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Theorem pexmidlem8N 34791
Description: Lemma for pexmidN 34783. The contradiction of pexmidlem6N 34789 and pexmidlem7N 34790 shows that there can be no atom  p that is not in  X  .+  (  ._|_  `  X ), which is therefore the whole atom space. (Contributed by NM, 3-Feb-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
pexmidALT.a  |-  A  =  ( Atoms `  K )
pexmidALT.p  |-  .+  =  ( +P `  K
)
pexmidALT.o  |-  ._|_  =  ( _|_P `  K
)
Assertion
Ref Expression
pexmidlem8N  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/) ) )  -> 
( X  .+  (  ._|_  `  X ) )  =  A )

Proof of Theorem pexmidlem8N
Dummy variable  p is distinct from all other variables.
StepHypRef Expression
1 nonconne 2675 . 2  |-  -.  ( X  =  X  /\  X  =/=  X )
2 simpll 753 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/) ) )  ->  K  e.  HL )
3 simplr 754 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/) ) )  ->  X  C_  A )
4 pexmidALT.a . . . . . . 7  |-  A  =  ( Atoms `  K )
5 pexmidALT.o . . . . . . 7  |-  ._|_  =  ( _|_P `  K
)
64, 5polssatN 34722 . . . . . 6  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
(  ._|_  `  X )  C_  A )
76adantr 465 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/) ) )  -> 
(  ._|_  `  X )  C_  A )
8 pexmidALT.p . . . . . 6  |-  .+  =  ( +P `  K
)
94, 8paddssat 34628 . . . . 5  |-  ( ( K  e.  HL  /\  X  C_  A  /\  (  ._|_  `  X )  C_  A )  ->  ( X  .+  (  ._|_  `  X
) )  C_  A
)
102, 3, 7, 9syl3anc 1228 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/) ) )  -> 
( X  .+  (  ._|_  `  X ) ) 
C_  A )
11 df-pss 3492 . . . . . . 7  |-  ( ( X  .+  (  ._|_  `  X ) )  C.  A 
<->  ( ( X  .+  (  ._|_  `  X )
)  C_  A  /\  ( X  .+  (  ._|_  `  X ) )  =/= 
A ) )
12 pssnel 3892 . . . . . . 7  |-  ( ( X  .+  (  ._|_  `  X ) )  C.  A  ->  E. p ( p  e.  A  /\  -.  p  e.  ( X  .+  (  ._|_  `  X
) ) ) )
1311, 12sylbir 213 . . . . . 6  |-  ( ( ( X  .+  (  ._|_  `  X ) ) 
C_  A  /\  ( X  .+  (  ._|_  `  X
) )  =/=  A
)  ->  E. p
( p  e.  A  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )
14 df-rex 2820 . . . . . 6  |-  ( E. p  e.  A  -.  p  e.  ( X  .+  (  ._|_  `  X
) )  <->  E. p
( p  e.  A  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )
1513, 14sylibr 212 . . . . 5  |-  ( ( ( X  .+  (  ._|_  `  X ) ) 
C_  A  /\  ( X  .+  (  ._|_  `  X
) )  =/=  A
)  ->  E. p  e.  A  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) )
16 simplll 757 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  C_  A )  /\  (
(  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/) ) )  /\  (
p  e.  A  /\  -.  p  e.  ( X  .+  (  ._|_  `  X
) ) ) )  ->  K  e.  HL )
17 simpllr 758 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  C_  A )  /\  (
(  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/) ) )  /\  (
p  e.  A  /\  -.  p  e.  ( X  .+  (  ._|_  `  X
) ) ) )  ->  X  C_  A
)
18 simprl 755 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  C_  A )  /\  (
(  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/) ) )  /\  (
p  e.  A  /\  -.  p  e.  ( X  .+  (  ._|_  `  X
) ) ) )  ->  p  e.  A
)
19 simplrl 759 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  C_  A )  /\  (
(  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/) ) )  /\  (
p  e.  A  /\  -.  p  e.  ( X  .+  (  ._|_  `  X
) ) ) )  ->  (  ._|_  `  (  ._|_  `  X ) )  =  X )
20 simplrr 760 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  C_  A )  /\  (
(  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/) ) )  /\  (
p  e.  A  /\  -.  p  e.  ( X  .+  (  ._|_  `  X
) ) ) )  ->  X  =/=  (/) )
21 simprr 756 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  C_  A )  /\  (
(  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/) ) )  /\  (
p  e.  A  /\  -.  p  e.  ( X  .+  (  ._|_  `  X
) ) ) )  ->  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) )
22 eqid 2467 . . . . . . . . . 10  |-  ( le
`  K )  =  ( le `  K
)
23 eqid 2467 . . . . . . . . . 10  |-  ( join `  K )  =  (
join `  K )
24 eqid 2467 . . . . . . . . . 10  |-  ( X 
.+  { p }
)  =  ( X 
.+  { p }
)
2522, 23, 4, 8, 5, 24pexmidlem6N 34789 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  ( X  .+  { p }
)  =  X )
2622, 23, 4, 8, 5, 24pexmidlem7N 34790 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  ( X  .+  { p }
)  =/=  X )
2725, 26jca 532 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  (
( X  .+  {
p } )  =  X  /\  ( X 
.+  { p }
)  =/=  X ) )
2816, 17, 18, 19, 20, 21, 27syl33anc 1243 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  C_  A )  /\  (
(  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/) ) )  /\  (
p  e.  A  /\  -.  p  e.  ( X  .+  (  ._|_  `  X
) ) ) )  ->  ( ( X 
.+  { p }
)  =  X  /\  ( X  .+  { p } )  =/=  X
) )
29 nonconne 2675 . . . . . . . 8  |-  -.  (
( X  .+  {
p } )  =  X  /\  ( X 
.+  { p }
)  =/=  X )
3029, 12false 350 . . . . . . 7  |-  ( ( ( X  .+  {
p } )  =  X  /\  ( X 
.+  { p }
)  =/=  X )  <-> 
( X  =  X  /\  X  =/=  X
) )
3128, 30sylib 196 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  C_  A )  /\  (
(  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/) ) )  /\  (
p  e.  A  /\  -.  p  e.  ( X  .+  (  ._|_  `  X
) ) ) )  ->  ( X  =  X  /\  X  =/= 
X ) )
3231rexlimdvaa 2956 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/) ) )  -> 
( E. p  e.  A  -.  p  e.  ( X  .+  (  ._|_  `  X ) )  ->  ( X  =  X  /\  X  =/= 
X ) ) )
3315, 32syl5 32 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/) ) )  -> 
( ( ( X 
.+  (  ._|_  `  X
) )  C_  A  /\  ( X  .+  (  ._|_  `  X ) )  =/=  A )  -> 
( X  =  X  /\  X  =/=  X
) ) )
3410, 33mpand 675 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/) ) )  -> 
( ( X  .+  (  ._|_  `  X )
)  =/=  A  -> 
( X  =  X  /\  X  =/=  X
) ) )
3534necon1bd 2685 . 2  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/) ) )  -> 
( -.  ( X  =  X  /\  X  =/=  X )  ->  ( X  .+  (  ._|_  `  X
) )  =  A ) )
361, 35mpi 17 1  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/) ) )  -> 
( X  .+  (  ._|_  `  X ) )  =  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379   E.wex 1596    e. wcel 1767    =/= wne 2662   E.wrex 2815    C_ wss 3476    C. wpss 3477   (/)c0 3785   {csn 4027   ` cfv 5588  (class class class)co 6284   lecple 14562   joincjn 15431   Atomscatm 34078   HLchlt 34165   +Pcpadd 34609   _|_PcpolN 34716
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-riotaBAD 33774
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-1st 6784  df-2nd 6785  df-undef 7002  df-poset 15433  df-plt 15445  df-lub 15461  df-glb 15462  df-join 15463  df-meet 15464  df-p0 15526  df-p1 15527  df-lat 15533  df-clat 15595  df-oposet 33991  df-ol 33993  df-oml 33994  df-covers 34081  df-ats 34082  df-atl 34113  df-cvlat 34137  df-hlat 34166  df-psubsp 34317  df-pmap 34318  df-padd 34610  df-polarityN 34717  df-psubclN 34749
This theorem is referenced by:  pexmidALTN  34792
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