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Theorem pexmidlem6N 33459
Description: Lemma for pexmidN 33453. (Contributed by NM, 3-Feb-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
pexmidlem.l  |-  .<_  =  ( le `  K )
pexmidlem.j  |-  .\/  =  ( join `  K )
pexmidlem.a  |-  A  =  ( Atoms `  K )
pexmidlem.p  |-  .+  =  ( +P `  K
)
pexmidlem.o  |-  ._|_  =  ( _|_P `  K
)
pexmidlem.m  |-  M  =  ( X  .+  {
p } )
Assertion
Ref Expression
pexmidlem6N  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  M  =  X )

Proof of Theorem pexmidlem6N
StepHypRef Expression
1 pexmidlem.l . . . . . . . 8  |-  .<_  =  ( le `  K )
2 pexmidlem.j . . . . . . . 8  |-  .\/  =  ( join `  K )
3 pexmidlem.a . . . . . . . 8  |-  A  =  ( Atoms `  K )
4 pexmidlem.p . . . . . . . 8  |-  .+  =  ( +P `  K
)
5 pexmidlem.o . . . . . . . 8  |-  ._|_  =  ( _|_P `  K
)
6 pexmidlem.m . . . . . . . 8  |-  M  =  ( X  .+  {
p } )
71, 2, 3, 4, 5, 6pexmidlem5N 33458 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X
) ) ) )  ->  ( (  ._|_  `  X )  i^i  M
)  =  (/) )
873adantr1 1147 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  (
(  ._|_  `  X )  i^i  M )  =  (/) )
98fveq2d 5690 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  (  ._|_  `  ( (  ._|_  `  X )  i^i  M
) )  =  ( 
._|_  `  (/) ) )
10 simpl1 991 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  K  e.  HL )
113, 5pol0N 33393 . . . . . 6  |-  ( K  e.  HL  ->  (  ._|_  `  (/) )  =  A )
1210, 11syl 16 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  (  ._|_  `  (/) )  =  A )
139, 12eqtrd 2470 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  (  ._|_  `  ( (  ._|_  `  X )  i^i  M
) )  =  A )
1413ineq1d 3546 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  (
(  ._|_  `  ( (  ._|_  `  X )  i^i 
M ) )  i^i 
M )  =  ( A  i^i  M ) )
15 simpl2 992 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  X  C_  A )
16 simpl3 993 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  p  e.  A )
1716snssd 4013 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  { p }  C_  A )
183, 4paddssat 33298 . . . . . . 7  |-  ( ( K  e.  HL  /\  X  C_  A  /\  {
p }  C_  A
)  ->  ( X  .+  { p } ) 
C_  A )
1910, 15, 17, 18syl3anc 1218 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  ( X  .+  { p }
)  C_  A )
206, 19syl5eqss 3395 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  M  C_  A )
2110, 15, 203jca 1168 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  ( K  e.  HL  /\  X  C_  A  /\  M  C_  A ) )
223, 4sspadd1 33299 . . . . . . 7  |-  ( ( K  e.  HL  /\  X  C_  A  /\  {
p }  C_  A
)  ->  X  C_  ( X  .+  { p }
) )
2310, 15, 17, 22syl3anc 1218 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  X  C_  ( X  .+  {
p } ) )
2423, 6syl6sseqr 3398 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  X  C_  M )
25 simpr1 994 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  (  ._|_  `  (  ._|_  `  X
) )  =  X )
26 eqid 2438 . . . . . . . . . . 11  |-  ( PSubCl `  K )  =  (
PSubCl `  K )
273, 5, 26ispsubclN 33421 . . . . . . . . . 10  |-  ( K  e.  HL  ->  ( X  e.  ( PSubCl `  K )  <->  ( X  C_  A  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X ) ) )
2810, 27syl 16 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  ( X  e.  ( PSubCl `  K )  <->  ( X  C_  A  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X ) ) )
2915, 25, 28mpbir2and 913 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  X  e.  ( PSubCl `  K )
)
303, 4, 26paddatclN 33433 . . . . . . . 8  |-  ( ( K  e.  HL  /\  X  e.  ( PSubCl `  K )  /\  p  e.  A )  ->  ( X  .+  { p }
)  e.  ( PSubCl `  K ) )
3110, 29, 16, 30syl3anc 1218 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  ( X  .+  { p }
)  e.  ( PSubCl `  K ) )
326, 31syl5eqel 2522 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  M  e.  ( PSubCl `  K )
)
335, 26psubcli2N 33423 . . . . . 6  |-  ( ( K  e.  HL  /\  M  e.  ( PSubCl `  K ) )  -> 
(  ._|_  `  (  ._|_  `  M ) )  =  M )
3410, 32, 33syl2anc 661 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  (  ._|_  `  (  ._|_  `  M
) )  =  M )
3524, 34jca 532 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  ( X  C_  M  /\  (  ._|_  `  (  ._|_  `  M
) )  =  M ) )
363, 5poml4N 33437 . . . 4  |-  ( ( K  e.  HL  /\  X  C_  A  /\  M  C_  A )  ->  (
( X  C_  M  /\  (  ._|_  `  (  ._|_  `  M ) )  =  M )  -> 
( (  ._|_  `  (
(  ._|_  `  X )  i^i  M ) )  i^i 
M )  =  ( 
._|_  `  (  ._|_  `  X
) ) ) )
3721, 35, 36sylc 60 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  (
(  ._|_  `  ( (  ._|_  `  X )  i^i 
M ) )  i^i 
M )  =  ( 
._|_  `  (  ._|_  `  X
) ) )
38 sseqin2 3564 . . . 4  |-  ( M 
C_  A  <->  ( A  i^i  M )  =  M )
3920, 38sylib 196 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  ( A  i^i  M )  =  M )
4014, 37, 393eqtr3rd 2479 . 2  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  M  =  (  ._|_  `  (  ._|_  `  X ) ) )
4140, 25eqtrd 2470 1  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  M  =  X )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2601    i^i cin 3322    C_ wss 3323   (/)c0 3632   {csn 3872   ` cfv 5413  (class class class)co 6086   lecple 14237   joincjn 15106   Atomscatm 32748   HLchlt 32835   +Pcpadd 33279   _|_PcpolN 33386   PSubClcpscN 33418
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-riotaBAD 32444
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-iin 4169  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-1st 6572  df-2nd 6573  df-undef 6784  df-poset 15108  df-plt 15120  df-lub 15136  df-glb 15137  df-join 15138  df-meet 15139  df-p0 15201  df-p1 15202  df-lat 15208  df-clat 15270  df-oposet 32661  df-ol 32663  df-oml 32664  df-covers 32751  df-ats 32752  df-atl 32783  df-cvlat 32807  df-hlat 32836  df-psubsp 32987  df-pmap 32988  df-padd 33280  df-polarityN 33387  df-psubclN 33419
This theorem is referenced by:  pexmidlem8N  33461
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