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Theorem pexmidlem4N 33713
Description: Lemma for pexmidN 33709. (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 991 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( X  =/=  (/)  /\  q  e.  ( (  ._|_  `  X
)  i^i  M )
) )  ->  K  e.  HL )
2 hllat 33104 . . . 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 992 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( X  =/=  (/)  /\  q  e.  ( (  ._|_  `  X
)  i^i  M )
) )  ->  X  C_  A )
5 simpl3 993 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( X  =/=  (/)  /\  q  e.  ( (  ._|_  `  X
)  i^i  M )
) )  ->  p  e.  A )
6 simprl 755 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( X  =/=  (/)  /\  q  e.  ( (  ._|_  `  X
)  i^i  M )
) )  ->  X  =/=  (/) )
7 inss2 3592 . . . . . 6  |-  ( ( 
._|_  `  X )  i^i 
M )  C_  M
87sseli 3373 . . . . 5  |-  ( q  e.  ( (  ._|_  `  X )  i^i  M
)  ->  q  e.  M )
9 pexmidlem.m . . . . 5  |-  M  =  ( X  .+  {
p } )
108, 9syl6eleq 2533 . . . 4  |-  ( q  e.  ( (  ._|_  `  X )  i^i  M
)  ->  q  e.  ( X  .+  { p } ) )
1110ad2antll 728 . . 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 33545 . . 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 1226 . 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 988 . . . . . 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 1016 . . . . . 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 3591 . . . . . . 7  |-  ( ( 
._|_  `  X )  i^i 
M )  C_  (  ._|_  `  X )
21 simp2r 1015 . . . . . . 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 3375 . . . . . 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 1017 . . . . . 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 33712 . . . . . 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 1223 . . . . 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 1189 . . . 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 ) ) ) )
2827expd 436 . . 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 2861 . 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2620   E.wrex 2737    i^i cin 3348    C_ wss 3349   (/)c0 3658   {csn 3898   class class class wbr 4313   ` cfv 5439  (class class class)co 6112   lecple 14266   joincjn 15135   Latclat 15236   Atomscatm 33004   HLchlt 33091   +Pcpadd 33535   _|_PcpolN 33642
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-riotaBAD 32700
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-iun 4194  df-iin 4195  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-1st 6598  df-2nd 6599  df-undef 6813  df-poset 15137  df-plt 15149  df-lub 15165  df-glb 15166  df-join 15167  df-meet 15168  df-p0 15230  df-p1 15231  df-lat 15237  df-clat 15299  df-oposet 32917  df-ol 32919  df-oml 32920  df-covers 33007  df-ats 33008  df-atl 33039  df-cvlat 33063  df-hlat 33092  df-psubsp 33243  df-pmap 33244  df-padd 33536  df-polarityN 33643
This theorem is referenced by:  pexmidlem5N  33714
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