Theorem lnatexN 35065

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 2622	. . . 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 35062	. . 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 1432	. 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 1115	. . . . . 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 1116	. . . . . . . 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 2849	. . . . . . 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 477	. . . . . . 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 2863	. . . . . 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 1136	. . . . . . . 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 1114	. . . . . . . 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 34662	. . . . . . . 8 |- ( ( K e. HL /\ r e. A /\ s e. A ) -> s .<_ ( r ( join ` K ) s ) )
17	13, 14, 8, 16	syl3anc 1326	. . . . . . 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 1117	. . . . . . 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 4681	. . . . . 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 2856	. . . . . . . 8 |- ( q = s -> ( q =/= P <-> s =/= P ) )
21		breq1 4656	. . . . . . . 8 |- ( q = s -> ( q .<_ X <-> s .<_ X ) )
22	20, 21	anbi12d 747	. . . . . . 7 |- ( q = s -> ( ( q =/= P /\ q .<_ X ) <-> ( s =/= P /\ s .<_ X ) ) )
23	22	rspcev 3309	. . . . . 6 |- ( ( s e. A /\ ( s =/= P /\ s .<_ X ) ) -> E. q e. A ( q =/= P /\ q .<_ X ) )
24	8, 12, 19, 23	syl12anc 1324	. . . . 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 1114	. . . . . 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 477	. . . . . 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 1136	. . . . . . . 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 1115	. . . . . . . 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 34661	. . . . . . . 8 |- ( ( K e. HL /\ r e. A /\ s e. A ) -> r .<_ ( r ( join ` K ) s ) )
30	27, 25, 28, 29	syl3anc 1326	. . . . . . 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 1117	. . . . . . 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 4681	. . . . . 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 2856	. . . . . . . 8 |- ( q = r -> ( q =/= P <-> r =/= P ) )
34		breq1 4656	. . . . . . . 8 |- ( q = r -> ( q .<_ X <-> r .<_ X ) )
35	33, 34	anbi12d 747	. . . . . . 7 |- ( q = r -> ( ( q =/= P /\ q .<_ X ) <-> ( r =/= P /\ r .<_ X ) ) )
36	35	rspcev 3309	. . . . . 6 |- ( ( r e. A /\ ( r =/= P /\ r .<_ X ) ) -> E. q e. A ( q =/= P /\ q .<_ X ) )
37	25, 26, 32, 36	syl12anc 1324	. . . . 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 2881	. . . 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 1264	. . 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 3037	. 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 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   lecple 15948   joincjn 16944   Atomscatm 34550   HLchlt 34637   Linesclines 34780   pmapcpmap 34783

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-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-lines 34787  df-pmap 34790

This theorem is referenced by:  lnjatN  35066