Theorem lnatexN 33263

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 2438	. . . 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 33260	. . 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 1318	. 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 1042	. . . . . 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 1043	. . . . . . . 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 2690	. . . . . . 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 2625	. . . . . 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 1063	. . . . . . . 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 1041	. . . . . . . 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 32860	. . . . . . . 8 |- ( ( K e. HL /\ r e. A /\ s e. A ) -> s .<_ ( r ( join ` K ) s ) )
17	13, 14, 8, 16	syl3anc 1218	. . . . . . 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 1044	. . . . . . 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 4313	. . . . . 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 2611	. . . . . . . 8 |- ( q = s -> ( q =/= P <-> s =/= P ) )
21		breq1 4290	. . . . . . . 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 3068	. . . . . 6 |- ( ( s e. A /\ ( s =/= P /\ s .<_ X ) ) -> E. q e. A ( q =/= P /\ q .<_ X ) )
24	8, 12, 19, 23	syl12anc 1216	. . . . 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 1041	. . . . . 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 1063	. . . . . . . 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 1042	. . . . . . . 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 32859	. . . . . . . 8 |- ( ( K e. HL /\ r e. A /\ s e. A ) -> r .<_ ( r ( join ` K ) s ) )
30	27, 25, 28, 29	syl3anc 1218	. . . . . . 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 1044	. . . . . . 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 4313	. . . . . 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 2611	. . . . . . . 8 |- ( q = r -> ( q =/= P <-> r =/= P ) )
34		breq1 4290	. . . . . . . 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 3068	. . . . . 6 |- ( ( r e. A /\ ( r =/= P /\ r .<_ X ) ) -> E. q e. A ( q =/= P /\ q .<_ X ) )
37	25, 26, 32, 36	syl12anc 1216	. . . . 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 2684	. . . 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 1186	. . 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 2842	. 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 965   = wceq 1369   e. wcel 1756   =/= wne 2601   E.wrex 2711   class class class wbr 4287   ` cfv 5413  (class class class)co 6086   Basecbs 14166   lecple 14237   joincjn 15106   Atomscatm 32748   HLchlt 32835   Linesclines 32978   pmapcpmap 32981

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-poset 15108  df-plt 15120  df-lub 15136  df-glb 15137  df-join 15138  df-meet 15139  df-p0 15201  df-lat 15208  df-clat 15270  df-oposet 32661  df-ol 32663  df-oml 32664  df-covers 32751  df-ats 32752  df-atl 32783  df-cvlat 32807  df-hlat 32836  df-lines 32985  df-pmap 32988

This theorem is referenced by:  lnjatN  33264