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Theorem atcvreq0 30049
Description: An element covered by an atom must be zero. (atcveq0 23843 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 2435 . . . . . . . 8  |-  ( le
`  K )  =  ( le `  K
)
3 atcvreq0.z . . . . . . . 8  |-  .0.  =  ( 0. `  K )
41, 2, 3atl0le 30039 . . . . . . 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 30024 . . . . . 6  |-  ( P  e.  A  ->  P  e.  B )
9 eqid 2435 . . . . . . 7  |-  ( lt
`  K )  =  ( lt `  K
)
10 atcvreq0.c . . . . . . 7  |-  C  =  (  <o  `  K )
111, 9, 10cvrlt 30005 . . . . . 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 30036 . . . . . . . 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 30038 . . . . . . . 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 30023 . . . . . . . 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 30011 . . . . . 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 2440 . . 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 4207 . . 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 1652    e. wcel 1725   class class class wbr 4204   ` cfv 5446   Basecbs 13461   lecple 13528   Posetcpo 14389   ltcplt 14390   0.cp0 14458    <o ccvr 29997   Atomscatm 29998   AtLatcal 29999
This theorem is referenced by:  atncvrN  30050  atcvrj0  30162  1cvrjat  30209
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-undef 6535  df-riota 6541  df-poset 14395  df-plt 14407  df-glb 14424  df-p0 14460  df-lat 14467  df-covers 30001  df-ats 30002  df-atl 30033
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