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Theorem isat3 29790
Description: The predicate "is an atom". (elat2 23796 analog.) (Contributed by NM, 27-Apr-2014.)
Hypotheses
Ref Expression
isat3.b  |-  B  =  ( Base `  K
)
isat3.l  |-  .<_  =  ( le `  K )
isat3.z  |-  .0.  =  ( 0. `  K )
isat3.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
isat3  |-  ( K  e.  AtLat  ->  ( P  e.  A  <->  ( P  e.  B  /\  P  =/= 
.0.  /\  A. x  e.  B  ( x  .<_  P  ->  ( x  =  P  \/  x  =  .0.  ) ) ) ) )
Distinct variable groups:    x, B    x, K    x, P    x,  .0.
Allowed substitution hints:    A( x)    .<_ ( x)

Proof of Theorem isat3
StepHypRef Expression
1 isat3.b . . . 4  |-  B  =  ( Base `  K
)
2 isat3.z . . . 4  |-  .0.  =  ( 0. `  K )
3 eqid 2404 . . . 4  |-  (  <o  `  K )  =  ( 
<o  `  K )
4 isat3.a . . . 4  |-  A  =  ( Atoms `  K )
51, 2, 3, 4isat 29769 . . 3  |-  ( K  e.  AtLat  ->  ( P  e.  A  <->  ( P  e.  B  /\  .0.  (  <o  `  K ) P ) ) )
6 simpl 444 . . . . . 6  |-  ( ( K  e.  AtLat  /\  P  e.  B )  ->  K  e.  AtLat )
71, 2atl0cl 29786 . . . . . . 7  |-  ( K  e.  AtLat  ->  .0.  e.  B )
87adantr 452 . . . . . 6  |-  ( ( K  e.  AtLat  /\  P  e.  B )  ->  .0.  e.  B )
9 simpr 448 . . . . . 6  |-  ( ( K  e.  AtLat  /\  P  e.  B )  ->  P  e.  B )
10 isat3.l . . . . . . 7  |-  .<_  =  ( le `  K )
11 eqid 2404 . . . . . . 7  |-  ( lt
`  K )  =  ( lt `  K
)
121, 10, 11, 3cvrval2 29757 . . . . . 6  |-  ( ( K  e.  AtLat  /\  .0.  e.  B  /\  P  e.  B )  ->  (  .0.  (  <o  `  K
) P  <->  (  .0.  ( lt `  K ) P  /\  A. x  e.  B  ( (  .0.  ( lt `  K
) x  /\  x  .<_  P )  ->  x  =  P ) ) ) )
136, 8, 9, 12syl3anc 1184 . . . . 5  |-  ( ( K  e.  AtLat  /\  P  e.  B )  ->  (  .0.  (  <o  `  K
) P  <->  (  .0.  ( lt `  K ) P  /\  A. x  e.  B  ( (  .0.  ( lt `  K
) x  /\  x  .<_  P )  ->  x  =  P ) ) ) )
141, 11, 2atlltn0 29789 . . . . . 6  |-  ( ( K  e.  AtLat  /\  P  e.  B )  ->  (  .0.  ( lt `  K
) P  <->  P  =/=  .0.  ) )
151, 11, 2atlltn0 29789 . . . . . . . . . . 11  |-  ( ( K  e.  AtLat  /\  x  e.  B )  ->  (  .0.  ( lt `  K
) x  <->  x  =/=  .0.  ) )
1615adantlr 696 . . . . . . . . . 10  |-  ( ( ( K  e.  AtLat  /\  P  e.  B )  /\  x  e.  B
)  ->  (  .0.  ( lt `  K ) x  <->  x  =/=  .0.  ) )
1716imbi1d 309 . . . . . . . . 9  |-  ( ( ( K  e.  AtLat  /\  P  e.  B )  /\  x  e.  B
)  ->  ( (  .0.  ( lt `  K
) x  ->  x  =  P )  <->  ( x  =/=  .0.  ->  x  =  P ) ) )
1817imbi2d 308 . . . . . . . 8  |-  ( ( ( K  e.  AtLat  /\  P  e.  B )  /\  x  e.  B
)  ->  ( (
x  .<_  P  ->  (  .0.  ( lt `  K
) x  ->  x  =  P ) )  <->  ( x  .<_  P  ->  ( x  =/=  .0.  ->  x  =  P ) ) ) )
19 impexp 434 . . . . . . . . 9  |-  ( ( (  .0.  ( lt
`  K ) x  /\  x  .<_  P )  ->  x  =  P )  <->  (  .0.  ( lt `  K ) x  ->  ( x  .<_  P  ->  x  =  P ) ) )
20 bi2.04 351 . . . . . . . . 9  |-  ( (  .0.  ( lt `  K ) x  -> 
( x  .<_  P  ->  x  =  P )
)  <->  ( x  .<_  P  ->  (  .0.  ( lt `  K ) x  ->  x  =  P ) ) )
2119, 20bitri 241 . . . . . . . 8  |-  ( ( (  .0.  ( lt
`  K ) x  /\  x  .<_  P )  ->  x  =  P )  <->  ( x  .<_  P  ->  (  .0.  ( lt `  K ) x  ->  x  =  P ) ) )
22 orcom 377 . . . . . . . . . 10  |-  ( ( x  =  P  \/  x  =  .0.  )  <->  ( x  =  .0.  \/  x  =  P )
)
23 neor 2651 . . . . . . . . . 10  |-  ( ( x  =  .0.  \/  x  =  P )  <->  ( x  =/=  .0.  ->  x  =  P ) )
2422, 23bitri 241 . . . . . . . . 9  |-  ( ( x  =  P  \/  x  =  .0.  )  <->  ( x  =/=  .0.  ->  x  =  P ) )
2524imbi2i 304 . . . . . . . 8  |-  ( ( x  .<_  P  ->  ( x  =  P  \/  x  =  .0.  )
)  <->  ( x  .<_  P  ->  ( x  =/= 
.0.  ->  x  =  P ) ) )
2618, 21, 253bitr4g 280 . . . . . . 7  |-  ( ( ( K  e.  AtLat  /\  P  e.  B )  /\  x  e.  B
)  ->  ( (
(  .0.  ( lt
`  K ) x  /\  x  .<_  P )  ->  x  =  P )  <->  ( x  .<_  P  ->  ( x  =  P  \/  x  =  .0.  ) ) ) )
2726ralbidva 2682 . . . . . 6  |-  ( ( K  e.  AtLat  /\  P  e.  B )  ->  ( A. x  e.  B  ( (  .0.  ( lt `  K ) x  /\  x  .<_  P )  ->  x  =  P )  <->  A. x  e.  B  ( x  .<_  P  -> 
( x  =  P  \/  x  =  .0.  ) ) ) )
2814, 27anbi12d 692 . . . . 5  |-  ( ( K  e.  AtLat  /\  P  e.  B )  ->  (
(  .0.  ( lt
`  K ) P  /\  A. x  e.  B  ( (  .0.  ( lt `  K
) x  /\  x  .<_  P )  ->  x  =  P ) )  <->  ( P  =/=  .0.  /\  A. x  e.  B  ( x  .<_  P  ->  ( x  =  P  \/  x  =  .0.  ) ) ) ) )
2913, 28bitr2d 246 . . . 4  |-  ( ( K  e.  AtLat  /\  P  e.  B )  ->  (
( P  =/=  .0.  /\ 
A. x  e.  B  ( x  .<_  P  -> 
( x  =  P  \/  x  =  .0.  ) ) )  <->  .0.  (  <o  `  K ) P ) )
3029pm5.32da 623 . . 3  |-  ( K  e.  AtLat  ->  ( ( P  e.  B  /\  ( P  =/=  .0.  /\ 
A. x  e.  B  ( x  .<_  P  -> 
( x  =  P  \/  x  =  .0.  ) ) ) )  <-> 
( P  e.  B  /\  .0.  (  <o  `  K
) P ) ) )
315, 30bitr4d 248 . 2  |-  ( K  e.  AtLat  ->  ( P  e.  A  <->  ( P  e.  B  /\  ( P  =/=  .0.  /\  A. x  e.  B  (
x  .<_  P  ->  (
x  =  P  \/  x  =  .0.  )
) ) ) ) )
32 3anass 940 . 2  |-  ( ( P  e.  B  /\  P  =/=  .0.  /\  A. x  e.  B  (
x  .<_  P  ->  (
x  =  P  \/  x  =  .0.  )
) )  <->  ( P  e.  B  /\  ( P  =/=  .0.  /\  A. x  e.  B  (
x  .<_  P  ->  (
x  =  P  \/  x  =  .0.  )
) ) ) )
3331, 32syl6bbr 255 1  |-  ( K  e.  AtLat  ->  ( P  e.  A  <->  ( P  e.  B  /\  P  =/= 
.0.  /\  A. x  e.  B  ( x  .<_  P  ->  ( x  =  P  \/  x  =  .0.  ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   A.wral 2666   class class class wbr 4172   ` cfv 5413   Basecbs 13424   lecple 13491   ltcplt 14353   0.cp0 14421    <o ccvr 29745   Atomscatm 29746   AtLatcal 29747
This theorem is referenced by:  atn0  29791  dihlspsnat  31816
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 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-undef 6502  df-riota 6508  df-plt 14370  df-glb 14387  df-p0 14423  df-lat 14430  df-covers 29749  df-ats 29750  df-atl 29781
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