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Theorem isat3 29497
Description: The predicate "is an atom". (elat2 22920 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 2283 . . . 4  |-  (  <o  `  K )  =  ( 
<o  `  K )
4 isat3.a . . . 4  |-  A  =  ( Atoms `  K )
51, 2, 3, 4isat 29476 . . 3  |-  ( K  e.  AtLat  ->  ( P  e.  A  <->  ( P  e.  B  /\  .0.  (  <o  `  K ) P ) ) )
6 simpl 443 . . . . . 6  |-  ( ( K  e.  AtLat  /\  P  e.  B )  ->  K  e.  AtLat )
71, 2atl0cl 29493 . . . . . . 7  |-  ( K  e.  AtLat  ->  .0.  e.  B )
87adantr 451 . . . . . 6  |-  ( ( K  e.  AtLat  /\  P  e.  B )  ->  .0.  e.  B )
9 simpr 447 . . . . . 6  |-  ( ( K  e.  AtLat  /\  P  e.  B )  ->  P  e.  B )
10 isat3.l . . . . . . 7  |-  .<_  =  ( le `  K )
11 eqid 2283 . . . . . . 7  |-  ( lt
`  K )  =  ( lt `  K
)
121, 10, 11, 3cvrval2 29464 . . . . . 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 1182 . . . . 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 29496 . . . . . 6  |-  ( ( K  e.  AtLat  /\  P  e.  B )  ->  (  .0.  ( lt `  K
) P  <->  P  =/=  .0.  ) )
151, 11, 2atlltn0 29496 . . . . . . . . . . 11  |-  ( ( K  e.  AtLat  /\  x  e.  B )  ->  (  .0.  ( lt `  K
) x  <->  x  =/=  .0.  ) )
1615adantlr 695 . . . . . . . . . 10  |-  ( ( ( K  e.  AtLat  /\  P  e.  B )  /\  x  e.  B
)  ->  (  .0.  ( lt `  K ) x  <->  x  =/=  .0.  ) )
1716imbi1d 308 . . . . . . . . 9  |-  ( ( ( K  e.  AtLat  /\  P  e.  B )  /\  x  e.  B
)  ->  ( (  .0.  ( lt `  K
) x  ->  x  =  P )  <->  ( x  =/=  .0.  ->  x  =  P ) ) )
1817imbi2d 307 . . . . . . . 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 433 . . . . . . . . 9  |-  ( ( (  .0.  ( lt
`  K ) x  /\  x  .<_  P )  ->  x  =  P )  <->  (  .0.  ( lt `  K ) x  ->  ( x  .<_  P  ->  x  =  P ) ) )
20 bi2.04 350 . . . . . . . . 9  |-  ( (  .0.  ( lt `  K ) x  -> 
( x  .<_  P  ->  x  =  P )
)  <->  ( x  .<_  P  ->  (  .0.  ( lt `  K ) x  ->  x  =  P ) ) )
2119, 20bitri 240 . . . . . . . 8  |-  ( ( (  .0.  ( lt
`  K ) x  /\  x  .<_  P )  ->  x  =  P )  <->  ( x  .<_  P  ->  (  .0.  ( lt `  K ) x  ->  x  =  P ) ) )
22 orcom 376 . . . . . . . . . 10  |-  ( ( x  =  P  \/  x  =  .0.  )  <->  ( x  =  .0.  \/  x  =  P )
)
23 neor 2530 . . . . . . . . . 10  |-  ( ( x  =  .0.  \/  x  =  P )  <->  ( x  =/=  .0.  ->  x  =  P ) )
2422, 23bitri 240 . . . . . . . . 9  |-  ( ( x  =  P  \/  x  =  .0.  )  <->  ( x  =/=  .0.  ->  x  =  P ) )
2524imbi2i 303 . . . . . . . 8  |-  ( ( x  .<_  P  ->  ( x  =  P  \/  x  =  .0.  )
)  <->  ( x  .<_  P  ->  ( x  =/= 
.0.  ->  x  =  P ) ) )
2618, 21, 253bitr4g 279 . . . . . . 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 2559 . . . . . 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 691 . . . . 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 245 . . . 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 622 . . 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 247 . 2  |-  ( K  e.  AtLat  ->  ( P  e.  A  <->  ( P  e.  B  /\  ( P  =/=  .0.  /\  A. x  e.  B  (
x  .<_  P  ->  (
x  =  P  \/  x  =  .0.  )
) ) ) ) )
32 3anass 938 . 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 254 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 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   class class class wbr 4023   ` cfv 5255   Basecbs 13148   lecple 13215   ltcplt 14075   0.cp0 14143    <o ccvr 29452   Atomscatm 29453   AtLatcal 29454
This theorem is referenced by:  atn0  29498  dihlspsnat  31523
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-undef 6298  df-riota 6304  df-plt 14092  df-glb 14109  df-p0 14145  df-lat 14152  df-covers 29456  df-ats 29457  df-atl 29488
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