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Metamath Proof Explorer

Theorem lplnn0N

Description: A lattice plane is nonzero. (Contributed by NM, 15-Jul-2012) (New usage is discouraged.)

Ref Expression
Hypotheses lplnn0.z 
|- .0. = ( 0. ` K )
lplnn0.p 
|- P = ( LPlanes ` K )
Assertion lplnn0N 
|- ( ( K e. HL /\ X e. P ) -> X =/= .0. )

Proof

Step Hyp Ref Expression
1 lplnn0.z 
 |-  .0. = ( 0. ` K )
2 lplnn0.p 
 |-  P = ( LPlanes ` K )
3 eqid 
 |-  ( Atoms ` K ) = ( Atoms ` K )
4 3 atex 
 |-  ( K e. HL -> ( Atoms ` K ) =/= (/) )
5 n0 
 |-  ( ( Atoms ` K ) =/= (/) <-> E. p p e. ( Atoms ` K ) )
6 4 5 sylib 
 |-  ( K e. HL -> E. p p e. ( Atoms ` K ) )
7 6 adantr 
 |-  ( ( K e. HL /\ X e. P ) -> E. p p e. ( Atoms ` K ) )
8 eqid 
 |-  ( le ` K ) = ( le ` K )
9 8 3 2 lplnnleat 
 |-  ( ( K e. HL /\ X e. P /\ p e. ( Atoms ` K ) ) -> -. X ( le ` K ) p )
10 9 3expa 
 |-  ( ( ( K e. HL /\ X e. P ) /\ p e. ( Atoms ` K ) ) -> -. X ( le ` K ) p )
11 hlop 
 |-  ( K e. HL -> K e. OP )
12 11 ad2antrr 
 |-  ( ( ( K e. HL /\ X e. P ) /\ p e. ( Atoms ` K ) ) -> K e. OP )
13 eqid 
 |-  ( Base ` K ) = ( Base ` K )
14 13 3 atbase 
 |-  ( p e. ( Atoms ` K ) -> p e. ( Base ` K ) )
15 14 adantl 
 |-  ( ( ( K e. HL /\ X e. P ) /\ p e. ( Atoms ` K ) ) -> p e. ( Base ` K ) )
16 13 8 1 op0le 
 |-  ( ( K e. OP /\ p e. ( Base ` K ) ) -> .0. ( le ` K ) p )
17 12 15 16 syl2anc 
 |-  ( ( ( K e. HL /\ X e. P ) /\ p e. ( Atoms ` K ) ) -> .0. ( le ` K ) p )
18 breq1 
 |-  ( X = .0. -> ( X ( le ` K ) p <-> .0. ( le ` K ) p ) )
19 17 18 syl5ibrcom 
 |-  ( ( ( K e. HL /\ X e. P ) /\ p e. ( Atoms ` K ) ) -> ( X = .0. -> X ( le ` K ) p ) )
20 19 necon3bd 
 |-  ( ( ( K e. HL /\ X e. P ) /\ p e. ( Atoms ` K ) ) -> ( -. X ( le ` K ) p -> X =/= .0. ) )
21 10 20 mpd 
 |-  ( ( ( K e. HL /\ X e. P ) /\ p e. ( Atoms ` K ) ) -> X =/= .0. )
22 7 21 exlimddv 
 |-  ( ( K e. HL /\ X e. P ) -> X =/= .0. )