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Theorem pexmidlem4N 30087
Description: Lemma for pexmidN 30083. (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
pexmidlem4N  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( X  =/=  (/)  /\  q  e.  ( (  ._|_  `  X
)  i^i  M )
) )  ->  p  e.  ( X  .+  (  ._|_  `  X ) ) )
Distinct variable groups:    A, q    K, q    M, q    ._|_ , q    .+ , q    X, q    q, p
Allowed substitution hints:    A( p)    .+ ( p)    .\/ ( q, p)    K( p)    .<_ ( q, p)    M( p)    ._|_ ( p)    X( p)

Proof of Theorem pexmidlem4N
Dummy variable  r is distinct from all other variables.
StepHypRef Expression
1 simpl1 960 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( X  =/=  (/)  /\  q  e.  ( (  ._|_  `  X
)  i^i  M )
) )  ->  K  e.  HL )
2 hllat 29478 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
31, 2syl 16 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( X  =/=  (/)  /\  q  e.  ( (  ._|_  `  X
)  i^i  M )
) )  ->  K  e.  Lat )
4 simpl2 961 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( X  =/=  (/)  /\  q  e.  ( (  ._|_  `  X
)  i^i  M )
) )  ->  X  C_  A )
5 simpl3 962 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( X  =/=  (/)  /\  q  e.  ( (  ._|_  `  X
)  i^i  M )
) )  ->  p  e.  A )
6 simprl 733 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( X  =/=  (/)  /\  q  e.  ( (  ._|_  `  X
)  i^i  M )
) )  ->  X  =/=  (/) )
7 inss2 3505 . . . . . 6  |-  ( ( 
._|_  `  X )  i^i 
M )  C_  M
87sseli 3287 . . . . 5  |-  ( q  e.  ( (  ._|_  `  X )  i^i  M
)  ->  q  e.  M )
9 pexmidlem.m . . . . 5  |-  M  =  ( X  .+  {
p } )
108, 9syl6eleq 2477 . . . 4  |-  ( q  e.  ( (  ._|_  `  X )  i^i  M
)  ->  q  e.  ( X  .+  { p } ) )
1110ad2antll 710 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( X  =/=  (/)  /\  q  e.  ( (  ._|_  `  X
)  i^i  M )
) )  ->  q  e.  ( X  .+  {
p } ) )
12 pexmidlem.l . . . 4  |-  .<_  =  ( le `  K )
13 pexmidlem.j . . . 4  |-  .\/  =  ( join `  K )
14 pexmidlem.a . . . 4  |-  A  =  ( Atoms `  K )
15 pexmidlem.p . . . 4  |-  .+  =  ( + P `  K
)
1612, 13, 14, 15elpaddatiN 29919 . . 3  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  p  e.  A )  /\  ( X  =/=  (/)  /\  q  e.  ( X  .+  {
p } ) ) )  ->  E. r  e.  X  q  .<_  ( r  .\/  p ) )
173, 4, 5, 6, 11, 16syl32anc 1192 . 2  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( X  =/=  (/)  /\  q  e.  ( (  ._|_  `  X
)  i^i  M )
) )  ->  E. r  e.  X  q  .<_  ( r  .\/  p ) )
18 simp1 957 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( X  =/=  (/)  /\  q  e.  ( (  ._|_  `  X
)  i^i  M )
)  /\  ( r  e.  X  /\  q  .<_  ( r  .\/  p
) ) )  -> 
( K  e.  HL  /\  X  C_  A  /\  p  e.  A )
)
19 simp3l 985 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( X  =/=  (/)  /\  q  e.  ( (  ._|_  `  X
)  i^i  M )
)  /\  ( r  e.  X  /\  q  .<_  ( r  .\/  p
) ) )  -> 
r  e.  X )
20 inss1 3504 . . . . . . 7  |-  ( ( 
._|_  `  X )  i^i 
M )  C_  (  ._|_  `  X )
21 simp2r 984 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( X  =/=  (/)  /\  q  e.  ( (  ._|_  `  X
)  i^i  M )
)  /\  ( r  e.  X  /\  q  .<_  ( r  .\/  p
) ) )  -> 
q  e.  ( ( 
._|_  `  X )  i^i 
M ) )
2220, 21sseldi 3289 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( X  =/=  (/)  /\  q  e.  ( (  ._|_  `  X
)  i^i  M )
)  /\  ( r  e.  X  /\  q  .<_  ( r  .\/  p
) ) )  -> 
q  e.  (  ._|_  `  X ) )
23 simp3r 986 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( X  =/=  (/)  /\  q  e.  ( (  ._|_  `  X
)  i^i  M )
)  /\  ( r  e.  X  /\  q  .<_  ( r  .\/  p
) ) )  -> 
q  .<_  ( r  .\/  p ) )
24 pexmidlem.o . . . . . . 7  |-  ._|_  =  ( _|_ P `  K
)
2512, 13, 14, 15, 24, 9pexmidlem3N 30086 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X ) )  /\  q  .<_  ( r 
.\/  p ) )  ->  p  e.  ( X  .+  (  ._|_  `  X ) ) )
2618, 19, 22, 23, 25syl121anc 1189 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( X  =/=  (/)  /\  q  e.  ( (  ._|_  `  X
)  i^i  M )
)  /\  ( r  e.  X  /\  q  .<_  ( r  .\/  p
) ) )  ->  p  e.  ( X  .+  (  ._|_  `  X
) ) )
27263expia 1155 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( X  =/=  (/)  /\  q  e.  ( (  ._|_  `  X
)  i^i  M )
) )  ->  (
( r  e.  X  /\  q  .<_  ( r 
.\/  p ) )  ->  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )
2827exp3a 426 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( X  =/=  (/)  /\  q  e.  ( (  ._|_  `  X
)  i^i  M )
) )  ->  (
r  e.  X  -> 
( q  .<_  ( r 
.\/  p )  ->  p  e.  ( X  .+  (  ._|_  `  X
) ) ) ) )
2928rexlimdv 2772 . 2  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( X  =/=  (/)  /\  q  e.  ( (  ._|_  `  X
)  i^i  M )
) )  ->  ( E. r  e.  X  q  .<_  ( r  .\/  p )  ->  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )
3017, 29mpd 15 1  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( X  =/=  (/)  /\  q  e.  ( (  ._|_  `  X
)  i^i  M )
) )  ->  p  e.  ( X  .+  (  ._|_  `  X ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2550   E.wrex 2650    i^i cin 3262    C_ wss 3263   (/)c0 3571   {csn 3757   class class class wbr 4153   ` cfv 5394  (class class class)co 6020   lecple 13463   joincjn 14328   Latclat 14401   Atomscatm 29378   HLchlt 29465   + Pcpadd 29909   _|_ PcpolN 30016
This theorem is referenced by:  pexmidlem5N  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 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-iin 4038  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-undef 6479  df-riota 6485  df-poset 14330  df-plt 14342  df-lub 14358  df-glb 14359  df-join 14360  df-meet 14361  df-p0 14395  df-p1 14396  df-lat 14402  df-clat 14464  df-oposet 29291  df-ol 29293  df-oml 29294  df-covers 29381  df-ats 29382  df-atl 29413  df-cvlat 29437  df-hlat 29466  df-psubsp 29617  df-pmap 29618  df-padd 29910  df-polarityN 30017
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