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Theorem pexmidlem4N 29535
Description: Lemma for pexmidN 29531. (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 958 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( X  =/=  (/)  /\  q  e.  ( (  ._|_  `  X
)  i^i  M )
) )  ->  K  e.  HL )
2 hllat 28926 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
31, 2syl 15 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( X  =/=  (/)  /\  q  e.  ( (  ._|_  `  X
)  i^i  M )
) )  ->  K  e.  Lat )
4 simpl2 959 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( X  =/=  (/)  /\  q  e.  ( (  ._|_  `  X
)  i^i  M )
) )  ->  X  C_  A )
5 simpl3 960 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( X  =/=  (/)  /\  q  e.  ( (  ._|_  `  X
)  i^i  M )
) )  ->  p  e.  A )
6 simprl 732 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( X  =/=  (/)  /\  q  e.  ( (  ._|_  `  X
)  i^i  M )
) )  ->  X  =/=  (/) )
7 inss2 3390 . . . . . 6  |-  ( ( 
._|_  `  X )  i^i 
M )  C_  M
87sseli 3176 . . . . 5  |-  ( q  e.  ( (  ._|_  `  X )  i^i  M
)  ->  q  e.  M )
9 pexmidlem.m . . . . 5  |-  M  =  ( X  .+  {
p } )
108, 9syl6eleq 2373 . . . 4  |-  ( q  e.  ( (  ._|_  `  X )  i^i  M
)  ->  q  e.  ( X  .+  { p } ) )
1110ad2antll 709 . . 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 29367 . . 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 1190 . 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 955 . . . . . 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 983 . . . . . 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 3389 . . . . . . 7  |-  ( ( 
._|_  `  X )  i^i 
M )  C_  (  ._|_  `  X )
21 simp2r 982 . . . . . . 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 3178 . . . . . 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 984 . . . . . 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 29534 . . . . . 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 1187 . . . . 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 1153 . . . 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 425 . . 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 2666 . 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 14 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544    i^i cin 3151    C_ wss 3152   (/)c0 3455   {csn 3640   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   lecple 13215   joincjn 14078   Latclat 14151   Atomscatm 28826   HLchlt 28913   + Pcpadd 29357   _|_ PcpolN 29464
This theorem is referenced by:  pexmidlem5N  29536
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-oposet 28739  df-ol 28741  df-oml 28742  df-covers 28829  df-ats 28830  df-atl 28861  df-cvlat 28885  df-hlat 28914  df-psubsp 29065  df-pmap 29066  df-padd 29358  df-polarityN 29465
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