Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pexmidlem4N Structured version   Unicode version

Theorem pexmidlem4N 35798
Description: Lemma for pexmidN 35794. (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 999 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( X  =/=  (/)  /\  q  e.  ( (  ._|_  `  X
)  i^i  M )
) )  ->  K  e.  HL )
2 hllat 35189 . . . 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 1000 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( X  =/=  (/)  /\  q  e.  ( (  ._|_  `  X
)  i^i  M )
) )  ->  X  C_  A )
5 simpl3 1001 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( X  =/=  (/)  /\  q  e.  ( (  ._|_  `  X
)  i^i  M )
) )  ->  p  e.  A )
6 simprl 756 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( X  =/=  (/)  /\  q  e.  ( (  ._|_  `  X
)  i^i  M )
) )  ->  X  =/=  (/) )
7 inss2 3715 . . . . . 6  |-  ( ( 
._|_  `  X )  i^i 
M )  C_  M
87sseli 3495 . . . . 5  |-  ( q  e.  ( (  ._|_  `  X )  i^i  M
)  ->  q  e.  M )
9 pexmidlem.m . . . . 5  |-  M  =  ( X  .+  {
p } )
108, 9syl6eleq 2555 . . . 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 35630 . . 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 1236 . 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 996 . . . . . 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 1024 . . . . . 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 3714 . . . . . . 7  |-  ( ( 
._|_  `  X )  i^i 
M )  C_  (  ._|_  `  X )
21 simp2r 1023 . . . . . . 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 3497 . . . . . 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 1025 . . . . . 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 35797 . . . . . 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 1233 . . . . 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 1198 . . . 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 2947 . 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 973    = wceq 1395    e. wcel 1819    =/= wne 2652   E.wrex 2808    i^i cin 3470    C_ wss 3471   (/)c0 3793   {csn 4032   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   lecple 14718   joincjn 15699   Latclat 15801   Atomscatm 35089   HLchlt 35176   +Pcpadd 35620   _|_PcpolN 35727
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-riotaBAD 34785
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-undef 7020  df-preset 15683  df-poset 15701  df-plt 15714  df-lub 15730  df-glb 15731  df-join 15732  df-meet 15733  df-p0 15795  df-p1 15796  df-lat 15802  df-clat 15864  df-oposet 35002  df-ol 35004  df-oml 35005  df-covers 35092  df-ats 35093  df-atl 35124  df-cvlat 35148  df-hlat 35177  df-psubsp 35328  df-pmap 35329  df-padd 35621  df-polarityN 35728
This theorem is referenced by:  pexmidlem5N  35799
  Copyright terms: Public domain W3C validator