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Theorem atcvreq0 29480
Description: An element covered by an atom must be zero. (atcveq0 23692 analog.) (Contributed by NM, 4-Nov-2011.)
Hypotheses
Ref Expression
atcvreq0.b  |-  B  =  ( Base `  K
)
atcvreq0.l  |-  .<_  =  ( le `  K )
atcvreq0.z  |-  .0.  =  ( 0. `  K )
atcvreq0.c  |-  C  =  (  <o  `  K )
atcvreq0.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
atcvreq0  |-  ( ( K  e.  AtLat  /\  X  e.  B  /\  P  e.  A )  ->  ( X C P  <->  X  =  .0.  ) )

Proof of Theorem atcvreq0
StepHypRef Expression
1 atcvreq0.b . . . . . . . 8  |-  B  =  ( Base `  K
)
2 eqid 2380 . . . . . . . 8  |-  ( le
`  K )  =  ( le `  K
)
3 atcvreq0.z . . . . . . . 8  |-  .0.  =  ( 0. `  K )
41, 2, 3atl0le 29470 . . . . . . 7  |-  ( ( K  e.  AtLat  /\  X  e.  B )  ->  .0.  ( le `  K ) X )
543adant3 977 . . . . . 6  |-  ( ( K  e.  AtLat  /\  X  e.  B  /\  P  e.  A )  ->  .0.  ( le `  K ) X )
65adantr 452 . . . . 5  |-  ( ( ( K  e.  AtLat  /\  X  e.  B  /\  P  e.  A )  /\  X C P )  ->  .0.  ( le `  K ) X )
7 atcvreq0.a . . . . . . 7  |-  A  =  ( Atoms `  K )
81, 7atbase 29455 . . . . . 6  |-  ( P  e.  A  ->  P  e.  B )
9 eqid 2380 . . . . . . 7  |-  ( lt
`  K )  =  ( lt `  K
)
10 atcvreq0.c . . . . . . 7  |-  C  =  (  <o  `  K )
111, 9, 10cvrlt 29436 . . . . . 6  |-  ( ( ( K  e.  AtLat  /\  X  e.  B  /\  P  e.  B )  /\  X C P )  ->  X ( lt
`  K ) P )
128, 11syl3anl3 1234 . . . . 5  |-  ( ( ( K  e.  AtLat  /\  X  e.  B  /\  P  e.  A )  /\  X C P )  ->  X ( lt
`  K ) P )
13 atlpos 29467 . . . . . . . 8  |-  ( K  e.  AtLat  ->  K  e.  Poset
)
14133ad2ant1 978 . . . . . . 7  |-  ( ( K  e.  AtLat  /\  X  e.  B  /\  P  e.  A )  ->  K  e.  Poset )
1514adantr 452 . . . . . 6  |-  ( ( ( K  e.  AtLat  /\  X  e.  B  /\  P  e.  A )  /\  X C P )  ->  K  e.  Poset )
161, 3atl0cl 29469 . . . . . . . 8  |-  ( K  e.  AtLat  ->  .0.  e.  B )
17163ad2ant1 978 . . . . . . 7  |-  ( ( K  e.  AtLat  /\  X  e.  B  /\  P  e.  A )  ->  .0.  e.  B )
1817adantr 452 . . . . . 6  |-  ( ( ( K  e.  AtLat  /\  X  e.  B  /\  P  e.  A )  /\  X C P )  ->  .0.  e.  B
)
1983ad2ant3 980 . . . . . . 7  |-  ( ( K  e.  AtLat  /\  X  e.  B  /\  P  e.  A )  ->  P  e.  B )
2019adantr 452 . . . . . 6  |-  ( ( ( K  e.  AtLat  /\  X  e.  B  /\  P  e.  A )  /\  X C P )  ->  P  e.  B
)
21 simpl2 961 . . . . . 6  |-  ( ( ( K  e.  AtLat  /\  X  e.  B  /\  P  e.  A )  /\  X C P )  ->  X  e.  B
)
223, 10, 7atcvr0 29454 . . . . . . . 8  |-  ( ( K  e.  AtLat  /\  P  e.  A )  ->  .0.  C P )
23223adant2 976 . . . . . . 7  |-  ( ( K  e.  AtLat  /\  X  e.  B  /\  P  e.  A )  ->  .0.  C P )
2423adantr 452 . . . . . 6  |-  ( ( ( K  e.  AtLat  /\  X  e.  B  /\  P  e.  A )  /\  X C P )  ->  .0.  C P
)
251, 2, 9, 10cvrnbtwn3 29442 . . . . . 6  |-  ( ( K  e.  Poset  /\  (  .0.  e.  B  /\  P  e.  B  /\  X  e.  B )  /\  .0.  C P )  ->  (
(  .0.  ( le
`  K ) X  /\  X ( lt
`  K ) P )  <->  .0.  =  X
) )
2615, 18, 20, 21, 24, 25syl131anc 1197 . . . . 5  |-  ( ( ( K  e.  AtLat  /\  X  e.  B  /\  P  e.  A )  /\  X C P )  ->  ( (  .0.  ( le `  K
) X  /\  X
( lt `  K
) P )  <->  .0.  =  X ) )
276, 12, 26mpbi2and 888 . . . 4  |-  ( ( ( K  e.  AtLat  /\  X  e.  B  /\  P  e.  A )  /\  X C P )  ->  .0.  =  X
)
2827eqcomd 2385 . . 3  |-  ( ( ( K  e.  AtLat  /\  X  e.  B  /\  P  e.  A )  /\  X C P )  ->  X  =  .0.  )
2928ex 424 . 2  |-  ( ( K  e.  AtLat  /\  X  e.  B  /\  P  e.  A )  ->  ( X C P  ->  X  =  .0.  ) )
30 breq1 4149 . . 3  |-  ( X  =  .0.  ->  ( X C P  <->  .0.  C P ) )
3123, 30syl5ibrcom 214 . 2  |-  ( ( K  e.  AtLat  /\  X  e.  B  /\  P  e.  A )  ->  ( X  =  .0.  ->  X C P ) )
3229, 31impbid 184 1  |-  ( ( K  e.  AtLat  /\  X  e.  B  /\  P  e.  A )  ->  ( X C P  <->  X  =  .0.  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   class class class wbr 4146   ` cfv 5387   Basecbs 13389   lecple 13456   Posetcpo 14317   ltcplt 14318   0.cp0 14386    <o ccvr 29428   Atomscatm 29429   AtLatcal 29430
This theorem is referenced by:  atncvrN  29481  atcvrj0  29593  1cvrjat  29640
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 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-undef 6472  df-riota 6478  df-poset 14323  df-plt 14335  df-glb 14352  df-p0 14388  df-lat 14395  df-covers 29432  df-ats 29433  df-atl 29464
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