Theorem lnatexN 33053

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 2429	. . . 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 33050	. . 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 1364	. 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 1059	. . . . . 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 1060	. . . . . . . 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 2702	. . . . . . 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 462	. . . . . . 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 2726	. . . . . 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 1080	. . . . . . . 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 1058	. . . . . . . 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 32650	. . . . . . . 8 |- ( ( K e. HL /\ r e. A /\ s e. A ) -> s .<_ ( r ( join ` K ) s ) )
17	13, 14, 8, 16	syl3anc 1264	. . . . . . 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 1061	. . . . . . 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 4452	. . . . . 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 2712	. . . . . . . 8 |- ( q = s -> ( q =/= P <-> s =/= P ) )
21		breq1 4429	. . . . . . . 8 |- ( q = s -> ( q .<_ X <-> s .<_ X ) )
22	20, 21	anbi12d 715	. . . . . . 7 |- ( q = s -> ( ( q =/= P /\ q .<_ X ) <-> ( s =/= P /\ s .<_ X ) ) )
23	22	rspcev 3188	. . . . . 6 |- ( ( s e. A /\ ( s =/= P /\ s .<_ X ) ) -> E. q e. A ( q =/= P /\ q .<_ X ) )
24	8, 12, 19, 23	syl12anc 1262	. . . . 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 1058	. . . . . 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 462	. . . . . 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 1080	. . . . . . . 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 1059	. . . . . . . 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 32649	. . . . . . . 8 |- ( ( K e. HL /\ r e. A /\ s e. A ) -> r .<_ ( r ( join ` K ) s ) )
30	27, 25, 28, 29	syl3anc 1264	. . . . . . 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 1061	. . . . . . 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 4452	. . . . . 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 2712	. . . . . . . 8 |- ( q = r -> ( q =/= P <-> r =/= P ) )
34		breq1 4429	. . . . . . . 8 |- ( q = r -> ( q .<_ X <-> r .<_ X ) )
35	33, 34	anbi12d 715	. . . . . . 7 |- ( q = r -> ( ( q =/= P /\ q .<_ X ) <-> ( r =/= P /\ r .<_ X ) ) )
36	35	rspcev 3188	. . . . . 6 |- ( ( r e. A /\ ( r =/= P /\ r .<_ X ) ) -> E. q e. A ( q =/= P /\ q .<_ X ) )
37	25, 26, 32, 36	syl12anc 1262	. . . . 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 2749	. . . 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 1204	. . 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 2930	. 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 370   /\ w3a 982   = wceq 1437   e. wcel 1870   =/= wne 2625   E.wrex 2783   class class class wbr 4426   ` cfv 5601  (class class class)co 6305   Basecbs 15084   lecple 15159   joincjn 16140   Atomscatm 32538   HLchlt 32625   Linesclines 32768   pmapcpmap 32771

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-preset 16124  df-poset 16142  df-plt 16155  df-lub 16171  df-glb 16172  df-join 16173  df-meet 16174  df-p0 16236  df-lat 16243  df-clat 16305  df-oposet 32451  df-ol 32453  df-oml 32454  df-covers 32541  df-ats 32542  df-atl 32573  df-cvlat 32597  df-hlat 32626  df-lines 32775  df-pmap 32778

This theorem is referenced by:  lnjatN  33054