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Theorem isat3 30119
Description: The predicate "is an atom". (elat2 22936 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 2296 . . . 4  |-  (  <o  `  K )  =  ( 
<o  `  K )
4 isat3.a . . . 4  |-  A  =  ( Atoms `  K )
51, 2, 3, 4isat 30098 . . 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 30115 . . . . . . 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 2296 . . . . . . 7  |-  ( lt
`  K )  =  ( lt `  K
)
121, 10, 11, 3cvrval2 30086 . . . . . 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 30118 . . . . . 6  |-  ( ( K  e.  AtLat  /\  P  e.  B )  ->  (  .0.  ( lt `  K
) P  <->  P  =/=  .0.  ) )
151, 11, 2atlltn0 30118 . . . . . . . . . . 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 2543 . . . . . . . . . 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 2572 . . . . . 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 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   class class class wbr 4039   ` cfv 5271   Basecbs 13164   lecple 13231   ltcplt 14091   0.cp0 14159    <o ccvr 30074   Atomscatm 30075   AtLatcal 30076
This theorem is referenced by:  atn0  30120  dihlspsnat  32145
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-undef 6314  df-riota 6320  df-plt 14108  df-glb 14125  df-p0 14161  df-lat 14168  df-covers 30078  df-ats 30079  df-atl 30110
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