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Theorem pexmidlem4N 30707
Description: Lemma for pexmidN 30703. (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 30098 . . . 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 3554 . . . . . 6  |-  ( ( 
._|_  `  X )  i^i 
M )  C_  M
87sseli 3336 . . . . 5  |-  ( q  e.  ( (  ._|_  `  X )  i^i  M
)  ->  q  e.  M )
9 pexmidlem.m . . . . 5  |-  M  =  ( X  .+  {
p } )
108, 9syl6eleq 2525 . . . 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 30539 . . 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 3553 . . . . . . 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 3338 . . . . . 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 30706 . . . . . 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 2821 . 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 1652    e. wcel 1725    =/= wne 2598   E.wrex 2698    i^i cin 3311    C_ wss 3312   (/)c0 3620   {csn 3806   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   lecple 13528   joincjn 14393   Latclat 14466   Atomscatm 29998   HLchlt 30085   + Pcpadd 30529   _|_ PcpolN 30636
This theorem is referenced by:  pexmidlem5N  30708
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-poset 14395  df-plt 14407  df-lub 14423  df-glb 14424  df-join 14425  df-meet 14426  df-p0 14460  df-p1 14461  df-lat 14467  df-clat 14529  df-oposet 29911  df-ol 29913  df-oml 29914  df-covers 30001  df-ats 30002  df-atl 30033  df-cvlat 30057  df-hlat 30086  df-psubsp 30237  df-pmap 30238  df-padd 30530  df-polarityN 30637
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