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Theorem pexmidlem8N 30225
Description: Lemma for pexmidN 30217. The contradiction of pexmidlem6N 30223 and pexmidlem7N 30224 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 2536 . 2  |-  -.  ( X  =  X  /\  X  =/=  X )
2 simpll 730 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/) ) )  ->  K  e.  HL )
3 simplr 731 . . . . 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 30156 . . . . . 6  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
(  ._|_  `  X )  C_  A )
76adantr 451 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/) ) )  -> 
(  ._|_  `  X )  C_  A )
8 pexmidALT.p . . . . . 6  |-  .+  =  ( + P `  K
)
94, 8paddssat 30062 . . . . 5  |-  ( ( K  e.  HL  /\  X  C_  A  /\  (  ._|_  `  X )  C_  A )  ->  ( X  .+  (  ._|_  `  X
) )  C_  A
)
102, 3, 7, 9syl3anc 1183 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/) ) )  -> 
( X  .+  (  ._|_  `  X ) ) 
C_  A )
11 df-pss 3254 . . . . . . 7  |-  ( ( X  .+  (  ._|_  `  X ) )  C.  A 
<->  ( ( X  .+  (  ._|_  `  X )
)  C_  A  /\  ( X  .+  (  ._|_  `  X ) )  =/= 
A ) )
12 pssnel 3608 . . . . . . 7  |-  ( ( X  .+  (  ._|_  `  X ) )  C.  A  ->  E. p ( p  e.  A  /\  -.  p  e.  ( X  .+  (  ._|_  `  X
) ) ) )
1311, 12sylbir 204 . . . . . 6  |-  ( ( ( X  .+  (  ._|_  `  X ) ) 
C_  A  /\  ( X  .+  (  ._|_  `  X
) )  =/=  A
)  ->  E. p
( p  e.  A  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )
14 df-rex 2634 . . . . . 6  |-  ( E. p  e.  A  -.  p  e.  ( X  .+  (  ._|_  `  X
) )  <->  E. p
( p  e.  A  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )
1513, 14sylibr 203 . . . . 5  |-  ( ( ( X  .+  (  ._|_  `  X ) ) 
C_  A  /\  ( X  .+  (  ._|_  `  X
) )  =/=  A
)  ->  E. p  e.  A  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) )
16 simplll 734 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  X  C_  A )  /\  (
(  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/) ) )  /\  (
p  e.  A  /\  -.  p  e.  ( X  .+  (  ._|_  `  X
) ) ) )  ->  K  e.  HL )
17 simpllr 735 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  X  C_  A )  /\  (
(  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/) ) )  /\  (
p  e.  A  /\  -.  p  e.  ( X  .+  (  ._|_  `  X
) ) ) )  ->  X  C_  A
)
18 simprl 732 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  X  C_  A )  /\  (
(  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/) ) )  /\  (
p  e.  A  /\  -.  p  e.  ( X  .+  (  ._|_  `  X
) ) ) )  ->  p  e.  A
)
19 simplrl 736 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  X  C_  A )  /\  (
(  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/) ) )  /\  (
p  e.  A  /\  -.  p  e.  ( X  .+  (  ._|_  `  X
) ) ) )  ->  (  ._|_  `  (  ._|_  `  X ) )  =  X )
20 simplrr 737 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  X  C_  A )  /\  (
(  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/) ) )  /\  (
p  e.  A  /\  -.  p  e.  ( X  .+  (  ._|_  `  X
) ) ) )  ->  X  =/=  (/) )
21 simprr 733 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  X  C_  A )  /\  (
(  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/) ) )  /\  (
p  e.  A  /\  -.  p  e.  ( X  .+  (  ._|_  `  X
) ) ) )  ->  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) )
22 eqid 2366 . . . . . . . . . . 11  |-  ( le
`  K )  =  ( le `  K
)
23 eqid 2366 . . . . . . . . . . 11  |-  ( join `  K )  =  (
join `  K )
24 eqid 2366 . . . . . . . . . . 11  |-  ( X 
.+  { p }
)  =  ( X 
.+  { p }
)
2522, 23, 4, 8, 5, 24pexmidlem6N 30223 . . . . . . . . . 10  |-  ( ( ( 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 30224 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  ( X  .+  { p }
)  =/=  X )
2725, 26jca 518 . . . . . . . . 9  |-  ( ( ( 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 1198 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  C_  A )  /\  (
(  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/) ) )  /\  (
p  e.  A  /\  -.  p  e.  ( X  .+  (  ._|_  `  X
) ) ) )  ->  ( ( X 
.+  { p }
)  =  X  /\  ( X  .+  { p } )  =/=  X
) )
29 nonconne 2536 . . . . . . . . 9  |-  -.  (
( X  .+  {
p } )  =  X  /\  ( X 
.+  { p }
)  =/=  X )
3029, 12false 339 . . . . . . . 8  |-  ( ( ( X  .+  {
p } )  =  X  /\  ( X 
.+  { p }
)  =/=  X )  <-> 
( X  =  X  /\  X  =/=  X
) )
3128, 30sylib 188 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  C_  A )  /\  (
(  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/) ) )  /\  (
p  e.  A  /\  -.  p  e.  ( X  .+  (  ._|_  `  X
) ) ) )  ->  ( X  =  X  /\  X  =/= 
X ) )
3231exp32 588 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/) ) )  -> 
( p  e.  A  ->  ( -.  p  e.  ( X  .+  (  ._|_  `  X ) )  ->  ( X  =  X  /\  X  =/= 
X ) ) ) )
3332rexlimdv 2751 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/) ) )  -> 
( E. p  e.  A  -.  p  e.  ( X  .+  (  ._|_  `  X ) )  ->  ( X  =  X  /\  X  =/= 
X ) ) )
3415, 33syl5 28 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/) ) )  -> 
( ( ( X 
.+  (  ._|_  `  X
) )  C_  A  /\  ( X  .+  (  ._|_  `  X ) )  =/=  A )  -> 
( X  =  X  /\  X  =/=  X
) ) )
3510, 34mpand 656 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/) ) )  -> 
( ( X  .+  (  ._|_  `  X )
)  =/=  A  -> 
( X  =  X  /\  X  =/=  X
) ) )
3635necon1bd 2597 . 2  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/) ) )  -> 
( -.  ( X  =  X  /\  X  =/=  X )  ->  ( X  .+  (  ._|_  `  X
) )  =  A ) )
371, 36mpi 16 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 358    /\ w3a 935   E.wex 1546    = wceq 1647    e. wcel 1715    =/= wne 2529   E.wrex 2629    C_ wss 3238    C. wpss 3239   (/)c0 3543   {csn 3729   ` cfv 5358  (class class class)co 5981   lecple 13423   joincjn 14288   Atomscatm 29512   HLchlt 29599   + Pcpadd 30043   _|_ PcpolN 30150
This theorem is referenced by:  pexmidALTN  30226
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-iin 4010  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-undef 6440  df-riota 6446  df-poset 14290  df-plt 14302  df-lub 14318  df-glb 14319  df-join 14320  df-meet 14321  df-p0 14355  df-p1 14356  df-lat 14362  df-clat 14424  df-oposet 29425  df-ol 29427  df-oml 29428  df-covers 29515  df-ats 29516  df-atl 29547  df-cvlat 29571  df-hlat 29600  df-psubsp 29751  df-pmap 29752  df-padd 30044  df-polarityN 30151  df-psubclN 30183
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