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Metamath Proof Explorer

Theorem pmapeq0

Description: A projective map value is zero iff its argument is lattice zero. (Contributed by NM, 27-Jan-2012)

Ref Expression
Hypotheses pmapeq0.b 
|- B = ( Base ` K )
pmapeq0.z 
|- .0. = ( 0. ` K )
pmapeq0.m 
|- M = ( pmap ` K )
Assertion pmapeq0 
|- ( ( K e. HL /\ X e. B ) -> ( ( M ` X ) = (/) <-> X = .0. ) )

Proof

Step Hyp Ref Expression
1 pmapeq0.b 
 |-  B = ( Base ` K )
2 pmapeq0.z 
 |-  .0. = ( 0. ` K )
3 pmapeq0.m 
 |-  M = ( pmap ` K )
4 hlatl 
 |-  ( K e. HL -> K e. AtLat )
5 4 adantr 
 |-  ( ( K e. HL /\ X e. B ) -> K e. AtLat )
6 2 3 pmap0 
 |-  ( K e. AtLat -> ( M ` .0. ) = (/) )
7 5 6 syl 
 |-  ( ( K e. HL /\ X e. B ) -> ( M ` .0. ) = (/) )
8 7 eqeq2d 
 |-  ( ( K e. HL /\ X e. B ) -> ( ( M ` X ) = ( M ` .0. ) <-> ( M ` X ) = (/) ) )
9 hlop 
 |-  ( K e. HL -> K e. OP )
10 9 adantr 
 |-  ( ( K e. HL /\ X e. B ) -> K e. OP )
11 1 2 op0cl 
 |-  ( K e. OP -> .0. e. B )
12 10 11 syl 
 |-  ( ( K e. HL /\ X e. B ) -> .0. e. B )
13 1 3 pmap11 
 |-  ( ( K e. HL /\ X e. B /\ .0. e. B ) -> ( ( M ` X ) = ( M ` .0. ) <-> X = .0. ) )
14 12 13 mpd3an3 
 |-  ( ( K e. HL /\ X e. B ) -> ( ( M ` X ) = ( M ` .0. ) <-> X = .0. ) )
15 8 14 bitr3d 
 |-  ( ( K e. HL /\ X e. B ) -> ( ( M ` X ) = (/) <-> X = .0. ) )