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Theorem pexmidlem3N 32969
Description: Lemma for pexmidN 32966. Use atom exchange hlatexch1 32392 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 997 . 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 1023 . 2  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X ) )  /\  q  .<_  ( r 
.\/  p ) )  ->  r  e.  X
)
3 simp2r 1024 . 2  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X ) )  /\  q  .<_  ( r 
.\/  p ) )  ->  q  e.  ( 
._|_  `  X ) )
4 simpl1 1000 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X ) ) )  ->  K  e.  HL )
5 simpl2 1001 . . . . . 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 32905 . . . . . 6  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
(  ._|_  `  X )  C_  A )
94, 5, 8syl2anc 659 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X ) ) )  ->  (  ._|_  `  X )  C_  A
)
10 simprr 758 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X ) ) )  ->  q  e.  (  ._|_  `  X )
)
119, 10sseldd 3442 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X ) ) )  ->  q  e.  A )
12 simpl3 1002 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X ) ) )  ->  p  e.  A )
13 simprl 756 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X ) ) )  ->  r  e.  X )
145, 13sseldd 3442 . . . 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 32967 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X ) ) )  ->  q  =/=  r )
20193adantl3 1155 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X ) ) )  ->  q  =/=  r )
2115, 16, 6hlatexch1 32392 . . . 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 1243 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X ) ) )  ->  ( q  .<_  ( r  .\/  p
)  ->  p  .<_  ( r  .\/  q ) ) )
23223impia 1194 . 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 32968 . 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 1232 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 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598    C_ wss 3413   {csn 3971   class class class wbr 4394   ` cfv 5568  (class class class)co 6277   lecple 14914   joincjn 15895   Atomscatm 32261   HLchlt 32348   +Pcpadd 32792   _|_PcpolN 32899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-riotaBAD 31957
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-iin 4273  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6783  df-2nd 6784  df-undef 7004  df-preset 15879  df-poset 15897  df-plt 15910  df-lub 15926  df-glb 15927  df-join 15928  df-meet 15929  df-p0 15991  df-p1 15992  df-lat 15998  df-clat 16060  df-oposet 32174  df-ol 32176  df-oml 32177  df-covers 32264  df-ats 32265  df-atl 32296  df-cvlat 32320  df-hlat 32349  df-psubsp 32500  df-pmap 32501  df-padd 32793  df-polarityN 32900
This theorem is referenced by:  pexmidlem4N  32970
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