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Theorem isat3 29423
Description: The predicate "is an atom". (elat2 23692 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 2388 . . . 4  |-  (  <o  `  K )  =  ( 
<o  `  K )
4 isat3.a . . . 4  |-  A  =  ( Atoms `  K )
51, 2, 3, 4isat 29402 . . 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 29419 . . . . . . 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 2388 . . . . . . 7  |-  ( lt
`  K )  =  ( lt `  K
)
121, 10, 11, 3cvrval2 29390 . . . . . 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 29422 . . . . . 6  |-  ( ( K  e.  AtLat  /\  P  e.  B )  ->  (  .0.  ( lt `  K
) P  <->  P  =/=  .0.  ) )
151, 11, 2atlltn0 29422 . . . . . . . . . . 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 2635 . . . . . . . . . 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 2666 . . . . . 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 1717    =/= wne 2551   A.wral 2650   class class class wbr 4154   ` cfv 5395   Basecbs 13397   lecple 13464   ltcplt 14326   0.cp0 14394    <o ccvr 29378   Atomscatm 29379   AtLatcal 29380
This theorem is referenced by:  atn0  29424  dihlspsnat  31449
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-undef 6480  df-riota 6486  df-plt 14343  df-glb 14360  df-p0 14396  df-lat 14403  df-covers 29382  df-ats 29383  df-atl 29414
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