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Theorem pexmidlem8N 30463
Description: Lemma for pexmidN 30455. The contradiction of pexmidlem6N 30461 and pexmidlem7N 30462 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 2578 . 2  |-  -.  ( X  =  X  /\  X  =/=  X )
2 simpll 731 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/) ) )  ->  K  e.  HL )
3 simplr 732 . . . . 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 30394 . . . . . 6  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
(  ._|_  `  X )  C_  A )
76adantr 452 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/) ) )  -> 
(  ._|_  `  X )  C_  A )
8 pexmidALT.p . . . . . 6  |-  .+  =  ( + P `  K
)
94, 8paddssat 30300 . . . . 5  |-  ( ( K  e.  HL  /\  X  C_  A  /\  (  ._|_  `  X )  C_  A )  ->  ( X  .+  (  ._|_  `  X
) )  C_  A
)
102, 3, 7, 9syl3anc 1184 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/) ) )  -> 
( X  .+  (  ._|_  `  X ) ) 
C_  A )
11 df-pss 3300 . . . . . . 7  |-  ( ( X  .+  (  ._|_  `  X ) )  C.  A 
<->  ( ( X  .+  (  ._|_  `  X )
)  C_  A  /\  ( X  .+  (  ._|_  `  X ) )  =/= 
A ) )
12 pssnel 3657 . . . . . . 7  |-  ( ( X  .+  (  ._|_  `  X ) )  C.  A  ->  E. p ( p  e.  A  /\  -.  p  e.  ( X  .+  (  ._|_  `  X
) ) ) )
1311, 12sylbir 205 . . . . . 6  |-  ( ( ( X  .+  (  ._|_  `  X ) ) 
C_  A  /\  ( X  .+  (  ._|_  `  X
) )  =/=  A
)  ->  E. p
( p  e.  A  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )
14 df-rex 2676 . . . . . 6  |-  ( E. p  e.  A  -.  p  e.  ( X  .+  (  ._|_  `  X
) )  <->  E. p
( p  e.  A  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )
1513, 14sylibr 204 . . . . 5  |-  ( ( ( X  .+  (  ._|_  `  X ) ) 
C_  A  /\  ( X  .+  (  ._|_  `  X
) )  =/=  A
)  ->  E. p  e.  A  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) )
16 simplll 735 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  C_  A )  /\  (
(  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/) ) )  /\  (
p  e.  A  /\  -.  p  e.  ( X  .+  (  ._|_  `  X
) ) ) )  ->  K  e.  HL )
17 simpllr 736 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  C_  A )  /\  (
(  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/) ) )  /\  (
p  e.  A  /\  -.  p  e.  ( X  .+  (  ._|_  `  X
) ) ) )  ->  X  C_  A
)
18 simprl 733 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  C_  A )  /\  (
(  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/) ) )  /\  (
p  e.  A  /\  -.  p  e.  ( X  .+  (  ._|_  `  X
) ) ) )  ->  p  e.  A
)
19 simplrl 737 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  C_  A )  /\  (
(  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/) ) )  /\  (
p  e.  A  /\  -.  p  e.  ( X  .+  (  ._|_  `  X
) ) ) )  ->  (  ._|_  `  (  ._|_  `  X ) )  =  X )
20 simplrr 738 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  C_  A )  /\  (
(  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/) ) )  /\  (
p  e.  A  /\  -.  p  e.  ( X  .+  (  ._|_  `  X
) ) ) )  ->  X  =/=  (/) )
21 simprr 734 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  C_  A )  /\  (
(  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/) ) )  /\  (
p  e.  A  /\  -.  p  e.  ( X  .+  (  ._|_  `  X
) ) ) )  ->  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) )
22 eqid 2408 . . . . . . . . . 10  |-  ( le
`  K )  =  ( le `  K
)
23 eqid 2408 . . . . . . . . . 10  |-  ( join `  K )  =  (
join `  K )
24 eqid 2408 . . . . . . . . . 10  |-  ( X 
.+  { p }
)  =  ( X 
.+  { p }
)
2522, 23, 4, 8, 5, 24pexmidlem6N 30461 . . . . . . . . 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 30462 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  ( X  .+  { p }
)  =/=  X )
2725, 26jca 519 . . . . . . . 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 1199 . . . . . . 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 2578 . . . . . . . 8  |-  -.  (
( X  .+  {
p } )  =  X  /\  ( X 
.+  { p }
)  =/=  X )
3029, 12false 340 . . . . . . 7  |-  ( ( ( X  .+  {
p } )  =  X  /\  ( X 
.+  { p }
)  =/=  X )  <-> 
( X  =  X  /\  X  =/=  X
) )
3128, 30sylib 189 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  C_  A )  /\  (
(  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/) ) )  /\  (
p  e.  A  /\  -.  p  e.  ( X  .+  (  ._|_  `  X
) ) ) )  ->  ( X  =  X  /\  X  =/= 
X ) )
3231rexlimdvaa 2795 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/) ) )  -> 
( E. p  e.  A  -.  p  e.  ( X  .+  (  ._|_  `  X ) )  ->  ( X  =  X  /\  X  =/= 
X ) ) )
3315, 32syl5 30 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/) ) )  -> 
( ( ( X 
.+  (  ._|_  `  X
) )  C_  A  /\  ( X  .+  (  ._|_  `  X ) )  =/=  A )  -> 
( X  =  X  /\  X  =/=  X
) ) )
3410, 33mpand 657 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/) ) )  -> 
( ( X  .+  (  ._|_  `  X )
)  =/=  A  -> 
( X  =  X  /\  X  =/=  X
) ) )
3534necon1bd 2639 . 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 359    /\ w3a 936   E.wex 1547    = wceq 1649    e. wcel 1721    =/= wne 2571   E.wrex 2671    C_ wss 3284    C. wpss 3285   (/)c0 3592   {csn 3778   ` cfv 5417  (class class class)co 6044   lecple 13495   joincjn 14360   Atomscatm 29750   HLchlt 29837   + Pcpadd 30281   _|_ PcpolN 30388
This theorem is referenced by:  pexmidALTN  30464
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
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 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-iin 4060  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-undef 6506  df-riota 6512  df-poset 14362  df-plt 14374  df-lub 14390  df-glb 14391  df-join 14392  df-meet 14393  df-p0 14427  df-p1 14428  df-lat 14434  df-clat 14496  df-oposet 29663  df-ol 29665  df-oml 29666  df-covers 29753  df-ats 29754  df-atl 29785  df-cvlat 29809  df-hlat 29838  df-psubsp 29989  df-pmap 29990  df-padd 30282  df-polarityN 30389  df-psubclN 30421
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