Theorem llnnleat 34799

Description: An atom cannot majorize a lattice line. (Contributed by NM, 8-Jul-2012.)

Hypotheses

Ref	Expression
llnnleat.l	|- .<_ = ( le ` K )
llnnleat.a	|- A = ( Atoms ` K )
llnnleat.n	|- N = ( LLines ` K )

Assertion

Ref	Expression
llnnleat	|- ( ( K e. HL /\ X e. N /\ P e. A ) -> -. X .<_ P )

Proof of Theorem llnnleat

Dummy variable q is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp2 1062	. . . 4 |- ( ( K e. HL /\ X e. N /\ P e. A ) -> X e. N )
2		eqid 2622	. . . . . 6 |- ( Base ` K ) = ( Base ` K )
3		eqid 2622	. . . . . 6 |- ( <o ` K ) = ( <o ` K )
4		llnnleat.a	. . . . . 6 |- A = ( Atoms ` K )
5		llnnleat.n	. . . . . 6 |- N = ( LLines ` K )
6	2, 3, 4, 5	islln 34792	. . . . 5 |- ( K e. HL -> ( X e. N <-> ( X e. ( Base ` K ) /\ E. q e. A q ( <o ` K ) X ) ) )
7	6	3ad2ant1 1082	. . . 4 |- ( ( K e. HL /\ X e. N /\ P e. A ) -> ( X e. N <-> ( X e. ( Base ` K ) /\ E. q e. A q ( <o ` K ) X ) ) )
8	1, 7	mpbid 222	. . 3 |- ( ( K e. HL /\ X e. N /\ P e. A ) -> ( X e. ( Base ` K ) /\ E. q e. A q ( <o ` K ) X ) )
9	8	simprd 479	. 2 |- ( ( K e. HL /\ X e. N /\ P e. A ) -> E. q e. A q ( <o ` K ) X )
10		simp11 1091	. . . . . 6 |- ( ( ( K e. HL /\ X e. N /\ P e. A ) /\ q e. A /\ q ( <o ` K ) X ) -> K e. HL )
11		hlatl 34647	. . . . . 6 |- ( K e. HL -> K e. AtLat )
12	10, 11	syl 17	. . . . 5 |- ( ( ( K e. HL /\ X e. N /\ P e. A ) /\ q e. A /\ q ( <o ` K ) X ) -> K e. AtLat )
13		simp2 1062	. . . . 5 |- ( ( ( K e. HL /\ X e. N /\ P e. A ) /\ q e. A /\ q ( <o ` K ) X ) -> q e. A )
14		simp13 1093	. . . . 5 |- ( ( ( K e. HL /\ X e. N /\ P e. A ) /\ q e. A /\ q ( <o ` K ) X ) -> P e. A )
15		eqid 2622	. . . . . 6 |- ( lt ` K ) = ( lt ` K )
16	15, 4	atnlt 34600	. . . . 5 |- ( ( K e. AtLat /\ q e. A /\ P e. A ) -> -. q ( lt ` K ) P )
17	12, 13, 14, 16	syl3anc 1326	. . . 4 |- ( ( ( K e. HL /\ X e. N /\ P e. A ) /\ q e. A /\ q ( <o ` K ) X ) -> -. q ( lt ` K ) P )
18	2, 4	atbase 34576	. . . . . . 7 |- ( q e. A -> q e. ( Base ` K ) )
19	18	3ad2ant2 1083	. . . . . 6 |- ( ( ( K e. HL /\ X e. N /\ P e. A ) /\ q e. A /\ q ( <o ` K ) X ) -> q e. ( Base ` K ) )
20		simp12 1092	. . . . . . 7 |- ( ( ( K e. HL /\ X e. N /\ P e. A ) /\ q e. A /\ q ( <o ` K ) X ) -> X e. N )
21	2, 5	llnbase 34795	. . . . . . 7 |- ( X e. N -> X e. ( Base ` K ) )
22	20, 21	syl 17	. . . . . 6 |- ( ( ( K e. HL /\ X e. N /\ P e. A ) /\ q e. A /\ q ( <o ` K ) X ) -> X e. ( Base ` K ) )
23		simp3 1063	. . . . . 6 |- ( ( ( K e. HL /\ X e. N /\ P e. A ) /\ q e. A /\ q ( <o ` K ) X ) -> q ( <o ` K ) X )
24	2, 15, 3	cvrlt 34557	. . . . . 6 |- ( ( ( K e. HL /\ q e. ( Base ` K ) /\ X e. ( Base ` K ) ) /\ q ( <o ` K ) X ) -> q ( lt ` K ) X )
25	10, 19, 22, 23, 24	syl31anc 1329	. . . . 5 |- ( ( ( K e. HL /\ X e. N /\ P e. A ) /\ q e. A /\ q ( <o ` K ) X ) -> q ( lt ` K ) X )
26		hlpos 34652	. . . . . . 7 |- ( K e. HL -> K e. Poset )
27	10, 26	syl 17	. . . . . 6 |- ( ( ( K e. HL /\ X e. N /\ P e. A ) /\ q e. A /\ q ( <o ` K ) X ) -> K e. Poset )
28	2, 4	atbase 34576	. . . . . . 7 |- ( P e. A -> P e. ( Base ` K ) )
29	14, 28	syl 17	. . . . . 6 |- ( ( ( K e. HL /\ X e. N /\ P e. A ) /\ q e. A /\ q ( <o ` K ) X ) -> P e. ( Base ` K ) )
30		llnnleat.l	. . . . . . 7 |- .<_ = ( le ` K )
31	2, 30, 15	pltletr 16971	. . . . . 6 |- ( ( K e. Poset /\ ( q e. ( Base ` K ) /\ X e. ( Base ` K ) /\ P e. ( Base ` K ) ) ) -> ( ( q ( lt ` K ) X /\ X .<_ P ) -> q ( lt ` K ) P ) )
32	27, 19, 22, 29, 31	syl13anc 1328	. . . . 5 |- ( ( ( K e. HL /\ X e. N /\ P e. A ) /\ q e. A /\ q ( <o ` K ) X ) -> ( ( q ( lt ` K ) X /\ X .<_ P ) -> q ( lt ` K ) P ) )
33	25, 32	mpand 711	. . . 4 |- ( ( ( K e. HL /\ X e. N /\ P e. A ) /\ q e. A /\ q ( <o ` K ) X ) -> ( X .<_ P -> q ( lt ` K ) P ) )
34	17, 33	mtod 189	. . 3 |- ( ( ( K e. HL /\ X e. N /\ P e. A ) /\ q e. A /\ q ( <o ` K ) X ) -> -. X .<_ P )
35	34	rexlimdv3a 3033	. 2 |- ( ( K e. HL /\ X e. N /\ P e. A ) -> ( E. q e. A q ( <o ` K ) X -> -. X .<_ P ) )
36	9, 35	mpd 15	1 |- ( ( K e. HL /\ X e. N /\ P e. A ) -> -. X .<_ P )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   class class class wbr 4653   ` cfv 5888   Basecbs 15857   lecple 15948   Posetcpo 16940   ltcplt 16941   <o ccvr 34549   Atomscatm 34550   AtLatcal 34551   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-preset 16928  df-poset 16946  df-plt 16958  df-glb 16975  df-p0 17039  df-lat 17046  df-covers 34553  df-ats 34554  df-atl 34585  df-cvlat 34609  df-hlat 34638  df-llines 34784

This theorem is referenced by:  llnneat  34800  llnn0  34802  lplnnle2at  34827