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Theorem atcvreq0 29797
Description: An element covered by an atom must be zero. (atcveq0 23804 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 2404 . . . . . . . 8  |-  ( le
`  K )  =  ( le `  K
)
3 atcvreq0.z . . . . . . . 8  |-  .0.  =  ( 0. `  K )
41, 2, 3atl0le 29787 . . . . . . 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 29772 . . . . . 6  |-  ( P  e.  A  ->  P  e.  B )
9 eqid 2404 . . . . . . 7  |-  ( lt
`  K )  =  ( lt `  K
)
10 atcvreq0.c . . . . . . 7  |-  C  =  (  <o  `  K )
111, 9, 10cvrlt 29753 . . . . . 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 29784 . . . . . . . 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 29786 . . . . . . . 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 29771 . . . . . . . 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 29759 . . . . . 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 2409 . . 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 4175 . . 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 1721   class class class wbr 4172   ` cfv 5413   Basecbs 13424   lecple 13491   Posetcpo 14352   ltcplt 14353   0.cp0 14421    <o ccvr 29745   Atomscatm 29746   AtLatcal 29747
This theorem is referenced by:  atncvrN  29798  atcvrj0  29910  1cvrjat  29957
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-undef 6502  df-riota 6508  df-poset 14358  df-plt 14370  df-glb 14387  df-p0 14423  df-lat 14430  df-covers 29749  df-ats 29750  df-atl 29781
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