Theorem 2llnm3N 34855

Description: Two lattice lines in a lattice plane always meet. (Contributed by NM, 5-Jul-2012.) (New usage is discouraged.)

Hypotheses

Ref	Expression
2llnm3.l	|- .<_ = ( le ` K )
2llnm3.m	|- ./\ = ( meet ` K )
2llnm3.z	|- .0. = ( 0. ` K )
2llnm3.n	|- N = ( LLines ` K )
2llnm3.p	|- P = ( LPlanes ` K )

Assertion

Ref	Expression
2llnm3N	|- ( ( K e. HL /\ ( X e. N /\ Y e. N /\ W e. P ) /\ ( X .<_ W /\ Y .<_ W ) ) -> ( X ./\ Y ) =/= .0. )

Proof of Theorem 2llnm3N

Step	Hyp	Ref	Expression
1		oveq1 6657	. . 3 |- ( X = Y -> ( X ./\ Y ) = ( Y ./\ Y ) )
2	1	neeq1d 2853	. 2 |- ( X = Y -> ( ( X ./\ Y ) =/= .0. <-> ( Y ./\ Y ) =/= .0. ) )
3		simpl1 1064	. . . 4 |- ( ( ( K e. HL /\ ( X e. N /\ Y e. N /\ W e. P ) /\ ( X .<_ W /\ Y .<_ W ) ) /\ X =/= Y ) -> K e. HL )
4		hlatl 34647	. . . 4 |- ( K e. HL -> K e. AtLat )
5	3, 4	syl 17	. . 3 |- ( ( ( K e. HL /\ ( X e. N /\ Y e. N /\ W e. P ) /\ ( X .<_ W /\ Y .<_ W ) ) /\ X =/= Y ) -> K e. AtLat )
6		simpl2 1065	. . . 4 |- ( ( ( K e. HL /\ ( X e. N /\ Y e. N /\ W e. P ) /\ ( X .<_ W /\ Y .<_ W ) ) /\ X =/= Y ) -> ( X e. N /\ Y e. N /\ W e. P ) )
7		simpl3l 1116	. . . 4 |- ( ( ( K e. HL /\ ( X e. N /\ Y e. N /\ W e. P ) /\ ( X .<_ W /\ Y .<_ W ) ) /\ X =/= Y ) -> X .<_ W )
8		simpl3r 1117	. . . 4 |- ( ( ( K e. HL /\ ( X e. N /\ Y e. N /\ W e. P ) /\ ( X .<_ W /\ Y .<_ W ) ) /\ X =/= Y ) -> Y .<_ W )
9		simpr 477	. . . 4 |- ( ( ( K e. HL /\ ( X e. N /\ Y e. N /\ W e. P ) /\ ( X .<_ W /\ Y .<_ W ) ) /\ X =/= Y ) -> X =/= Y )
10		2llnm3.l	. . . . 5 |- .<_ = ( le ` K )
11		2llnm3.m	. . . . 5 |- ./\ = ( meet ` K )
12		eqid 2622	. . . . 5 |- ( Atoms ` K ) = ( Atoms ` K )
13		2llnm3.n	. . . . 5 |- N = ( LLines ` K )
14		2llnm3.p	. . . . 5 |- P = ( LPlanes ` K )
15	10, 11, 12, 13, 14	2llnm2N 34854	. . . 4 |- ( ( K e. HL /\ ( X e. N /\ Y e. N /\ W e. P ) /\ ( X .<_ W /\ Y .<_ W /\ X =/= Y ) ) -> ( X ./\ Y ) e. ( Atoms ` K ) )
16	3, 6, 7, 8, 9, 15	syl113anc 1338	. . 3 |- ( ( ( K e. HL /\ ( X e. N /\ Y e. N /\ W e. P ) /\ ( X .<_ W /\ Y .<_ W ) ) /\ X =/= Y ) -> ( X ./\ Y ) e. ( Atoms ` K ) )
17		2llnm3.z	. . . 4 |- .0. = ( 0. ` K )
18	17, 12	atn0 34595	. . 3 |- ( ( K e. AtLat /\ ( X ./\ Y ) e. ( Atoms ` K ) ) -> ( X ./\ Y ) =/= .0. )
19	5, 16, 18	syl2anc 693	. 2 |- ( ( ( K e. HL /\ ( X e. N /\ Y e. N /\ W e. P ) /\ ( X .<_ W /\ Y .<_ W ) ) /\ X =/= Y ) -> ( X ./\ Y ) =/= .0. )
20		hllat 34650	. . . . 5 |- ( K e. HL -> K e. Lat )
21	20	3ad2ant1 1082	. . . 4 |- ( ( K e. HL /\ ( X e. N /\ Y e. N /\ W e. P ) /\ ( X .<_ W /\ Y .<_ W ) ) -> K e. Lat )
22		simp22 1095	. . . . 5 |- ( ( K e. HL /\ ( X e. N /\ Y e. N /\ W e. P ) /\ ( X .<_ W /\ Y .<_ W ) ) -> Y e. N )
23		eqid 2622	. . . . . 6 |- ( Base ` K ) = ( Base ` K )
24	23, 13	llnbase 34795	. . . . 5 |- ( Y e. N -> Y e. ( Base ` K ) )
25	22, 24	syl 17	. . . 4 |- ( ( K e. HL /\ ( X e. N /\ Y e. N /\ W e. P ) /\ ( X .<_ W /\ Y .<_ W ) ) -> Y e. ( Base ` K ) )
26	23, 11	latmidm 17086	. . . 4 |- ( ( K e. Lat /\ Y e. ( Base ` K ) ) -> ( Y ./\ Y ) = Y )
27	21, 25, 26	syl2anc 693	. . 3 |- ( ( K e. HL /\ ( X e. N /\ Y e. N /\ W e. P ) /\ ( X .<_ W /\ Y .<_ W ) ) -> ( Y ./\ Y ) = Y )
28		simp1 1061	. . . 4 |- ( ( K e. HL /\ ( X e. N /\ Y e. N /\ W e. P ) /\ ( X .<_ W /\ Y .<_ W ) ) -> K e. HL )
29	17, 13	llnn0 34802	. . . 4 |- ( ( K e. HL /\ Y e. N ) -> Y =/= .0. )
30	28, 22, 29	syl2anc 693	. . 3 |- ( ( K e. HL /\ ( X e. N /\ Y e. N /\ W e. P ) /\ ( X .<_ W /\ Y .<_ W ) ) -> Y =/= .0. )
31	27, 30	eqnetrd 2861	. 2 |- ( ( K e. HL /\ ( X e. N /\ Y e. N /\ W e. P ) /\ ( X .<_ W /\ Y .<_ W ) ) -> ( Y ./\ Y ) =/= .0. )
32	2, 19, 31	pm2.61ne 2879	1 |- ( ( K e. HL /\ ( X e. N /\ Y e. N /\ W e. P ) /\ ( X .<_ W /\ Y .<_ W ) ) -> ( X ./\ Y ) =/= .0. )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   lecple 15948   meetcmee 16945   0.cp0 17037   Latclat 17045   Atomscatm 34550   AtLatcal 34551   HLchlt 34637   LLinesclln 34777   LPlanesclpl 34778

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  df-lplanes 34785

This theorem is referenced by: (None)