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Metamath Proof Explorer

Theorem dia0eldmN

Description: The lattice zero belongs to the domain of partial isomorphism A. (Contributed by NM, 5-Dec-2013) (New usage is discouraged.)

Ref Expression
Hypotheses dia0eldm.z 
|- .0. = ( 0. ` K )
dia0eldm.h 
|- H = ( LHyp ` K )
dia0eldm.i 
|- I = ( ( DIsoA ` K ) ` W )
Assertion dia0eldmN 
|- ( ( K e. HL /\ W e. H ) -> .0. e. dom I )

Proof

Step Hyp Ref Expression
1 dia0eldm.z 
 |-  .0. = ( 0. ` K )
2 dia0eldm.h 
 |-  H = ( LHyp ` K )
3 dia0eldm.i 
 |-  I = ( ( DIsoA ` K ) ` W )
4 hlop 
 |-  ( K e. HL -> K e. OP )
5 4 adantr 
 |-  ( ( K e. HL /\ W e. H ) -> K e. OP )
6 eqid 
 |-  ( Base ` K ) = ( Base ` K )
7 6 1 op0cl 
 |-  ( K e. OP -> .0. e. ( Base ` K ) )
8 5 7 syl 
 |-  ( ( K e. HL /\ W e. H ) -> .0. e. ( Base ` K ) )
9 6 2 lhpbase 
 |-  ( W e. H -> W e. ( Base ` K ) )
10 eqid 
 |-  ( le ` K ) = ( le ` K )
11 6 10 1 op0le 
 |-  ( ( K e. OP /\ W e. ( Base ` K ) ) -> .0. ( le ` K ) W )
12 4 9 11 syl2an 
 |-  ( ( K e. HL /\ W e. H ) -> .0. ( le ` K ) W )
13 6 10 2 3 diaeldm 
 |-  ( ( K e. HL /\ W e. H ) -> ( .0. e. dom I <-> ( .0. e. ( Base ` K ) /\ .0. ( le ` K ) W ) ) )
14 8 12 13 mpbir2and 
 |-  ( ( K e. HL /\ W e. H ) -> .0. e. dom I )