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Theorem osumcllem7N 30456
Description: Lemma for osumclN 30461. (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 987 . . . 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 29858 . . . 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 988 . . 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 992 . . 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 991 . . 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 3530 . . . . . 6  |-  ( Y  i^i  M )  C_  M
87sseli 3312 . . . . 5  |-  ( q  e.  ( Y  i^i  M )  ->  q  e.  M )
983ad2ant3 980 . . . 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 2502 . . 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 30299 . . 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 1192 . 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 987 . . . 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 1089 . . . 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 1091 . . . 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 958 . . . 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 3529 . . . . 5  |-  ( Y  i^i  M )  C_  Y
23 simp13 989 . . . . 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 3314 . . . 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 959 . . . 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 30455 . . . 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 1201 . . 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 2800 . 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 set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2575   E.wrex 2675    i^i cin 3287    C_ wss 3288   (/)c0 3596   {csn 3782   class class class wbr 4180   ` cfv 5421  (class class class)co 6048   lecple 13499   joincjn 14364   Latclat 14437   Atomscatm 29758   HLchlt 29845   + Pcpadd 30289   _|_ PcpolN 30396   PSubClcpscN 30428
This theorem is referenced by:  osumcllem8N  30457
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-iin 4064  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-undef 6510  df-riota 6516  df-poset 14366  df-plt 14378  df-lub 14394  df-glb 14395  df-join 14396  df-meet 14397  df-p0 14431  df-p1 14432  df-lat 14438  df-clat 14500  df-oposet 29671  df-ol 29673  df-oml 29674  df-covers 29761  df-ats 29762  df-atl 29793  df-cvlat 29817  df-hlat 29846  df-pmap 29998  df-padd 30290  df-polarityN 30397
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