->  q  e.  (  ._|_  `  X )
)
119, 10sseldd 3505 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X ) ) )  ->  q  e.  A )
12 simpl3 1001 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X ) ) )  ->  p  e.  A )
13 simprl 755 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X ) ) )  ->  r  e.  X )
145, 13sseldd 3505 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X ) ) )  ->  r  e.  A )
15 pexmidlem.l . . . . . 6  |-  .<_  =  ( le `  K )
16 pexmidlem.j . . . . . 6  |-  .\/  =  ( join `  K )
17 pexmidlem.p . . . . . 6  |-  .+  =  ( +P `  K
)
18 pexmidlem.m . . . . . 6  |-  M  =  ( X  .+  {
p } )
1915, 16, 6, 17, 7, 18pexmidlem1N 34766 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X ) ) )  ->  q  =/=  r )
20193adantl3 1154 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X ) ) )  ->  q  =/=  r )
2115, 16, 6hlatexch1 34191 . . . 4  |-  ( ( K  e.  HL  /\  ( q  e.  A  /\  p  e.  A  /\  r  e.  A
)  /\  q  =/=  r )  ->  (
q  .<_  ( r  .\/  p )  ->  p  .<_  ( r  .\/  q
) ) )
224, 11, 12, 14, 20, 21syl131anc 1241 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X ) ) )  ->  ( q  .<_  ( r  .\/  p
)  ->  p  .<_  ( r  .\/  q ) ) )
23223impia 1193 . 2  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X ) )  /\  q  .<_  ( r 
.\/  p ) )  ->  p  .<_  ( r 
.\/  q ) )
2415, 16, 6, 17, 7, 18pexmidlem2N 34767 . 2  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X )  /\  p  .<_  ( r  .\/  q ) ) )  ->  p  e.  ( X  .+  (  ._|_  `  X ) ) )
251, 2, 3, 23, 24syl13anc 1230 1  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X ) )  /\  q  .<_  ( r 
.\/  p ) )  ->  p  e.  ( X  .+  (  ._|_  `  X ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662    C_ wss 3476   {csn 4027   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   lecple 14558   joincjn 15427   Atomscatm 34060   HLchlt 34147   +Pcpadd 34591   _|_PcpolN 34698
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 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-riotaBAD 33756
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-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-1st 6781  df-2nd 6782  df-undef 6999  df-poset 15429  df-plt 15441  df-lub 15457  df-glb 15458  df-join 15459  df-meet 15460  df-p0 15522  df-p1 15523  df-lat 15529  df-clat 15591  df-oposet 33973  df-ol 33975  df-oml 33976  df-covers 34063  df-ats 34064  df-atl 34095  df-cvlat 34119  df-hlat 34148  df-psubsp 34299  df-pmap 34300  df-padd 34592  df-polarityN 34699
This theorem is referenced by:  pexmidlem4N  34769
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