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Theorem osumcllem2N 29964
Description: Lemma for osumclN 29974. (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 958 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  U )  ->  K  e.  HL )
2 simpl2 959 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  U )  ->  X  C_  A )
3 simpr 447 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  U )  ->  p  e.  U )
43snssd 3797 . . . . 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 29821 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  ->  ( X  .+  Y )  C_  A )
98adantr 451 . . . . . . . 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 29915 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( X  .+  Y ) 
C_  A )  -> 
(  ._|_  `  ( X  .+  Y ) )  C_  A )
121, 9, 11syl2anc 642 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  U )  ->  (  ._|_  `  ( X 
.+  Y ) ) 
C_  A )
136, 10polssatN 29915 . . . . . . 7  |-  ( ( K  e.  HL  /\  (  ._|_  `  ( X  .+  Y ) )  C_  A )  ->  (  ._|_  `  (  ._|_  `  ( X  .+  Y ) ) )  C_  A )
141, 12, 13syl2anc 642 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  U )  ->  (  ._|_  `  (  ._|_  `  ( X  .+  Y
) ) )  C_  A )
155, 14syl5eqss 3256 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  U )  ->  U  C_  A )
164, 15sstrd 3223 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  U )  ->  { p }  C_  A )
176, 7sspadd1 29822 . . . 4  |-  ( ( K  e.  HL  /\  X  C_  A  /\  {
p }  C_  A
)  ->  X  C_  ( X  .+  { p }
) )
181, 2, 16, 17syl3anc 1182 . . 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 3259 . 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 29963 . 2  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  U )  ->  ( U  i^i  M
)  =  M )
2520, 24sseqtr4d 3249 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 358    /\ w3a 934    = wceq 1633    e. wcel 1701    i^i cin 3185    C_ wss 3186   {csn 3674   ` cfv 5292  (class class class)co 5900   lecple 13262   joincjn 14127   Atomscatm 29271   HLchlt 29358   + Pcpadd 29802   _|_ PcpolN 29909   PSubClcpscN 29941
This theorem is referenced by:  osumcllem9N  29971
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-iin 3945  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-undef 6340  df-riota 6346  df-poset 14129  df-plt 14141  df-lub 14157  df-glb 14158  df-join 14159  df-meet 14160  df-p0 14194  df-p1 14195  df-lat 14201  df-clat 14263  df-oposet 29184  df-ol 29186  df-oml 29187  df-covers 29274  df-ats 29275  df-atl 29306  df-cvlat 29330  df-hlat 29359  df-psubsp 29510  df-pmap 29511  df-padd 29803  df-polarityN 29910
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