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Theorem pexmidlem3N 30087
Description: Lemma for pexmidN 30084. Use atom exchange hlatexch1 29510 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 957 . 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 983 . 2  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X ) )  /\  q  .<_  ( r 
.\/  p ) )  ->  r  e.  X
)
3 simp2r 984 . 2  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X ) )  /\  q  .<_  ( r 
.\/  p ) )  ->  q  e.  ( 
._|_  `  X ) )
4 simpl1 960 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X ) ) )  ->  K  e.  HL )
5 simpl2 961 . . . . . 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 30023 . . . . . 6  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
(  ._|_  `  X )  C_  A )
94, 5, 8syl2anc 643 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X ) ) )  ->  (  ._|_  `  X )  C_  A
)
10 simprr 734 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X ) ) )  ->  q  e.  (  ._|_  `  X )
)
119, 10sseldd 3293 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X ) ) )  ->  q  e.  A )
12 simpl3 962 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X ) ) )  ->  p  e.  A )
13 simprl 733 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X ) ) )  ->  r  e.  X )
145, 13sseldd 3293 . . . 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 30085 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X ) ) )  ->  q  =/=  r )
20193adantl3 1115 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X ) ) )  ->  q  =/=  r )
2115, 16, 6hlatexch1 29510 . . . 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 1197 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X ) ) )  ->  ( q  .<_  ( r  .\/  p
)  ->  p  .<_  ( r  .\/  q ) ) )
23223impia 1150 . 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 30086 . 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 1186 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 set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2551    C_ wss 3264   {csn 3758   class class class wbr 4154   ` cfv 5395  (class class class)co 6021   lecple 13464   joincjn 14329   Atomscatm 29379   HLchlt 29466   + Pcpadd 29910   _|_ PcpolN 30017
This theorem is referenced by:  pexmidlem4N  30088
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-iin 4039  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-undef 6480  df-riota 6486  df-poset 14331  df-plt 14343  df-lub 14359  df-glb 14360  df-join 14361  df-meet 14362  df-p0 14396  df-p1 14397  df-lat 14403  df-clat 14465  df-oposet 29292  df-ol 29294  df-oml 29295  df-covers 29382  df-ats 29383  df-atl 29414  df-cvlat 29438  df-hlat 29467  df-psubsp 29618  df-pmap 29619  df-padd 29911  df-polarityN 30018
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