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Theorem atcvreq0 29504
Description: An element covered by an atom must be zero. (atcveq0 22928 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 2283 . . . . . . . 8  |-  ( le
`  K )  =  ( le `  K
)
3 atcvreq0.z . . . . . . . 8  |-  .0.  =  ( 0. `  K )
41, 2, 3atl0le 29494 . . . . . . 7  |-  ( ( K  e.  AtLat  /\  X  e.  B )  ->  .0.  ( le `  K ) X )
543adant3 975 . . . . . 6  |-  ( ( K  e.  AtLat  /\  X  e.  B  /\  P  e.  A )  ->  .0.  ( le `  K ) X )
65adantr 451 . . . . 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 29479 . . . . . 6  |-  ( P  e.  A  ->  P  e.  B )
9 eqid 2283 . . . . . . 7  |-  ( lt
`  K )  =  ( lt `  K
)
10 atcvreq0.c . . . . . . 7  |-  C  =  (  <o  `  K )
111, 9, 10cvrlt 29460 . . . . . 6  |-  ( ( ( K  e.  AtLat  /\  X  e.  B  /\  P  e.  B )  /\  X C P )  ->  X ( lt
`  K ) P )
128, 11syl3anl3 1232 . . . . 5  |-  ( ( ( K  e.  AtLat  /\  X  e.  B  /\  P  e.  A )  /\  X C P )  ->  X ( lt
`  K ) P )
13 atlpos 29491 . . . . . . . 8  |-  ( K  e.  AtLat  ->  K  e.  Poset
)
14133ad2ant1 976 . . . . . . 7  |-  ( ( K  e.  AtLat  /\  X  e.  B  /\  P  e.  A )  ->  K  e.  Poset )
1514adantr 451 . . . . . 6  |-  ( ( ( K  e.  AtLat  /\  X  e.  B  /\  P  e.  A )  /\  X C P )  ->  K  e.  Poset )
161, 3atl0cl 29493 . . . . . . . 8  |-  ( K  e.  AtLat  ->  .0.  e.  B )
17163ad2ant1 976 . . . . . . 7  |-  ( ( K  e.  AtLat  /\  X  e.  B  /\  P  e.  A )  ->  .0.  e.  B )
1817adantr 451 . . . . . 6  |-  ( ( ( K  e.  AtLat  /\  X  e.  B  /\  P  e.  A )  /\  X C P )  ->  .0.  e.  B
)
1983ad2ant3 978 . . . . . . 7  |-  ( ( K  e.  AtLat  /\  X  e.  B  /\  P  e.  A )  ->  P  e.  B )
2019adantr 451 . . . . . 6  |-  ( ( ( K  e.  AtLat  /\  X  e.  B  /\  P  e.  A )  /\  X C P )  ->  P  e.  B
)
21 simpl2 959 . . . . . 6  |-  ( ( ( K  e.  AtLat  /\  X  e.  B  /\  P  e.  A )  /\  X C P )  ->  X  e.  B
)
223, 10, 7atcvr0 29478 . . . . . . . 8  |-  ( ( K  e.  AtLat  /\  P  e.  A )  ->  .0.  C P )
23223adant2 974 . . . . . . 7  |-  ( ( K  e.  AtLat  /\  X  e.  B  /\  P  e.  A )  ->  .0.  C P )
2423adantr 451 . . . . . 6  |-  ( ( ( K  e.  AtLat  /\  X  e.  B  /\  P  e.  A )  /\  X C P )  ->  .0.  C P
)
251, 2, 9, 10cvrnbtwn3 29466 . . . . . 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 1195 . . . . 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 887 . . . 4  |-  ( ( ( K  e.  AtLat  /\  X  e.  B  /\  P  e.  A )  /\  X C P )  ->  .0.  =  X
)
2827eqcomd 2288 . . 3  |-  ( ( ( K  e.  AtLat  /\  X  e.  B  /\  P  e.  A )  /\  X C P )  ->  X  =  .0.  )
2928ex 423 . 2  |-  ( ( K  e.  AtLat  /\  X  e.  B  /\  P  e.  A )  ->  ( X C P  ->  X  =  .0.  ) )
30 breq1 4026 . . 3  |-  ( X  =  .0.  ->  ( X C P  <->  .0.  C P ) )
3123, 30syl5ibrcom 213 . 2  |-  ( ( K  e.  AtLat  /\  X  e.  B  /\  P  e.  A )  ->  ( X  =  .0.  ->  X C P ) )
3229, 31impbid 183 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   class class class wbr 4023   ` cfv 5255   Basecbs 13148   lecple 13215   Posetcpo 14074   ltcplt 14075   0.cp0 14143    <o ccvr 29452   Atomscatm 29453   AtLatcal 29454
This theorem is referenced by:  atncvrN  29505  atcvrj0  29617  1cvrjat  29664
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-poset 14080  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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