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Theorem atcvr0 30023
Description: An atom covers zero. (atcv0 23837 analog.) (Contributed by NM, 4-Nov-2011.)
Hypotheses
Ref Expression
atomcvr0.z  |-  .0.  =  ( 0. `  K )
atomcvr0.c  |-  C  =  (  <o  `  K )
atomcvr0.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
atcvr0  |-  ( ( K  e.  D  /\  P  e.  A )  ->  .0.  C P )

Proof of Theorem atcvr0
StepHypRef Expression
1 eqid 2435 . . 3  |-  ( Base `  K )  =  (
Base `  K )
2 atomcvr0.z . . 3  |-  .0.  =  ( 0. `  K )
3 atomcvr0.c . . 3  |-  C  =  (  <o  `  K )
4 atomcvr0.a . . 3  |-  A  =  ( Atoms `  K )
51, 2, 3, 4isat 30021 . 2  |-  ( K  e.  D  ->  ( P  e.  A  <->  ( P  e.  ( Base `  K
)  /\  .0.  C P ) ) )
65simplbda 608 1  |-  ( ( K  e.  D  /\  P  e.  A )  ->  .0.  C P )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   class class class wbr 4204   ` cfv 5446   Basecbs 13461   0.cp0 14458    <o ccvr 29997   Atomscatm 29998
This theorem is referenced by:  0ltat  30026  leatb  30027  atnle0  30044  atlen0  30045  atcmp  30046  atcvreq0  30049  atcvr0eq  30160  lnnat  30161  athgt  30190  ps-2  30212  lhp0lt  30737
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-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
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-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  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-iota 5410  df-fun 5448  df-fv 5454  df-ats 30002
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