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Theorem osumcllem7N 34776
Description: Lemma for osumclN 34781. (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 1026 . . . 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 34178 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
31, 2syl 16 . . 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 1027 . . 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 1031 . . 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 1030 . . 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 3719 . . . . . 6  |-  ( Y  i^i  M )  C_  M
87sseli 3500 . . . . 5  |-  ( q  e.  ( Y  i^i  M )  ->  q  e.  M )
983ad2ant3 1019 . . . 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 2565 . . 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 34619 . . 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  /\  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 1026 . . . 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 1128 . . . 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 1130 . . . 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 997 . . . 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 3718 . . . . 5  |-  ( Y  i^i  M )  C_  Y
23 simp13 1028 . . . . 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 3502 . . . 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 998 . . . 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 34775 . . . 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 1245 . . 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 2957 . 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   E.wrex 2815    i^i cin 3475    C_ wss 3476   (/)c0 3785   {csn 4027   class class class wbr 4447   ` cfv 5588  (class class class)co 6284   lecple 14562   joincjn 15431   Latclat 15532   Atomscatm 34078   HLchlt 34165   +Pcpadd 34609   _|_PcpolN 34716   PSubClcpscN 34748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-riotaBAD 33774
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-1st 6784  df-2nd 6785  df-undef 7002  df-poset 15433  df-plt 15445  df-lub 15461  df-glb 15462  df-join 15463  df-meet 15464  df-p0 15526  df-p1 15527  df-lat 15533  df-clat 15595  df-oposet 33991  df-ol 33993  df-oml 33994  df-covers 34081  df-ats 34082  df-atl 34113  df-cvlat 34137  df-hlat 34166  df-pmap 34318  df-padd 34610  df-polarityN 34717
This theorem is referenced by:  osumcllem8N  34777
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