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Theorem pexmidlem3N 33935
Description: Lemma for pexmidN 33932. Use atom exchange hlatexch1 33358 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 988 . 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 1014 . 2  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X ) )  /\  q  .<_  ( r 
.\/  p ) )  ->  r  e.  X
)
3 simp2r 1015 . 2  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X ) )  /\  q  .<_  ( r 
.\/  p ) )  ->  q  e.  ( 
._|_  `  X ) )
4 simpl1 991 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X ) ) )  ->  K  e.  HL )
5 simpl2 992 . . . . . 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 33871 . . . . . 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 3460 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X ) ) )  ->  q  e.  A )
12 simpl3 993 . . . 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 3460 . . . 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 33933 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X ) ) )  ->  q  =/=  r )
20193adantl3 1146 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X ) ) )  ->  q  =/=  r )
2115, 16, 6hlatexch1 33358 . . . 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 1232 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X ) ) )  ->  ( q  .<_  ( r  .\/  p
)  ->  p  .<_  ( r  .\/  q ) ) )
23223impia 1185 . 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 33934 . 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 1221 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 965    = wceq 1370    e. wcel 1758    =/= wne 2645    C_ wss 3431   {csn 3980   class class class wbr 4395   ` cfv 5521  (class class class)co 6195   lecple 14359   joincjn 15228   Atomscatm 33227   HLchlt 33314   +Pcpadd 33758   _|_PcpolN 33865
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-riotaBAD 32923
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-iun 4276  df-iin 4277  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-1st 6682  df-2nd 6683  df-undef 6897  df-poset 15230  df-plt 15242  df-lub 15258  df-glb 15259  df-join 15260  df-meet 15261  df-p0 15323  df-p1 15324  df-lat 15330  df-clat 15392  df-oposet 33140  df-ol 33142  df-oml 33143  df-covers 33230  df-ats 33231  df-atl 33262  df-cvlat 33286  df-hlat 33315  df-psubsp 33466  df-pmap 33467  df-padd 33759  df-polarityN 33866
This theorem is referenced by:  pexmidlem4N  33936
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