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Theorem osumcllem2N 30755
Description: Lemma for osumclN 30765. (Contributed by NM, 25-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
osumcllem2N  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  U )  ->  X  C_  ( U  i^i  M ) )

Proof of Theorem osumcllem2N
StepHypRef Expression
1 simpl1 961 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  U )  ->  K  e.  HL )
2 simpl2 962 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  U )  ->  X  C_  A )
3 simpr 449 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  U )  ->  p  e.  U )
43snssd 3944 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  U )  ->  { p }  C_  U )
5 osumcllem.u . . . . . 6  |-  U  =  (  ._|_  `  (  ._|_  `  ( X  .+  Y
) ) )
6 osumcllem.a . . . . . . . . . 10  |-  A  =  ( Atoms `  K )
7 osumcllem.p . . . . . . . . . 10  |-  .+  =  ( + P `  K
)
86, 7paddssat 30612 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  ->  ( X  .+  Y )  C_  A )
98adantr 453 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  U )  ->  ( X  .+  Y
)  C_  A )
10 osumcllem.o . . . . . . . . 9  |-  ._|_  =  ( _|_ P `  K
)
116, 10polssatN 30706 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( X  .+  Y ) 
C_  A )  -> 
(  ._|_  `  ( X  .+  Y ) )  C_  A )
121, 9, 11syl2anc 644 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  U )  ->  (  ._|_  `  ( X 
.+  Y ) ) 
C_  A )
136, 10polssatN 30706 . . . . . . 7  |-  ( ( K  e.  HL  /\  (  ._|_  `  ( X  .+  Y ) )  C_  A )  ->  (  ._|_  `  (  ._|_  `  ( X  .+  Y ) ) )  C_  A )
141, 12, 13syl2anc 644 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  U )  ->  (  ._|_  `  (  ._|_  `  ( X  .+  Y
) ) )  C_  A )
155, 14syl5eqss 3393 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  U )  ->  U  C_  A )
164, 15sstrd 3359 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  U )  ->  { p }  C_  A )
176, 7sspadd1 30613 . . . 4  |-  ( ( K  e.  HL  /\  X  C_  A  /\  {
p }  C_  A
)  ->  X  C_  ( X  .+  { p }
) )
181, 2, 16, 17syl3anc 1185 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  U )  ->  X  C_  ( X  .+  { p } ) )
19 osumcllem.m . . 3  |-  M  =  ( X  .+  {
p } )
2018, 19syl6sseqr 3396 . 2  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  U )  ->  X  C_  M )
21 osumcllem.l . . 3  |-  .<_  =  ( le `  K )
22 osumcllem.j . . 3  |-  .\/  =  ( join `  K )
23 osumcllem.c . . 3  |-  C  =  ( PSubCl `  K )
2421, 22, 6, 7, 10, 23, 19, 5osumcllem1N 30754 . 2  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  U )  ->  ( U  i^i  M
)  =  M )
2520, 24sseqtr4d 3386 1  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  U )  ->  X  C_  ( U  i^i  M ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    i^i cin 3320    C_ wss 3321   {csn 3815   ` cfv 5455  (class class class)co 6082   lecple 13537   joincjn 14402   Atomscatm 30062   HLchlt 30149   + Pcpadd 30593   _|_ PcpolN 30700   PSubClcpscN 30732
This theorem is referenced by:  osumcllem9N  30762
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-rmo 2714  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-iin 4097  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-poset 14404  df-plt 14416  df-lub 14432  df-glb 14433  df-join 14434  df-meet 14435  df-p0 14469  df-p1 14470  df-lat 14476  df-clat 14538  df-oposet 29975  df-ol 29977  df-oml 29978  df-covers 30065  df-ats 30066  df-atl 30097  df-cvlat 30121  df-hlat 30150  df-psubsp 30301  df-pmap 30302  df-padd 30594  df-polarityN 30701
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