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Theorem osumcllem1N 30753
Description: Lemma for osumclN 30764. (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
osumcllem1N  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  U )  ->  ( U  i^i  M
)  =  M )

Proof of Theorem osumcllem1N
StepHypRef Expression
1 osumcllem.m . . 3  |-  M  =  ( X  .+  {
p } )
2 osumcllem.a . . . . . . 7  |-  A  =  ( Atoms `  K )
3 osumcllem.p . . . . . . 7  |-  .+  =  ( + P `  K
)
42, 3sspadd1 30612 . . . . . 6  |-  ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  ->  X  C_  ( X  .+  Y
) )
54adantr 452 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  U )  ->  X  C_  ( X  .+  Y ) )
6 simpl1 960 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  U )  ->  K  e.  HL )
72, 3paddssat 30611 . . . . . . . 8  |-  ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  ->  ( X  .+  Y )  C_  A )
87adantr 452 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  U )  ->  ( X  .+  Y
)  C_  A )
9 osumcllem.o . . . . . . . 8  |-  ._|_  =  ( _|_ P `  K
)
102, 92polssN 30712 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( X  .+  Y ) 
C_  A )  -> 
( X  .+  Y
)  C_  (  ._|_  `  (  ._|_  `  ( X 
.+  Y ) ) ) )
116, 8, 10syl2anc 643 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  U )  ->  ( X  .+  Y
)  C_  (  ._|_  `  (  ._|_  `  ( X 
.+  Y ) ) ) )
12 osumcllem.u . . . . . 6  |-  U  =  (  ._|_  `  (  ._|_  `  ( X  .+  Y
) ) )
1311, 12syl6sseqr 3395 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  U )  ->  ( X  .+  Y
)  C_  U )
145, 13sstrd 3358 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  U )  ->  X  C_  U )
15 simpr 448 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  U )  ->  p  e.  U )
1615snssd 3943 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  U )  ->  { p }  C_  U )
17 simpl2 961 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  U )  ->  X  C_  A )
182, 9polssatN 30705 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  ( X  .+  Y ) 
C_  A )  -> 
(  ._|_  `  ( X  .+  Y ) )  C_  A )
196, 8, 18syl2anc 643 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  U )  ->  (  ._|_  `  ( X 
.+  Y ) ) 
C_  A )
202, 9polssatN 30705 . . . . . . . 8  |-  ( ( K  e.  HL  /\  (  ._|_  `  ( X  .+  Y ) )  C_  A )  ->  (  ._|_  `  (  ._|_  `  ( X  .+  Y ) ) )  C_  A )
216, 19, 20syl2anc 643 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  U )  ->  (  ._|_  `  (  ._|_  `  ( X  .+  Y
) ) )  C_  A )
2212, 21syl5eqss 3392 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  U )  ->  U  C_  A )
2316, 22sstrd 3358 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  U )  ->  { p }  C_  A )
24 eqid 2436 . . . . . . . 8  |-  ( PSubSp `  K )  =  (
PSubSp `  K )
252, 24, 9polsubN 30704 . . . . . . 7  |-  ( ( K  e.  HL  /\  (  ._|_  `  ( X  .+  Y ) )  C_  A )  ->  (  ._|_  `  (  ._|_  `  ( X  .+  Y ) ) )  e.  ( PSubSp `  K ) )
266, 19, 25syl2anc 643 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  U )  ->  (  ._|_  `  (  ._|_  `  ( X  .+  Y
) ) )  e.  ( PSubSp `  K )
)
2712, 26syl5eqel 2520 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  U )  ->  U  e.  ( PSubSp `  K ) )
282, 24, 3paddss 30642 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  { p }  C_  A  /\  U  e.  ( PSubSp `
 K ) ) )  ->  ( ( X  C_  U  /\  {
p }  C_  U
)  <->  ( X  .+  { p } )  C_  U ) )
296, 17, 23, 27, 28syl13anc 1186 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  U )  ->  ( ( X  C_  U  /\  { p }  C_  U )  <->  ( X  .+  { p } ) 
C_  U ) )
3014, 16, 29mpbi2and 888 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  U )  ->  ( X  .+  {
p } )  C_  U )
311, 30syl5eqss 3392 . 2  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  U )  ->  M  C_  U )
32 sseqin2 3560 . 2  |-  ( M 
C_  U  <->  ( U  i^i  M )  =  M )
3331, 32sylib 189 1  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  U )  ->  ( U  i^i  M
)  =  M )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    i^i cin 3319    C_ wss 3320   {csn 3814   ` cfv 5454  (class class class)co 6081   lecple 13536   joincjn 14401   Atomscatm 30061   HLchlt 30148   PSubSpcpsubsp 30293   + Pcpadd 30592   _|_ PcpolN 30699   PSubClcpscN 30731
This theorem is referenced by:  osumcllem2N  30754  osumcllem9N  30761
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-undef 6543  df-riota 6549  df-poset 14403  df-plt 14415  df-lub 14431  df-glb 14432  df-join 14433  df-meet 14434  df-p0 14468  df-p1 14469  df-lat 14475  df-clat 14537  df-oposet 29974  df-ol 29976  df-oml 29977  df-covers 30064  df-ats 30065  df-atl 30096  df-cvlat 30120  df-hlat 30149  df-psubsp 30300  df-pmap 30301  df-padd 30593  df-polarityN 30700
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