Theorem lnjatN 35066

Description: Given an atom in a line, there is another atom which when joined equals the line. (Contributed by NM, 30-Apr-2012.) (New usage is discouraged.)

Hypotheses

Ref	Expression
lnjat.b	|- B = ( Base ` K )
lnjat.l	|- .<_ = ( le ` K )
lnjat.j	|- .\/ = ( join ` K )
lnjat.a	|- A = ( Atoms ` K )
lnjat.n	|- N = ( Lines ` K )
lnjat.m	|- M = ( pmap ` K )

Assertion

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

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

Allowed substitution hint:   .\/ ( q)

Proof of Theorem lnjatN

Step	Hyp	Ref	Expression
1		simpl1 1064	. . 3 |- ( ( ( K e. HL /\ X e. B /\ P e. A ) /\ ( ( M ` X ) e. N /\ P .<_ X ) ) -> K e. HL )
2		simpl2 1065	. . 3 |- ( ( ( K e. HL /\ X e. B /\ P e. A ) /\ ( ( M ` X ) e. N /\ P .<_ X ) ) -> X e. B )
3		simprl 794	. . 3 |- ( ( ( K e. HL /\ X e. B /\ P e. A ) /\ ( ( M ` X ) e. N /\ P .<_ X ) ) -> ( M ` X ) e. N )
4		lnjat.b	. . . 4 |- B = ( Base ` K )
5		lnjat.l	. . . 4 |- .<_ = ( le ` K )
6		lnjat.a	. . . 4 |- A = ( Atoms ` K )
7		lnjat.n	. . . 4 |- N = ( Lines ` K )
8		lnjat.m	. . . 4 |- M = ( pmap ` K )
9	4, 5, 6, 7, 8	lnatexN 35065	. . 3 |- ( ( K e. HL /\ X e. B /\ ( M ` X ) e. N ) -> E. q e. A ( q =/= P /\ q .<_ X ) )
10	1, 2, 3, 9	syl3anc 1326	. 2 |- ( ( ( K e. HL /\ X e. B /\ P e. A ) /\ ( ( M ` X ) e. N /\ P .<_ X ) ) -> E. q e. A ( q =/= P /\ q .<_ X ) )
11		simp3l 1089	. . . . 5 |- ( ( ( ( K e. HL /\ X e. B /\ P e. A ) /\ ( ( M ` X ) e. N /\ P .<_ X ) ) /\ q e. A /\ ( q =/= P /\ q .<_ X ) ) -> q =/= P )
12		simp1l1 1154	. . . . . 6 |- ( ( ( ( K e. HL /\ X e. B /\ P e. A ) /\ ( ( M ` X ) e. N /\ P .<_ X ) ) /\ q e. A /\ ( q =/= P /\ q .<_ X ) ) -> K e. HL )
13		simp1l2 1155	. . . . . 6 |- ( ( ( ( K e. HL /\ X e. B /\ P e. A ) /\ ( ( M ` X ) e. N /\ P .<_ X ) ) /\ q e. A /\ ( q =/= P /\ q .<_ X ) ) -> X e. B )
14		simp1rl 1126	. . . . . 6 |- ( ( ( ( K e. HL /\ X e. B /\ P e. A ) /\ ( ( M ` X ) e. N /\ P .<_ X ) ) /\ q e. A /\ ( q =/= P /\ q .<_ X ) ) -> ( M ` X ) e. N )
15		simp1l3 1156	. . . . . 6 |- ( ( ( ( K e. HL /\ X e. B /\ P e. A ) /\ ( ( M ` X ) e. N /\ P .<_ X ) ) /\ q e. A /\ ( q =/= P /\ q .<_ X ) ) -> P e. A )
16		simp2 1062	. . . . . 6 |- ( ( ( ( K e. HL /\ X e. B /\ P e. A ) /\ ( ( M ` X ) e. N /\ P .<_ X ) ) /\ q e. A /\ ( q =/= P /\ q .<_ X ) ) -> q e. A )
17	11	necomd 2849	. . . . . 6 |- ( ( ( ( K e. HL /\ X e. B /\ P e. A ) /\ ( ( M ` X ) e. N /\ P .<_ X ) ) /\ q e. A /\ ( q =/= P /\ q .<_ X ) ) -> P =/= q )
18		simp1rr 1127	. . . . . 6 |- ( ( ( ( K e. HL /\ X e. B /\ P e. A ) /\ ( ( M ` X ) e. N /\ P .<_ X ) ) /\ q e. A /\ ( q =/= P /\ q .<_ X ) ) -> P .<_ X )
19		simp3r 1090	. . . . . 6 |- ( ( ( ( K e. HL /\ X e. B /\ P e. A ) /\ ( ( M ` X ) e. N /\ P .<_ X ) ) /\ q e. A /\ ( q =/= P /\ q .<_ X ) ) -> q .<_ X )
20		lnjat.j	. . . . . . 7 |- .\/ = ( join ` K )
21	4, 5, 20, 6, 7, 8	lneq2at 35064	. . . . . 6 |- ( ( ( K e. HL /\ X e. B /\ ( M ` X ) e. N ) /\ ( P e. A /\ q e. A /\ P =/= q ) /\ ( P .<_ X /\ q .<_ X ) ) -> X = ( P .\/ q ) )
22	12, 13, 14, 15, 16, 17, 18, 19, 21	syl332anc 1357	. . . . 5 |- ( ( ( ( K e. HL /\ X e. B /\ P e. A ) /\ ( ( M ` X ) e. N /\ P .<_ X ) ) /\ q e. A /\ ( q =/= P /\ q .<_ X ) ) -> X = ( P .\/ q ) )
23	11, 22	jca 554	. . . 4 |- ( ( ( ( K e. HL /\ X e. B /\ P e. A ) /\ ( ( M ` X ) e. N /\ P .<_ X ) ) /\ q e. A /\ ( q =/= P /\ q .<_ X ) ) -> ( q =/= P /\ X = ( P .\/ q ) ) )
24	23	3exp 1264	. . 3 |- ( ( ( K e. HL /\ X e. B /\ P e. A ) /\ ( ( M ` X ) e. N /\ P .<_ X ) ) -> ( q e. A -> ( ( q =/= P /\ q .<_ X ) -> ( q =/= P /\ X = ( P .\/ q ) ) ) ) )
25	24	reximdvai 3015	. 2 |- ( ( ( K e. HL /\ X e. B /\ P e. A ) /\ ( ( M ` X ) e. N /\ P .<_ X ) ) -> ( E. q e. A ( q =/= P /\ q .<_ X ) -> E. q e. A ( q =/= P /\ X = ( P .\/ q ) ) ) )
26	10, 25	mpd 15	1 |- ( ( ( K e. HL /\ X e. B /\ P e. A ) /\ ( ( M ` X ) e. N /\ P .<_ X ) ) -> E. q e. A ( q =/= P /\ X = ( P .\/ q ) ) )

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: (None)