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Theorem pmapeq0 30625
Description: A projective map value is zero iff its argument is lattice zero. (Contributed by NM, 27-Jan-2012.)
Hypotheses
Ref Expression
pmapeq0.b  |-  B  =  ( Base `  K
)
pmapeq0.z  |-  .0.  =  ( 0. `  K )
pmapeq0.m  |-  M  =  ( pmap `  K
)
Assertion
Ref Expression
pmapeq0  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( ( M `  X )  =  (/)  <->  X  =  .0.  ) )

Proof of Theorem pmapeq0
StepHypRef Expression
1 hlatl 30220 . . . . 5  |-  ( K  e.  HL  ->  K  e.  AtLat )
21adantr 453 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  K  e.  AtLat )
3 pmapeq0.z . . . . 5  |-  .0.  =  ( 0. `  K )
4 pmapeq0.m . . . . 5  |-  M  =  ( pmap `  K
)
53, 4pmap0 30624 . . . 4  |-  ( K  e.  AtLat  ->  ( M `  .0.  )  =  (/) )
62, 5syl 16 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( M `  .0.  )  =  (/) )
76eqeq2d 2449 . 2  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( ( M `  X )  =  ( M `  .0.  )  <->  ( M `  X )  =  (/) ) )
8 hlop 30222 . . . . 5  |-  ( K  e.  HL  ->  K  e.  OP )
98adantr 453 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  K  e.  OP )
10 pmapeq0.b . . . . 5  |-  B  =  ( Base `  K
)
1110, 3op0cl 30044 . . . 4  |-  ( K  e.  OP  ->  .0.  e.  B )
129, 11syl 16 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  .0.  e.  B )
1310, 4pmap11 30621 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  .0.  e.  B )  -> 
( ( M `  X )  =  ( M `  .0.  )  <->  X  =  .0.  ) )
1412, 13mpd3an3 1281 . 2  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( ( M `  X )  =  ( M `  .0.  )  <->  X  =  .0.  ) )
157, 14bitr3d 248 1  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( ( M `  X )  =  (/)  <->  X  =  .0.  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   (/)c0 3630   ` cfv 5456   Basecbs 13471   0.cp0 14468   OPcops 30032   AtLatcal 30124   HLchlt 30210   pmapcpmap 30356
This theorem is referenced by:  pmapjat1  30712
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-undef 6545  df-riota 6551  df-poset 14405  df-plt 14417  df-lub 14433  df-glb 14434  df-join 14435  df-meet 14436  df-p0 14470  df-lat 14477  df-clat 14539  df-oposet 30036  df-ol 30038  df-oml 30039  df-covers 30126  df-ats 30127  df-atl 30158  df-cvlat 30182  df-hlat 30211  df-pmap 30363
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