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Theorem pexmidlem6N 30089
Description: Lemma for pexmidN 30083. (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 30088 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X
) ) ) )  ->  ( (  ._|_  `  X )  i^i  M
)  =  (/) )
873adantr1 1116 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  (
(  ._|_  `  X )  i^i  M )  =  (/) )
98fveq2d 5672 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  (  ._|_  `  ( (  ._|_  `  X )  i^i  M
) )  =  ( 
._|_  `  (/) ) )
10 simpl1 960 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  K  e.  HL )
113, 5pol0N 30023 . . . . . 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 2419 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  (  ._|_  `  ( (  ._|_  `  X )  i^i  M
) )  =  A )
1413ineq1d 3484 . . 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 961 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  X  C_  A )
16 simpl3 962 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  p  e.  A )
1716snssd 3886 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  { p }  C_  A )
183, 4paddssat 29928 . . . . . . 7  |-  ( ( K  e.  HL  /\  X  C_  A  /\  {
p }  C_  A
)  ->  ( X  .+  { p } ) 
C_  A )
1910, 15, 17, 18syl3anc 1184 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  ( X  .+  { p }
)  C_  A )
206, 19syl5eqss 3335 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  M  C_  A )
2110, 15, 203jca 1134 . . . 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 29929 . . . . . . 7  |-  ( ( K  e.  HL  /\  X  C_  A  /\  {
p }  C_  A
)  ->  X  C_  ( X  .+  { p }
) )
2310, 15, 17, 22syl3anc 1184 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  X  C_  ( X  .+  {
p } ) )
2423, 6syl6sseqr 3338 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  X  C_  M )
25 simpr1 963 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  (  ._|_  `  (  ._|_  `  X
) )  =  X )
26 eqid 2387 . . . . . . . . . . 11  |-  ( PSubCl `  K )  =  (
PSubCl `  K )
273, 5, 26ispsubclN 30051 . . . . . . . . . 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 889 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  X  e.  ( PSubCl `  K )
)
303, 4, 26paddatclN 30063 . . . . . . . 8  |-  ( ( K  e.  HL  /\  X  e.  ( PSubCl `  K )  /\  p  e.  A )  ->  ( X  .+  { p }
)  e.  ( PSubCl `  K ) )
3110, 29, 16, 30syl3anc 1184 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  ( X  .+  { p }
)  e.  ( PSubCl `  K ) )
326, 31syl5eqel 2471 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  M  e.  ( PSubCl `  K )
)
335, 26psubcli2N 30053 . . . . . 6  |-  ( ( K  e.  HL  /\  M  e.  ( PSubCl `  K ) )  -> 
(  ._|_  `  (  ._|_  `  M ) )  =  M )
3410, 32, 33syl2anc 643 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  (  ._|_  `  (  ._|_  `  M
) )  =  M )
3524, 34jca 519 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  ( X  C_  M  /\  (  ._|_  `  (  ._|_  `  M
) )  =  M ) )
363, 5poml4N 30067 . . . 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 58 . . 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 3503 . . . 4  |-  ( M 
C_  A  <->  ( A  i^i  M )  =  M )
3920, 38sylib 189 . . 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 2428 . 2  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  M  =  (  ._|_  `  (  ._|_  `  X ) ) )
4140, 25eqtrd 2419 1  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  M  =  X )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2550    i^i cin 3262    C_ wss 3263   (/)c0 3571   {csn 3757   ` cfv 5394  (class class class)co 6020   lecple 13463   joincjn 14328   Atomscatm 29378   HLchlt 29465   + Pcpadd 29909   _|_ PcpolN 30016   PSubClcpscN 30048
This theorem is referenced by:  pexmidlem8N  30091
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-iin 4038  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-undef 6479  df-riota 6485  df-poset 14330  df-plt 14342  df-lub 14358  df-glb 14359  df-join 14360  df-meet 14361  df-p0 14395  df-p1 14396  df-lat 14402  df-clat 14464  df-oposet 29291  df-ol 29293  df-oml 29294  df-covers 29381  df-ats 29382  df-atl 29413  df-cvlat 29437  df-hlat 29466  df-psubsp 29617  df-pmap 29618  df-padd 29910  df-polarityN 30017  df-psubclN 30049
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