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Theorem osumcllem5N 30758
Description: Lemma for osumclN 30765. (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
osumcllem5N  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  A  /\  ( r  e.  X  /\  q  e.  Y  /\  p  .<_  ( r 
.\/  q ) ) )  ->  p  e.  ( X  .+  Y ) )

Proof of Theorem osumcllem5N
StepHypRef Expression
1 simp11 988 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  A  /\  ( r  e.  X  /\  q  e.  Y  /\  p  .<_  ( r 
.\/  q ) ) )  ->  K  e.  HL )
2 hllat 30162 . . 3  |-  ( K  e.  HL  ->  K  e.  Lat )
31, 2syl 16 . 2  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  A  /\  ( r  e.  X  /\  q  e.  Y  /\  p  .<_  ( r 
.\/  q ) ) )  ->  K  e.  Lat )
4 simp12 989 . 2  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  A  /\  ( r  e.  X  /\  q  e.  Y  /\  p  .<_  ( r 
.\/  q ) ) )  ->  X  C_  A
)
5 simp13 990 . 2  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  A  /\  ( r  e.  X  /\  q  e.  Y  /\  p  .<_  ( r 
.\/  q ) ) )  ->  Y  C_  A
)
6 simp31 994 . 2  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  A  /\  ( r  e.  X  /\  q  e.  Y  /\  p  .<_  ( r 
.\/  q ) ) )  ->  r  e.  X )
7 simp32 995 . 2  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  A  /\  ( r  e.  X  /\  q  e.  Y  /\  p  .<_  ( r 
.\/  q ) ) )  ->  q  e.  Y )
8 simp2 959 . 2  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  A  /\  ( r  e.  X  /\  q  e.  Y  /\  p  .<_  ( r 
.\/  q ) ) )  ->  p  e.  A )
9 simp33 996 . 2  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  A  /\  ( r  e.  X  /\  q  e.  Y  /\  p  .<_  ( r 
.\/  q ) ) )  ->  p  .<_  ( r  .\/  q ) )
10 osumcllem.l . . 3  |-  .<_  =  ( le `  K )
11 osumcllem.j . . 3  |-  .\/  =  ( join `  K )
12 osumcllem.a . . 3  |-  A  =  ( Atoms `  K )
13 osumcllem.p . . 3  |-  .+  =  ( + P `  K
)
1410, 11, 12, 13elpaddri 30600 . 2  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Y  C_  A )  /\  ( r  e.  X  /\  q  e.  Y
)  /\  ( p  e.  A  /\  p  .<_  ( r  .\/  q
) ) )  ->  p  e.  ( X  .+  Y ) )
153, 4, 5, 6, 7, 8, 9, 14syl322anc 1213 1  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  A  /\  ( r  e.  X  /\  q  e.  Y  /\  p  .<_  ( r 
.\/  q ) ) )  ->  p  e.  ( X  .+  Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 937    = wceq 1653    e. wcel 1726    C_ wss 3321   {csn 3815   class class class wbr 4213   ` cfv 5455  (class class class)co 6082   lecple 13537   joincjn 14402   Latclat 14475   Atomscatm 30062   HLchlt 30149   + Pcpadd 30593   _|_ PcpolN 30700   PSubClcpscN 30732
This theorem is referenced by:  osumcllem6N  30759
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-undef 6544  df-riota 6550  df-lub 14432  df-join 14434  df-lat 14476  df-ats 30066  df-atl 30097  df-cvlat 30121  df-hlat 30150  df-padd 30594
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