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Theorem pexmidlem8N 30848
Description: Lemma for pexmidN 30840. The contradiction of pexmidlem6N 30846 and pexmidlem7N 30847 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 2610 . 2  |-  -.  ( X  =  X  /\  X  =/=  X )
2 simpll 732 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/) ) )  ->  K  e.  HL )
3 simplr 733 . . . . 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 30779 . . . . . 6  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
(  ._|_  `  X )  C_  A )
76adantr 453 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/) ) )  -> 
(  ._|_  `  X )  C_  A )
8 pexmidALT.p . . . . . 6  |-  .+  =  ( + P `  K
)
94, 8paddssat 30685 . . . . 5  |-  ( ( K  e.  HL  /\  X  C_  A  /\  (  ._|_  `  X )  C_  A )  ->  ( X  .+  (  ._|_  `  X
) )  C_  A
)
102, 3, 7, 9syl3anc 1185 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/) ) )  -> 
( X  .+  (  ._|_  `  X ) ) 
C_  A )
11 df-pss 3338 . . . . . . 7  |-  ( ( X  .+  (  ._|_  `  X ) )  C.  A 
<->  ( ( X  .+  (  ._|_  `  X )
)  C_  A  /\  ( X  .+  (  ._|_  `  X ) )  =/= 
A ) )
12 pssnel 3695 . . . . . . 7  |-  ( ( X  .+  (  ._|_  `  X ) )  C.  A  ->  E. p ( p  e.  A  /\  -.  p  e.  ( X  .+  (  ._|_  `  X
) ) ) )
1311, 12sylbir 206 . . . . . 6  |-  ( ( ( X  .+  (  ._|_  `  X ) ) 
C_  A  /\  ( X  .+  (  ._|_  `  X
) )  =/=  A
)  ->  E. p
( p  e.  A  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )
14 df-rex 2713 . . . . . 6  |-  ( E. p  e.  A  -.  p  e.  ( X  .+  (  ._|_  `  X
) )  <->  E. p
( p  e.  A  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )
1513, 14sylibr 205 . . . . 5  |-  ( ( ( X  .+  (  ._|_  `  X ) ) 
C_  A  /\  ( X  .+  (  ._|_  `  X
) )  =/=  A
)  ->  E. p  e.  A  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) )
16 simplll 736 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  C_  A )  /\  (
(  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/) ) )  /\  (
p  e.  A  /\  -.  p  e.  ( X  .+  (  ._|_  `  X
) ) ) )  ->  K  e.  HL )
17 simpllr 737 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  C_  A )  /\  (
(  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/) ) )  /\  (
p  e.  A  /\  -.  p  e.  ( X  .+  (  ._|_  `  X
) ) ) )  ->  X  C_  A
)
18 simprl 734 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  C_  A )  /\  (
(  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/) ) )  /\  (
p  e.  A  /\  -.  p  e.  ( X  .+  (  ._|_  `  X
) ) ) )  ->  p  e.  A
)
19 simplrl 738 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  C_  A )  /\  (
(  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/) ) )  /\  (
p  e.  A  /\  -.  p  e.  ( X  .+  (  ._|_  `  X
) ) ) )  ->  (  ._|_  `  (  ._|_  `  X ) )  =  X )
20 simplrr 739 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  C_  A )  /\  (
(  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/) ) )  /\  (
p  e.  A  /\  -.  p  e.  ( X  .+  (  ._|_  `  X
) ) ) )  ->  X  =/=  (/) )
21 simprr 735 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  C_  A )  /\  (
(  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/) ) )  /\  (
p  e.  A  /\  -.  p  e.  ( X  .+  (  ._|_  `  X
) ) ) )  ->  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) )
22 eqid 2438 . . . . . . . . . 10  |-  ( le
`  K )  =  ( le `  K
)
23 eqid 2438 . . . . . . . . . 10  |-  ( join `  K )  =  (
join `  K )
24 eqid 2438 . . . . . . . . . 10  |-  ( X 
.+  { p }
)  =  ( X 
.+  { p }
)
2522, 23, 4, 8, 5, 24pexmidlem6N 30846 . . . . . . . . 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 30847 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  ( X  .+  { p }
)  =/=  X )
2725, 26jca 520 . . . . . . . 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 1200 . . . . . . 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 2610 . . . . . . . 8  |-  -.  (
( X  .+  {
p } )  =  X  /\  ( X 
.+  { p }
)  =/=  X )
3029, 12false 341 . . . . . . 7  |-  ( ( ( X  .+  {
p } )  =  X  /\  ( X 
.+  { p }
)  =/=  X )  <-> 
( X  =  X  /\  X  =/=  X
) )
3128, 30sylib 190 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  C_  A )  /\  (
(  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/) ) )  /\  (
p  e.  A  /\  -.  p  e.  ( X  .+  (  ._|_  `  X
) ) ) )  ->  ( X  =  X  /\  X  =/= 
X ) )
3231rexlimdvaa 2833 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/) ) )  -> 
( E. p  e.  A  -.  p  e.  ( X  .+  (  ._|_  `  X ) )  ->  ( X  =  X  /\  X  =/= 
X ) ) )
3315, 32syl5 31 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/) ) )  -> 
( ( ( X 
.+  (  ._|_  `  X
) )  C_  A  /\  ( X  .+  (  ._|_  `  X ) )  =/=  A )  -> 
( X  =  X  /\  X  =/=  X
) ) )
3410, 33mpand 658 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/) ) )  -> 
( ( X  .+  (  ._|_  `  X )
)  =/=  A  -> 
( X  =  X  /\  X  =/=  X
) ) )
3534necon1bd 2674 . 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 set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360    /\ w3a 937   E.wex 1551    = wceq 1653    e. wcel 1726    =/= wne 2601   E.wrex 2708    C_ wss 3322    C. wpss 3323   (/)c0 3630   {csn 3816   ` cfv 5457  (class class class)co 6084   lecple 13541   joincjn 14406   Atomscatm 30135   HLchlt 30222   + Pcpadd 30666   _|_ PcpolN 30773
This theorem is referenced by:  pexmidALTN  30849
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-iin 4098  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-undef 6546  df-riota 6552  df-poset 14408  df-plt 14420  df-lub 14436  df-glb 14437  df-join 14438  df-meet 14439  df-p0 14473  df-p1 14474  df-lat 14480  df-clat 14542  df-oposet 30048  df-ol 30050  df-oml 30051  df-covers 30138  df-ats 30139  df-atl 30170  df-cvlat 30194  df-hlat 30223  df-psubsp 30374  df-pmap 30375  df-padd 30667  df-polarityN 30774  df-psubclN 30806
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