Theorem llnexatN 34807

Description: Given an atom on a line, there is another atom whose join equals the line. (Contributed by NM, 26-Jun-2012.) (New usage is discouraged.)

Hypotheses

Ref	Expression
llnexat.l	|- .<_ = ( le ` K )
llnexat.j	|- .\/ = ( join ` K )
llnexat.a	|- A = ( Atoms ` K )
llnexat.n	|- N = ( LLines ` K )

Assertion

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

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

Allowed substitution hint:   .\/ ( q)

Proof of Theorem llnexatN

Step	Hyp	Ref	Expression
1		simp1 1061	. . . 4 |- ( ( K e. HL /\ X e. N /\ P e. A ) -> K e. HL )
2		simp3 1063	. . . 4 |- ( ( K e. HL /\ X e. N /\ P e. A ) -> P e. A )
3		simp2 1062	. . . 4 |- ( ( K e. HL /\ X e. N /\ P e. A ) -> X e. N )
4	1, 2, 3	3jca 1242	. . 3 |- ( ( K e. HL /\ X e. N /\ P e. A ) -> ( K e. HL /\ P e. A /\ X e. N ) )
5		llnexat.l	. . . 4 |- .<_ = ( le ` K )
6		eqid 2622	. . . 4 |- ( <o ` K ) = ( <o ` K )
7		llnexat.a	. . . 4 |- A = ( Atoms ` K )
8		llnexat.n	. . . 4 |- N = ( LLines ` K )
9	5, 6, 7, 8	atcvrlln2 34805	. . 3 |- ( ( ( K e. HL /\ P e. A /\ X e. N ) /\ P .<_ X ) -> P ( <o ` K ) X )
10	4, 9	sylan 488	. 2 |- ( ( ( K e. HL /\ X e. N /\ P e. A ) /\ P .<_ X ) -> P ( <o ` K ) X )
11		simpl1 1064	. . . 4 |- ( ( ( K e. HL /\ X e. N /\ P e. A ) /\ P .<_ X ) -> K e. HL )
12		simpl3 1066	. . . . 5 |- ( ( ( K e. HL /\ X e. N /\ P e. A ) /\ P .<_ X ) -> P e. A )
13		eqid 2622	. . . . . 6 |- ( Base ` K ) = ( Base ` K )
14	13, 7	atbase 34576	. . . . 5 |- ( P e. A -> P e. ( Base ` K ) )
15	12, 14	syl 17	. . . 4 |- ( ( ( K e. HL /\ X e. N /\ P e. A ) /\ P .<_ X ) -> P e. ( Base ` K ) )
16		simpl2 1065	. . . . 5 |- ( ( ( K e. HL /\ X e. N /\ P e. A ) /\ P .<_ X ) -> X e. N )
17	13, 8	llnbase 34795	. . . . 5 |- ( X e. N -> X e. ( Base ` K ) )
18	16, 17	syl 17	. . . 4 |- ( ( ( K e. HL /\ X e. N /\ P e. A ) /\ P .<_ X ) -> X e. ( Base ` K ) )
19		llnexat.j	. . . . 5 |- .\/ = ( join ` K )
20	13, 5, 19, 6, 7	cvrval3 34699	. . . 4 |- ( ( K e. HL /\ P e. ( Base ` K ) /\ X e. ( Base ` K ) ) -> ( P ( <o ` K ) X <-> E. q e. A ( -. q .<_ P /\ ( P .\/ q ) = X ) ) )
21	11, 15, 18, 20	syl3anc 1326	. . 3 |- ( ( ( K e. HL /\ X e. N /\ P e. A ) /\ P .<_ X ) -> ( P ( <o ` K ) X <-> E. q e. A ( -. q .<_ P /\ ( P .\/ q ) = X ) ) )
22		simpll1 1100	. . . . . . . 8 |- ( ( ( ( K e. HL /\ X e. N /\ P e. A ) /\ P .<_ X ) /\ q e. A ) -> K e. HL )
23		hlatl 34647	. . . . . . . 8 |- ( K e. HL -> K e. AtLat )
24	22, 23	syl 17	. . . . . . 7 |- ( ( ( ( K e. HL /\ X e. N /\ P e. A ) /\ P .<_ X ) /\ q e. A ) -> K e. AtLat )
25		simpr 477	. . . . . . 7 |- ( ( ( ( K e. HL /\ X e. N /\ P e. A ) /\ P .<_ X ) /\ q e. A ) -> q e. A )
26		simpll3 1102	. . . . . . 7 |- ( ( ( ( K e. HL /\ X e. N /\ P e. A ) /\ P .<_ X ) /\ q e. A ) -> P e. A )
27	5, 7	atncmp 34599	. . . . . . 7 |- ( ( K e. AtLat /\ q e. A /\ P e. A ) -> ( -. q .<_ P <-> q =/= P ) )
28	24, 25, 26, 27	syl3anc 1326	. . . . . 6 |- ( ( ( ( K e. HL /\ X e. N /\ P e. A ) /\ P .<_ X ) /\ q e. A ) -> ( -. q .<_ P <-> q =/= P ) )
29	28	anbi1d 741	. . . . 5 |- ( ( ( ( K e. HL /\ X e. N /\ P e. A ) /\ P .<_ X ) /\ q e. A ) -> ( ( -. q .<_ P /\ ( P .\/ q ) = X ) <-> ( q =/= P /\ ( P .\/ q ) = X ) ) )
30		necom 2847	. . . . . 6 |- ( q =/= P <-> P =/= q )
31		eqcom 2629	. . . . . 6 |- ( ( P .\/ q ) = X <-> X = ( P .\/ q ) )
32	30, 31	anbi12i 733	. . . . 5 |- ( ( q =/= P /\ ( P .\/ q ) = X ) <-> ( P =/= q /\ X = ( P .\/ q ) ) )
33	29, 32	syl6bb 276	. . . 4 |- ( ( ( ( K e. HL /\ X e. N /\ P e. A ) /\ P .<_ X ) /\ q e. A ) -> ( ( -. q .<_ P /\ ( P .\/ q ) = X ) <-> ( P =/= q /\ X = ( P .\/ q ) ) ) )
34	33	rexbidva 3049	. . 3 |- ( ( ( K e. HL /\ X e. N /\ P e. A ) /\ P .<_ X ) -> ( E. q e. A ( -. q .<_ P /\ ( P .\/ q ) = X ) <-> E. q e. A ( P =/= q /\ X = ( P .\/ q ) ) ) )
35	21, 34	bitrd 268	. 2 |- ( ( ( K e. HL /\ X e. N /\ P e. A ) /\ P .<_ X ) -> ( P ( <o ` K ) X <-> E. q e. A ( P =/= q /\ X = ( P .\/ q ) ) ) )
36	10, 35	mpbid 222	1 |- ( ( ( K e. HL /\ X e. N /\ P e. A ) /\ P .<_ X ) -> E. q e. A ( P =/= q /\ X = ( P .\/ q ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ 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   <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-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-llines 34784

This theorem is referenced by:  lplnexllnN  34850