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Theorem atcvreq0 30126
Description: An element covered by an atom must be zero. (atcveq0 22944 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 2296 . . . . . . . 8  |-  ( le
`  K )  =  ( le `  K
)
3 atcvreq0.z . . . . . . . 8  |-  .0.  =  ( 0. `  K )
41, 2, 3atl0le 30116 . . . . . . 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 30101 . . . . . 6  |-  ( P  e.  A  ->  P  e.  B )
9 eqid 2296 . . . . . . 7  |-  ( lt
`  K )  =  ( lt `  K
)
10 atcvreq0.c . . . . . . 7  |-  C  =  (  <o  `  K )
111, 9, 10cvrlt 30082 . . . . . 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 30113 . . . . . . . 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 30115 . . . . . . . 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 30100 . . . . . . . 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 30088 . . . . . 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 2301 . . 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 4042 . . 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 1632    e. wcel 1696   class class class wbr 4039   ` cfv 5271   Basecbs 13164   lecple 13231   Posetcpo 14090   ltcplt 14091   0.cp0 14159    <o ccvr 30074   Atomscatm 30075   AtLatcal 30076
This theorem is referenced by:  atncvrN  30127  atcvrj0  30239  1cvrjat  30286
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-poset 14096  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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