Theorem llnn0 34802

Description: A lattice line is nonzero. (Contributed by NM, 15-Jul-2012.)

Hypotheses

Ref	Expression
llnn0.z	|- .0. = ( 0. ` K )
llnn0.n	|- N = ( LLines ` K )

Assertion

Ref	Expression
llnn0	|- ( ( K e. HL /\ X e. N ) -> X =/= .0. )

Proof of Theorem llnn0

Dummy variable p is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . . . 5 |- ( Atoms ` K ) = ( Atoms ` K )
2	1	atex 34692	. . . 4 |- ( K e. HL -> ( Atoms ` K ) =/= (/) )
3		n0 3931	. . . 4 |- ( ( Atoms ` K ) =/= (/) <-> E. p p e. ( Atoms ` K ) )
4	2, 3	sylib 208	. . 3 |- ( K e. HL -> E. p p e. ( Atoms ` K ) )
5	4	adantr 481	. 2 |- ( ( K e. HL /\ X e. N ) -> E. p p e. ( Atoms ` K ) )
6		eqid 2622	. . . . 5 |- ( le ` K ) = ( le ` K )
7		llnn0.n	. . . . 5 |- N = ( LLines ` K )
8	6, 1, 7	llnnleat 34799	. . . 4 |- ( ( K e. HL /\ X e. N /\ p e. ( Atoms ` K ) ) -> -. X ( le ` K ) p )
9	8	3expa 1265	. . 3 |- ( ( ( K e. HL /\ X e. N ) /\ p e. ( Atoms ` K ) ) -> -. X ( le ` K ) p )
10		hlop 34649	. . . . . . 7 |- ( K e. HL -> K e. OP )
11	10	ad2antrr 762	. . . . . 6 |- ( ( ( K e. HL /\ X e. N ) /\ p e. ( Atoms ` K ) ) -> K e. OP )
12		eqid 2622	. . . . . . . 8 |- ( Base ` K ) = ( Base ` K )
13	12, 1	atbase 34576	. . . . . . 7 |- ( p e. ( Atoms ` K ) -> p e. ( Base ` K ) )
14	13	adantl 482	. . . . . 6 |- ( ( ( K e. HL /\ X e. N ) /\ p e. ( Atoms ` K ) ) -> p e. ( Base ` K ) )
15		llnn0.z	. . . . . . 7 |- .0. = ( 0. ` K )
16	12, 6, 15	op0le 34473	. . . . . 6 |- ( ( K e. OP /\ p e. ( Base ` K ) ) -> .0. ( le ` K ) p )
17	11, 14, 16	syl2anc 693	. . . . 5 |- ( ( ( K e. HL /\ X e. N ) /\ p e. ( Atoms ` K ) ) -> .0. ( le ` K ) p )
18		breq1 4656	. . . . 5 |- ( X = .0. -> ( X ( le ` K ) p <-> .0. ( le ` K ) p ) )
19	17, 18	syl5ibrcom 237	. . . 4 |- ( ( ( K e. HL /\ X e. N ) /\ p e. ( Atoms ` K ) ) -> ( X = .0. -> X ( le ` K ) p ) )
20	19	necon3bd 2808	. . 3 |- ( ( ( K e. HL /\ X e. N ) /\ p e. ( Atoms ` K ) ) -> ( -. X ( le ` K ) p -> X =/= .0. ) )
21	9, 20	mpd 15	. 2 |- ( ( ( K e. HL /\ X e. N ) /\ p e. ( Atoms ` K ) ) -> X =/= .0. )
22	5, 21	exlimddv 1863	1 |- ( ( K e. HL /\ X e. N ) -> X =/= .0. )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   (/)c0 3915   class class class wbr 4653   ` cfv 5888   Basecbs 15857   lecple 15948   0.cp0 17037   OPcops 34459   Atomscatm 34550   HLchlt 34637   LLinesclln 34777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-preset 16928  df-poset 16946  df-plt 16958  df-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-p0 17039  df-p1 17040  df-lat 17046  df-clat 17108  df-oposet 34463  df-ol 34465  df-oml 34466  df-covers 34553  df-ats 34554  df-atl 34585  df-cvlat 34609  df-hlat 34638  df-llines 34784

This theorem is referenced by:  2llnm3N  34855  cdleme22b  35629