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Theorem osumcllem7N 33236
Description: Lemma for osumclN 33241. (Contributed by NM, 24-Mar-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
osumcllem.l  |-  .<_  =  ( le `  K )
osumcllem.j  |-  .\/  =  ( join `  K )
osumcllem.a  |-  A  =  ( Atoms `  K )
osumcllem.p  |-  .+  =  ( +P `  K
)
osumcllem.o  |-  ._|_  =  ( _|_P `  K
)
osumcllem.c  |-  C  =  ( PSubCl `  K )
osumcllem.m  |-  M  =  ( X  .+  {
p } )
osumcllem.u  |-  U  =  (  ._|_  `  (  ._|_  `  ( X  .+  Y
) ) )
Assertion
Ref Expression
osumcllem7N  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  ( X  C_  (  ._|_  `  Y )  /\  X  =/=  (/)  /\  p  e.  A )  /\  q  e.  ( Y  i^i  M
) )  ->  p  e.  ( X  .+  Y
) )
Distinct variable groups:    A, q    K, q    M, q    ._|_ , q    .+ , q    X, q    Y, q    q, p
Allowed substitution hints:    A( p)    C( q, p)    .+ ( p)    U( q, p)    .\/ ( q, p)    K( p)    .<_ ( q, p)    M( p)    ._|_ ( p)    X( p)    Y( p)

Proof of Theorem osumcllem7N
Dummy variable  r is distinct from all other variables.
StepHypRef Expression
1 simp11 1035 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  ( X  C_  (  ._|_  `  Y )  /\  X  =/=  (/)  /\  p  e.  A )  /\  q  e.  ( Y  i^i  M
) )  ->  K  e.  HL )
2 hllat 32638 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
31, 2syl 17 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  ( X  C_  (  ._|_  `  Y )  /\  X  =/=  (/)  /\  p  e.  A )  /\  q  e.  ( Y  i^i  M
) )  ->  K  e.  Lat )
4 simp12 1036 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  ( X  C_  (  ._|_  `  Y )  /\  X  =/=  (/)  /\  p  e.  A )  /\  q  e.  ( Y  i^i  M
) )  ->  X  C_  A )
5 simp23 1040 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  ( X  C_  (  ._|_  `  Y )  /\  X  =/=  (/)  /\  p  e.  A )  /\  q  e.  ( Y  i^i  M
) )  ->  p  e.  A )
6 simp22 1039 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  ( X  C_  (  ._|_  `  Y )  /\  X  =/=  (/)  /\  p  e.  A )  /\  q  e.  ( Y  i^i  M
) )  ->  X  =/=  (/) )
7 inss2 3689 . . . . . 6  |-  ( Y  i^i  M )  C_  M
87sseli 3466 . . . . 5  |-  ( q  e.  ( Y  i^i  M )  ->  q  e.  M )
983ad2ant3 1028 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  ( X  C_  (  ._|_  `  Y )  /\  X  =/=  (/)  /\  p  e.  A )  /\  q  e.  ( Y  i^i  M
) )  ->  q  e.  M )
10 osumcllem.m . . . 4  |-  M  =  ( X  .+  {
p } )
119, 10syl6eleq 2527 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  ( X  C_  (  ._|_  `  Y )  /\  X  =/=  (/)  /\  p  e.  A )  /\  q  e.  ( Y  i^i  M
) )  ->  q  e.  ( X  .+  {
p } ) )
12 osumcllem.l . . . 4  |-  .<_  =  ( le `  K )
13 osumcllem.j . . . 4  |-  .\/  =  ( join `  K )
14 osumcllem.a . . . 4  |-  A  =  ( Atoms `  K )
15 osumcllem.p . . . 4  |-  .+  =  ( +P `  K
)
1612, 13, 14, 15elpaddatiN 33079 . . 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 1272 . 2  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  ( X  C_  (  ._|_  `  Y )  /\  X  =/=  (/)  /\  p  e.  A )  /\  q  e.  ( Y  i^i  M
) )  ->  E. r  e.  X  q  .<_  ( r  .\/  p ) )
18 simp11 1035 . . . 4  |-  ( ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A
)  /\  ( X  C_  (  ._|_  `  Y )  /\  X  =/=  (/)  /\  p  e.  A )  /\  q  e.  ( Y  i^i  M
) )  /\  r  e.  X  /\  q  .<_  ( r  .\/  p
) )  ->  ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A ) )
19 simp121 1137 . . . 4  |-  ( ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A
)  /\  ( X  C_  (  ._|_  `  Y )  /\  X  =/=  (/)  /\  p  e.  A )  /\  q  e.  ( Y  i^i  M
) )  /\  r  e.  X  /\  q  .<_  ( r  .\/  p
) )  ->  X  C_  (  ._|_  `  Y ) )
20 simp123 1139 . . . 4  |-  ( ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A
)  /\  ( X  C_  (  ._|_  `  Y )  /\  X  =/=  (/)  /\  p  e.  A )  /\  q  e.  ( Y  i^i  M
) )  /\  r  e.  X  /\  q  .<_  ( r  .\/  p
) )  ->  p  e.  A )
21 simp2 1006 . . . 4  |-  ( ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A
)  /\  ( X  C_  (  ._|_  `  Y )  /\  X  =/=  (/)  /\  p  e.  A )  /\  q  e.  ( Y  i^i  M
) )  /\  r  e.  X  /\  q  .<_  ( r  .\/  p
) )  ->  r  e.  X )
22 inss1 3688 . . . . 5  |-  ( Y  i^i  M )  C_  Y
23 simp13 1037 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A
)  /\  ( X  C_  (  ._|_  `  Y )  /\  X  =/=  (/)  /\  p  e.  A )  /\  q  e.  ( Y  i^i  M
) )  /\  r  e.  X  /\  q  .<_  ( r  .\/  p
) )  ->  q  e.  ( Y  i^i  M
) )
2422, 23sseldi 3468 . . . 4  |-  ( ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A
)  /\  ( X  C_  (  ._|_  `  Y )  /\  X  =/=  (/)  /\  p  e.  A )  /\  q  e.  ( Y  i^i  M
) )  /\  r  e.  X  /\  q  .<_  ( r  .\/  p
) )  ->  q  e.  Y )
25 simp3 1007 . . . 4  |-  ( ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A
)  /\  ( X  C_  (  ._|_  `  Y )  /\  X  =/=  (/)  /\  p  e.  A )  /\  q  e.  ( Y  i^i  M
) )  /\  r  e.  X  /\  q  .<_  ( r  .\/  p
) )  ->  q  .<_  ( r  .\/  p
) )
26 osumcllem.o . . . . 5  |-  ._|_  =  ( _|_P `  K
)
27 osumcllem.c . . . . 5  |-  C  =  ( PSubCl `  K )
28 osumcllem.u . . . . 5  |-  U  =  (  ._|_  `  (  ._|_  `  ( X  .+  Y
) ) )
2912, 13, 14, 15, 26, 27, 10, 28osumcllem6N 33235 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  ( X  C_  (  ._|_  `  Y )  /\  p  e.  A )  /\  (
r  e.  X  /\  q  e.  Y  /\  q  .<_  ( r  .\/  p ) ) )  ->  p  e.  ( X  .+  Y ) )
3018, 19, 20, 21, 24, 25, 29syl123anc 1281 . . 3  |-  ( ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A
)  /\  ( X  C_  (  ._|_  `  Y )  /\  X  =/=  (/)  /\  p  e.  A )  /\  q  e.  ( Y  i^i  M
) )  /\  r  e.  X  /\  q  .<_  ( r  .\/  p
) )  ->  p  e.  ( X  .+  Y
) )
3130rexlimdv3a 2926 . 2  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  ( X  C_  (  ._|_  `  Y )  /\  X  =/=  (/)  /\  p  e.  A )  /\  q  e.  ( Y  i^i  M
) )  ->  ( E. r  e.  X  q  .<_  ( r  .\/  p )  ->  p  e.  ( X  .+  Y
) ) )
3217, 31mpd 15 1  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  ( X  C_  (  ._|_  `  Y )  /\  X  =/=  (/)  /\  p  e.  A )  /\  q  e.  ( Y  i^i  M
) )  ->  p  e.  ( X  .+  Y
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625   E.wrex 2783    i^i cin 3441    C_ wss 3442   (/)c0 3767   {csn 4002   class class class wbr 4426   ` cfv 5601  (class class class)co 6305   lecple 15159   joincjn 16140   Latclat 16242   Atomscatm 32538   HLchlt 32625   +Pcpadd 33069   _|_PcpolN 33176   PSubClcpscN 33208
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-riotaBAD 32234
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-iin 4305  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-undef 7028  df-preset 16124  df-poset 16142  df-plt 16155  df-lub 16171  df-glb 16172  df-join 16173  df-meet 16174  df-p0 16236  df-p1 16237  df-lat 16243  df-clat 16305  df-oposet 32451  df-ol 32453  df-oml 32454  df-covers 32541  df-ats 32542  df-atl 32573  df-cvlat 32597  df-hlat 32626  df-pmap 32778  df-padd 33070  df-polarityN 33177
This theorem is referenced by:  osumcllem8N  33237
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