Theorem lnatexN 34931

Description: There is an atom in a line different from any other. (Contributed by NM, 30-Apr-2012.) (New usage is discouraged.)

Hypotheses

Ref	Expression
lnatex.b	|- B = ( Base ` K )
lnatex.l	|- .<_ = ( le ` K )
lnatex.a	|- A = ( Atoms ` K )
lnatex.n	|- N = ( Lines ` K )
lnatex.m	|- M = ( pmap ` K )

Assertion

Ref	Expression
lnatexN	|- ( ( K e. HL /\ X e. B /\ ( M ` X ) e. N ) -> E. q e. A ( q =/= P /\ q .<_ X ) )

Distinct variable groups:   A, q   .<_ , q   P, q   X, q

Allowed substitution hints:   B( q)   K( q)   M( q)   N( q)

Proof of Theorem lnatexN

Dummy variables r s are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lnatex.b	. . . 4 |- B = ( Base ` K )
2		eqid 2467	. . . 4 |- ( join ` K ) = ( join ` K )
3		lnatex.a	. . . 4 |- A = ( Atoms ` K )
4		lnatex.n	. . . 4 |- N = ( Lines ` K )
5		lnatex.m	. . . 4 |- M = ( pmap ` K )
6	1, 2, 3, 4, 5	isline3 34928	. . 3 |- ( ( K e. HL /\ X e. B ) -> ( ( M ` X ) e. N <-> E. r e. A E. s e. A ( r =/= s /\ X = ( r ( join ` K ) s ) ) ) )
7	6	biimp3a 1328	. 2 |- ( ( K e. HL /\ X e. B /\ ( M ` X ) e. N ) -> E. r e. A E. s e. A ( r =/= s /\ X = ( r ( join ` K ) s ) ) )
8		simpl2r 1050	. . . . . 6 |- ( ( ( ( K e. HL /\ X e. B /\ ( M ` X ) e. N ) /\ ( r e. A /\ s e. A ) /\ ( r =/= s /\ X = ( r ( join ` K ) s ) ) ) /\ r = P ) -> s e. A )
9		simpl3l 1051	. . . . . . . 8 |- ( ( ( ( K e. HL /\ X e. B /\ ( M ` X ) e. N ) /\ ( r e. A /\ s e. A ) /\ ( r =/= s /\ X = ( r ( join ` K ) s ) ) ) /\ r = P ) -> r =/= s )
10	9	necomd 2738	. . . . . . 7 |- ( ( ( ( K e. HL /\ X e. B /\ ( M ` X ) e. N ) /\ ( r e. A /\ s e. A ) /\ ( r =/= s /\ X = ( r ( join ` K ) s ) ) ) /\ r = P ) -> s =/= r )
11		simpr 461	. . . . . . 7 |- ( ( ( ( K e. HL /\ X e. B /\ ( M ` X ) e. N ) /\ ( r e. A /\ s e. A ) /\ ( r =/= s /\ X = ( r ( join ` K ) s ) ) ) /\ r = P ) -> r = P )
12	10, 11	neeqtrd 2762	. . . . . 6 |- ( ( ( ( K e. HL /\ X e. B /\ ( M ` X ) e. N ) /\ ( r e. A /\ s e. A ) /\ ( r =/= s /\ X = ( r ( join ` K ) s ) ) ) /\ r = P ) -> s =/= P )
13		simpl11 1071	. . . . . . . 8 |- ( ( ( ( K e. HL /\ X e. B /\ ( M ` X ) e. N ) /\ ( r e. A /\ s e. A ) /\ ( r =/= s /\ X = ( r ( join ` K ) s ) ) ) /\ r = P ) -> K e. HL )
14		simpl2l 1049	. . . . . . . 8 |- ( ( ( ( K e. HL /\ X e. B /\ ( M ` X ) e. N ) /\ ( r e. A /\ s e. A ) /\ ( r =/= s /\ X = ( r ( join ` K ) s ) ) ) /\ r = P ) -> r e. A )
15		lnatex.l	. . . . . . . . 9 |- .<_ = ( le ` K )
16	15, 2, 3	hlatlej2 34528	. . . . . . . 8 |- ( ( K e. HL /\ r e. A /\ s e. A ) -> s .<_ ( r ( join ` K ) s ) )
17	13, 14, 8, 16	syl3anc 1228	. . . . . . 7 |- ( ( ( ( K e. HL /\ X e. B /\ ( M ` X ) e. N ) /\ ( r e. A /\ s e. A ) /\ ( r =/= s /\ X = ( r ( join ` K ) s ) ) ) /\ r = P ) -> s .<_ ( r ( join ` K ) s ) )
18		simpl3r 1052	. . . . . . 7 |- ( ( ( ( K e. HL /\ X e. B /\ ( M ` X ) e. N ) /\ ( r e. A /\ s e. A ) /\ ( r =/= s /\ X = ( r ( join ` K ) s ) ) ) /\ r = P ) -> X = ( r ( join ` K ) s ) )
19	17, 18	breqtrrd 4479	. . . . . 6 |- ( ( ( ( K e. HL /\ X e. B /\ ( M ` X ) e. N ) /\ ( r e. A /\ s e. A ) /\ ( r =/= s /\ X = ( r ( join ` K ) s ) ) ) /\ r = P ) -> s .<_ X )
20		neeq1 2748	. . . . . . . 8 |- ( q = s -> ( q =/= P <-> s =/= P ) )
21		breq1 4456	. . . . . . . 8 |- ( q = s -> ( q .<_ X <-> s .<_ X ) )
22	20, 21	anbi12d 710	. . . . . . 7 |- ( q = s -> ( ( q =/= P /\ q .<_ X ) <-> ( s =/= P /\ s .<_ X ) ) )
23	22	rspcev 3219	. . . . . 6 |- ( ( s e. A /\ ( s =/= P /\ s .<_ X ) ) -> E. q e. A ( q =/= P /\ q .<_ X ) )
24	8, 12, 19, 23	syl12anc 1226	. . . . 5 |- ( ( ( ( K e. HL /\ X e. B /\ ( M ` X ) e. N ) /\ ( r e. A /\ s e. A ) /\ ( r =/= s /\ X = ( r ( join ` K ) s ) ) ) /\ r = P ) -> E. q e. A ( q =/= P /\ q .<_ X ) )
25		simpl2l 1049	. . . . . 6 |- ( ( ( ( K e. HL /\ X e. B /\ ( M ` X ) e. N ) /\ ( r e. A /\ s e. A ) /\ ( r =/= s /\ X = ( r ( join ` K ) s ) ) ) /\ r =/= P ) -> r e. A )
26		simpr 461	. . . . . 6 |- ( ( ( ( K e. HL /\ X e. B /\ ( M ` X ) e. N ) /\ ( r e. A /\ s e. A ) /\ ( r =/= s /\ X = ( r ( join ` K ) s ) ) ) /\ r =/= P ) -> r =/= P )
27		simpl11 1071	. . . . . . . 8 |- ( ( ( ( K e. HL /\ X e. B /\ ( M ` X ) e. N ) /\ ( r e. A /\ s e. A ) /\ ( r =/= s /\ X = ( r ( join ` K ) s ) ) ) /\ r =/= P ) -> K e. HL )
28		simpl2r 1050	. . . . . . . 8 |- ( ( ( ( K e. HL /\ X e. B /\ ( M ` X ) e. N ) /\ ( r e. A /\ s e. A ) /\ ( r =/= s /\ X = ( r ( join ` K ) s ) ) ) /\ r =/= P ) -> s e. A )
29	15, 2, 3	hlatlej1 34527	. . . . . . . 8 |- ( ( K e. HL /\ r e. A /\ s e. A ) -> r .<_ ( r ( join ` K ) s ) )
30	27, 25, 28, 29	syl3anc 1228	. . . . . . 7 |- ( ( ( ( K e. HL /\ X e. B /\ ( M ` X ) e. N ) /\ ( r e. A /\ s e. A ) /\ ( r =/= s /\ X = ( r ( join ` K ) s ) ) ) /\ r =/= P ) -> r .<_ ( r ( join ` K ) s ) )
31		simpl3r 1052	. . . . . . 7 |- ( ( ( ( K e. HL /\ X e. B /\ ( M ` X ) e. N ) /\ ( r e. A /\ s e. A ) /\ ( r =/= s /\ X = ( r ( join ` K ) s ) ) ) /\ r =/= P ) -> X = ( r ( join ` K ) s ) )
32	30, 31	breqtrrd 4479	. . . . . 6 |- ( ( ( ( K e. HL /\ X e. B /\ ( M ` X ) e. N ) /\ ( r e. A /\ s e. A ) /\ ( r =/= s /\ X = ( r ( join ` K ) s ) ) ) /\ r =/= P ) -> r .<_ X )
33		neeq1 2748	. . . . . . . 8 |- ( q = r -> ( q =/= P <-> r =/= P ) )
34		breq1 4456	. . . . . . . 8 |- ( q = r -> ( q .<_ X <-> r .<_ X ) )
35	33, 34	anbi12d 710	. . . . . . 7 |- ( q = r -> ( ( q =/= P /\ q .<_ X ) <-> ( r =/= P /\ r .<_ X ) ) )
36	35	rspcev 3219	. . . . . 6 |- ( ( r e. A /\ ( r =/= P /\ r .<_ X ) ) -> E. q e. A ( q =/= P /\ q .<_ X ) )
37	25, 26, 32, 36	syl12anc 1226	. . . . 5 |- ( ( ( ( K e. HL /\ X e. B /\ ( M ` X ) e. N ) /\ ( r e. A /\ s e. A ) /\ ( r =/= s /\ X = ( r ( join ` K ) s ) ) ) /\ r =/= P ) -> E. q e. A ( q =/= P /\ q .<_ X ) )
38	24, 37	pm2.61dane 2785	. . . 4 |- ( ( ( K e. HL /\ X e. B /\ ( M ` X ) e. N ) /\ ( r e. A /\ s e. A ) /\ ( r =/= s /\ X = ( r ( join ` K ) s ) ) ) -> E. q e. A ( q =/= P /\ q .<_ X ) )
39	38	3exp 1195	. . 3 |- ( ( K e. HL /\ X e. B /\ ( M ` X ) e. N ) -> ( ( r e. A /\ s e. A ) -> ( ( r =/= s /\ X = ( r ( join ` K ) s ) ) -> E. q e. A ( q =/= P /\ q .<_ X ) ) ) )
40	39	rexlimdvv 2965	. 2 |- ( ( K e. HL /\ X e. B /\ ( M ` X ) e. N ) -> ( E. r e. A E. s e. A ( r =/= s /\ X = ( r ( join ` K ) s ) ) -> E. q e. A ( q =/= P /\ q .<_ X ) ) )
41	7, 40	mpd 15	1 |- ( ( K e. HL /\ X e. B /\ ( M ` X ) e. N ) -> E. q e. A ( q =/= P /\ q .<_ X ) )

Syntax hints:   -> wi 4   /\ wa 369   /\ w3a 973   = wceq 1379   e. wcel 1767   =/= wne 2662   E.wrex 2818   class class class wbr 4453   ` cfv 5594  (class class class)co 6295   Basecbs 14507   lecple 14579   joincjn 15448   Atomscatm 34416   HLchlt 34503   Linesclines 34646   pmapcpmap 34649

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-poset 15450  df-plt 15462  df-lub 15478  df-glb 15479  df-join 15480  df-meet 15481  df-p0 15543  df-lat 15550  df-clat 15612  df-oposet 34329  df-ol 34331  df-oml 34332  df-covers 34419  df-ats 34420  df-atl 34451  df-cvlat 34475  df-hlat 34504  df-lines 34653  df-pmap 34656

This theorem is referenced by:  lnjatN  34932