Theorem atbtwnexOLDN 34733

Description: There exists a 3rd atom r on a line P .\/ Q intersecting element X at P, such that r is different from Q and not in X. (Contributed by NM, 30-Jul-2012.) (New usage is discouraged.)

Hypotheses

Ref	Expression
atbtwn.b	|- B = ( Base ` K )
atbtwn.l	|- .<_ = ( le ` K )
atbtwn.j	|- .\/ = ( join ` K )
atbtwn.a	|- A = ( Atoms ` K )

Assertion

Ref	Expression
atbtwnexOLDN	|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( X e. B /\ P .<_ X /\ -. Q .<_ X ) ) -> E. r e. A ( r =/= Q /\ -. r .<_ X /\ r .<_ ( P .\/ Q ) ) )

Distinct variable groups:   A, r   B, r   K, r   .<_ , r   P, r   Q, r   X, r

Allowed substitution hint:   .\/ ( r)

Proof of Theorem atbtwnexOLDN

Step	Hyp	Ref	Expression
1		simpr2 1068	. . . 4 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( X e. B /\ P .<_ X /\ -. Q .<_ X ) ) -> P .<_ X )
2		simpr3 1069	. . . 4 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( X e. B /\ P .<_ X /\ -. Q .<_ X ) ) -> -. Q .<_ X )
3		nbrne2 4673	. . . 4 |- ( ( P .<_ X /\ -. Q .<_ X ) -> P =/= Q )
4	1, 2, 3	syl2anc 693	. . 3 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( X e. B /\ P .<_ X /\ -. Q .<_ X ) ) -> P =/= Q )
5		atbtwn.l	. . . 4 |- .<_ = ( le ` K )
6		atbtwn.j	. . . 4 |- .\/ = ( join ` K )
7		atbtwn.a	. . . 4 |- A = ( Atoms ` K )
8	5, 6, 7	hlsupr 34672	. . 3 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ P =/= Q ) -> E. r e. A ( r =/= P /\ r =/= Q /\ r .<_ ( P .\/ Q ) ) )
9	4, 8	syldan 487	. 2 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( X e. B /\ P .<_ X /\ -. Q .<_ X ) ) -> E. r e. A ( r =/= P /\ r =/= Q /\ r .<_ ( P .\/ Q ) ) )
10		simp32 1098	. . . . 5 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( X e. B /\ P .<_ X /\ -. Q .<_ X ) ) /\ r e. A /\ ( r =/= P /\ r =/= Q /\ r .<_ ( P .\/ Q ) ) ) -> r =/= Q )
11		simp31 1097	. . . . . 6 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( X e. B /\ P .<_ X /\ -. Q .<_ X ) ) /\ r e. A /\ ( r =/= P /\ r =/= Q /\ r .<_ ( P .\/ Q ) ) ) -> r =/= P )
12		simp1l 1085	. . . . . . 7 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( X e. B /\ P .<_ X /\ -. Q .<_ X ) ) /\ r e. A /\ ( r =/= P /\ r =/= Q /\ r .<_ ( P .\/ Q ) ) ) -> ( K e. HL /\ P e. A /\ Q e. A ) )
13		simp2 1062	. . . . . . 7 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( X e. B /\ P .<_ X /\ -. Q .<_ X ) ) /\ r e. A /\ ( r =/= P /\ r =/= Q /\ r .<_ ( P .\/ Q ) ) ) -> r e. A )
14		simp1r1 1157	. . . . . . 7 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( X e. B /\ P .<_ X /\ -. Q .<_ X ) ) /\ r e. A /\ ( r =/= P /\ r =/= Q /\ r .<_ ( P .\/ Q ) ) ) -> X e. B )
15		simp1r2 1158	. . . . . . 7 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( X e. B /\ P .<_ X /\ -. Q .<_ X ) ) /\ r e. A /\ ( r =/= P /\ r =/= Q /\ r .<_ ( P .\/ Q ) ) ) -> P .<_ X )
16		simp1r3 1159	. . . . . . 7 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( X e. B /\ P .<_ X /\ -. Q .<_ X ) ) /\ r e. A /\ ( r =/= P /\ r =/= Q /\ r .<_ ( P .\/ Q ) ) ) -> -. Q .<_ X )
17		simp33 1099	. . . . . . 7 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( X e. B /\ P .<_ X /\ -. Q .<_ X ) ) /\ r e. A /\ ( r =/= P /\ r =/= Q /\ r .<_ ( P .\/ Q ) ) ) -> r .<_ ( P .\/ Q ) )
18		atbtwn.b	. . . . . . . 8 |- B = ( Base ` K )
19	18, 5, 6, 7	atbtwn 34732	. . . . . . 7 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( r e. A /\ X e. B ) /\ ( P .<_ X /\ -. Q .<_ X /\ r .<_ ( P .\/ Q ) ) ) -> ( r =/= P <-> -. r .<_ X ) )
20	12, 13, 14, 15, 16, 17, 19	syl123anc 1343	. . . . . 6 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( X e. B /\ P .<_ X /\ -. Q .<_ X ) ) /\ r e. A /\ ( r =/= P /\ r =/= Q /\ r .<_ ( P .\/ Q ) ) ) -> ( r =/= P <-> -. r .<_ X ) )
21	11, 20	mpbid 222	. . . . 5 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( X e. B /\ P .<_ X /\ -. Q .<_ X ) ) /\ r e. A /\ ( r =/= P /\ r =/= Q /\ r .<_ ( P .\/ Q ) ) ) -> -. r .<_ X )
22	10, 21, 17	3jca 1242	. . . 4 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( X e. B /\ P .<_ X /\ -. Q .<_ X ) ) /\ r e. A /\ ( r =/= P /\ r =/= Q /\ r .<_ ( P .\/ Q ) ) ) -> ( r =/= Q /\ -. r .<_ X /\ r .<_ ( P .\/ Q ) ) )
23	22	3exp 1264	. . 3 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( X e. B /\ P .<_ X /\ -. Q .<_ X ) ) -> ( r e. A -> ( ( r =/= P /\ r =/= Q /\ r .<_ ( P .\/ Q ) ) -> ( r =/= Q /\ -. r .<_ X /\ r .<_ ( P .\/ Q ) ) ) ) )
24	23	reximdvai 3015	. 2 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( X e. B /\ P .<_ X /\ -. Q .<_ X ) ) -> ( E. r e. A ( r =/= P /\ r =/= Q /\ r .<_ ( P .\/ Q ) ) -> E. r e. A ( r =/= Q /\ -. r .<_ X /\ r .<_ ( P .\/ Q ) ) ) )
25	9, 24	mpd 15	1 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( X e. B /\ P .<_ X /\ -. Q .<_ X ) ) -> E. r e. A ( r =/= Q /\ -. r .<_ X /\ r .<_ ( 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   Atomscatm 34550   HLchlt 34637

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

This theorem is referenced by: (None)