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Theorem pexmidlem6N 30164
Description: Lemma for pexmidN 30158. (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 30163 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X
) ) ) )  ->  ( (  ._|_  `  X )  i^i  M
)  =  (/) )
873adantr1 1114 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  (
(  ._|_  `  X )  i^i  M )  =  (/) )
98fveq2d 5529 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  (  ._|_  `  ( (  ._|_  `  X )  i^i  M
) )  =  ( 
._|_  `  (/) ) )
10 simpl1 958 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  K  e.  HL )
113, 5pol0N 30098 . . . . . 6  |-  ( K  e.  HL  ->  (  ._|_  `  (/) )  =  A )
1210, 11syl 15 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  (  ._|_  `  (/) )  =  A )
139, 12eqtrd 2315 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  (  ._|_  `  ( (  ._|_  `  X )  i^i  M
) )  =  A )
1413ineq1d 3369 . . 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 959 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  X  C_  A )
16 simpl3 960 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  p  e.  A )
1716snssd 3760 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  { p }  C_  A )
183, 4paddssat 30003 . . . . . . 7  |-  ( ( K  e.  HL  /\  X  C_  A  /\  {
p }  C_  A
)  ->  ( X  .+  { p } ) 
C_  A )
1910, 15, 17, 18syl3anc 1182 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  ( X  .+  { p }
)  C_  A )
206, 19syl5eqss 3222 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  M  C_  A )
2110, 15, 203jca 1132 . . . 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 30004 . . . . . . 7  |-  ( ( K  e.  HL  /\  X  C_  A  /\  {
p }  C_  A
)  ->  X  C_  ( X  .+  { p }
) )
2310, 15, 17, 22syl3anc 1182 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  X  C_  ( X  .+  {
p } ) )
2423, 6syl6sseqr 3225 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  X  C_  M )
25 simpr1 961 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  (  ._|_  `  (  ._|_  `  X
) )  =  X )
26 eqid 2283 . . . . . . . . . . 11  |-  ( PSubCl `  K )  =  (
PSubCl `  K )
273, 5, 26ispsubclN 30126 . . . . . . . . . 10  |-  ( K  e.  HL  ->  ( X  e.  ( PSubCl `  K )  <->  ( X  C_  A  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X ) ) )
2810, 27syl 15 . . . . . . . . 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 888 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  X  e.  ( PSubCl `  K )
)
303, 4, 26paddatclN 30138 . . . . . . . 8  |-  ( ( K  e.  HL  /\  X  e.  ( PSubCl `  K )  /\  p  e.  A )  ->  ( X  .+  { p }
)  e.  ( PSubCl `  K ) )
3110, 29, 16, 30syl3anc 1182 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  ( X  .+  { p }
)  e.  ( PSubCl `  K ) )
326, 31syl5eqel 2367 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  M  e.  ( PSubCl `  K )
)
335, 26psubcli2N 30128 . . . . . 6  |-  ( ( K  e.  HL  /\  M  e.  ( PSubCl `  K ) )  -> 
(  ._|_  `  (  ._|_  `  M ) )  =  M )
3410, 32, 33syl2anc 642 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  (  ._|_  `  (  ._|_  `  M
) )  =  M )
3524, 34jca 518 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  ( X  C_  M  /\  (  ._|_  `  (  ._|_  `  M
) )  =  M ) )
363, 5poml4N 30142 . . . 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 56 . . 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 3388 . . . 4  |-  ( M 
C_  A  <->  ( A  i^i  M )  =  M )
3920, 38sylib 188 . . 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 2324 . 2  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  M  =  (  ._|_  `  (  ._|_  `  X ) ) )
4140, 25eqtrd 2315 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446    i^i cin 3151    C_ wss 3152   (/)c0 3455   {csn 3640   ` cfv 5255  (class class class)co 5858   lecple 13215   joincjn 14078   Atomscatm 29453   HLchlt 29540   + Pcpadd 29984   _|_ PcpolN 30091   PSubClcpscN 30123
This theorem is referenced by:  pexmidlem8N  30166
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-psubsp 29692  df-pmap 29693  df-padd 29985  df-polarityN 30092  df-psubclN 30124
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