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Theorem osumcllem7N 30151
Description: Lemma for osumclN 30156. (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 985 . . . 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 29553 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
31, 2syl 15 . . 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 986 . . 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 990 . . 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 989 . . 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 3390 . . . . . 6  |-  ( Y  i^i  M )  C_  M
87sseli 3176 . . . . 5  |-  ( q  e.  ( Y  i^i  M )  ->  q  e.  M )
983ad2ant3 978 . . . 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 2373 . . 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 29994 . . 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  /\  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 985 . . . 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 1087 . . . 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 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
) )  ->  p  e.  A )
21 simp2 956 . . . 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 3389 . . . . 5  |-  ( Y  i^i  M )  C_  Y
23 simp13 987 . . . . 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 3178 . . . 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 957 . . . 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 30150 . . . 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 1199 . . 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 2669 . 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 14 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 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 29453   HLchlt 29540   + Pcpadd 29984   _|_ PcpolN 30091   PSubClcpscN 30123
This theorem is referenced by:  osumcllem8N  30152
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 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-pmap 29693  df-padd 29985  df-polarityN 30092
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