Theorem lplnexllnN 34850

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

Hypotheses

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

Assertion

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

Distinct variable groups:   y, .\/   y, .<_   y, N   y, Q   y, X

Allowed substitution hints:   A( y)   P( y)   K( y)

Proof of Theorem lplnexllnN

Dummy variables s r z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl2 1065	. . 3 |- ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) -> X e. P )
2		simpl1 1064	. . . 4 |- ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) -> K e. HL )
3		eqid 2622	. . . . . 6 |- ( Base ` K ) = ( Base ` K )
4		lplnexat.p	. . . . . 6 |- P = ( LPlanes ` K )
5	3, 4	lplnbase 34820	. . . . 5 |- ( X e. P -> X e. ( Base ` K ) )
6	1, 5	syl 17	. . . 4 |- ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) -> X e. ( Base ` K ) )
7		lplnexat.l	. . . . 5 |- .<_ = ( le ` K )
8		lplnexat.j	. . . . 5 |- .\/ = ( join ` K )
9		lplnexat.a	. . . . 5 |- A = ( Atoms ` K )
10		lplnexat.n	. . . . 5 |- N = ( LLines ` K )
11	3, 7, 8, 9, 10, 4	islpln3 34819	. . . 4 |- ( ( K e. HL /\ X e. ( Base ` K ) ) -> ( X e. P <-> E. z e. N E. r e. A ( -. r .<_ z /\ X = ( z .\/ r ) ) ) )
12	2, 6, 11	syl2anc 693	. . 3 |- ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) -> ( X e. P <-> E. z e. N E. r e. A ( -. r .<_ z /\ X = ( z .\/ r ) ) ) )
13	1, 12	mpbid 222	. 2 |- ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) -> E. z e. N E. r e. A ( -. r .<_ z /\ X = ( z .\/ r ) ) )
14		simpll1 1100	. . . . . . 7 |- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) ) -> K e. HL )
15		simpr2l 1120	. . . . . . 7 |- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) ) -> z e. N )
16		simpll3 1102	. . . . . . 7 |- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) ) -> Q e. A )
17		simpr1 1067	. . . . . . 7 |- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) ) -> Q .<_ z )
18	7, 8, 9, 10	llnexatN 34807	. . . . . . 7 |- ( ( ( K e. HL /\ z e. N /\ Q e. A ) /\ Q .<_ z ) -> E. s e. A ( Q =/= s /\ z = ( Q .\/ s ) ) )
19	14, 15, 16, 17, 18	syl31anc 1329	. . . . . 6 |- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) ) -> E. s e. A ( Q =/= s /\ z = ( Q .\/ s ) ) )
20		simp1l1 1154	. . . . . . . . . . 11 |- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) /\ ( s e. A /\ ( Q =/= s /\ z = ( Q .\/ s ) ) ) ) -> K e. HL )
21		simp22r 1181	. . . . . . . . . . 11 |- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) /\ ( s e. A /\ ( Q =/= s /\ z = ( Q .\/ s ) ) ) ) -> r e. A )
22		simp3l 1089	. . . . . . . . . . 11 |- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) /\ ( s e. A /\ ( Q =/= s /\ z = ( Q .\/ s ) ) ) ) -> s e. A )
23		simp1l3 1156	. . . . . . . . . . . 12 |- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) /\ ( s e. A /\ ( Q =/= s /\ z = ( Q .\/ s ) ) ) ) -> Q e. A )
24		simp23l 1182	. . . . . . . . . . . . 13 |- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) /\ ( s e. A /\ ( Q =/= s /\ z = ( Q .\/ s ) ) ) ) -> -. r .<_ z )
25		simp3rr 1135	. . . . . . . . . . . . . 14 |- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) /\ ( s e. A /\ ( Q =/= s /\ z = ( Q .\/ s ) ) ) ) -> z = ( Q .\/ s ) )
26	25	breq2d 4665	. . . . . . . . . . . . 13 |- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) /\ ( s e. A /\ ( Q =/= s /\ z = ( Q .\/ s ) ) ) ) -> ( r .<_ z <-> r .<_ ( Q .\/ s ) ) )
27	24, 26	mtbid 314	. . . . . . . . . . . 12 |- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) /\ ( s e. A /\ ( Q =/= s /\ z = ( Q .\/ s ) ) ) ) -> -. r .<_ ( Q .\/ s ) )
28	7, 8, 9	atnlej2 34666	. . . . . . . . . . . 12 |- ( ( K e. HL /\ ( r e. A /\ Q e. A /\ s e. A ) /\ -. r .<_ ( Q .\/ s ) ) -> r =/= s )
29	20, 21, 23, 22, 27, 28	syl131anc 1339	. . . . . . . . . . 11 |- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) /\ ( s e. A /\ ( Q =/= s /\ z = ( Q .\/ s ) ) ) ) -> r =/= s )
30	8, 9, 10	llni2 34798	. . . . . . . . . . 11 |- ( ( ( K e. HL /\ r e. A /\ s e. A ) /\ r =/= s ) -> ( r .\/ s ) e. N )
31	20, 21, 22, 29, 30	syl31anc 1329	. . . . . . . . . 10 |- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) /\ ( s e. A /\ ( Q =/= s /\ z = ( Q .\/ s ) ) ) ) -> ( r .\/ s ) e. N )
32		simp3rl 1134	. . . . . . . . . . 11 |- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) /\ ( s e. A /\ ( Q =/= s /\ z = ( Q .\/ s ) ) ) ) -> Q =/= s )
33	7, 8, 9	hlatcon2 34738	. . . . . . . . . . 11 |- ( ( K e. HL /\ ( Q e. A /\ s e. A /\ r e. A ) /\ ( Q =/= s /\ -. r .<_ ( Q .\/ s ) ) ) -> -. Q .<_ ( r .\/ s ) )
34	20, 23, 22, 21, 32, 27, 33	syl132anc 1344	. . . . . . . . . 10 |- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) /\ ( s e. A /\ ( Q =/= s /\ z = ( Q .\/ s ) ) ) ) -> -. Q .<_ ( r .\/ s ) )
35		simp23r 1183	. . . . . . . . . . 11 |- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) /\ ( s e. A /\ ( Q =/= s /\ z = ( Q .\/ s ) ) ) ) -> X = ( z .\/ r ) )
36	25	oveq1d 6665	. . . . . . . . . . 11 |- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) /\ ( s e. A /\ ( Q =/= s /\ z = ( Q .\/ s ) ) ) ) -> ( z .\/ r ) = ( ( Q .\/ s ) .\/ r ) )
37		hllat 34650	. . . . . . . . . . . . 13 |- ( K e. HL -> K e. Lat )
38	20, 37	syl 17	. . . . . . . . . . . 12 |- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) /\ ( s e. A /\ ( Q =/= s /\ z = ( Q .\/ s ) ) ) ) -> K e. Lat )
39	3, 9	atbase 34576	. . . . . . . . . . . . 13 |- ( Q e. A -> Q e. ( Base ` K ) )
40	23, 39	syl 17	. . . . . . . . . . . 12 |- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) /\ ( s e. A /\ ( Q =/= s /\ z = ( Q .\/ s ) ) ) ) -> Q e. ( Base ` K ) )
41	3, 9	atbase 34576	. . . . . . . . . . . . 13 |- ( s e. A -> s e. ( Base ` K ) )
42	22, 41	syl 17	. . . . . . . . . . . 12 |- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) /\ ( s e. A /\ ( Q =/= s /\ z = ( Q .\/ s ) ) ) ) -> s e. ( Base ` K ) )
43	3, 9	atbase 34576	. . . . . . . . . . . . 13 |- ( r e. A -> r e. ( Base ` K ) )
44	21, 43	syl 17	. . . . . . . . . . . 12 |- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) /\ ( s e. A /\ ( Q =/= s /\ z = ( Q .\/ s ) ) ) ) -> r e. ( Base ` K ) )
45	3, 8	latj31 17099	. . . . . . . . . . . 12 |- ( ( K e. Lat /\ ( Q e. ( Base ` K ) /\ s e. ( Base ` K ) /\ r e. ( Base ` K ) ) ) -> ( ( Q .\/ s ) .\/ r ) = ( ( r .\/ s ) .\/ Q ) )
46	38, 40, 42, 44, 45	syl13anc 1328	. . . . . . . . . . 11 |- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) /\ ( s e. A /\ ( Q =/= s /\ z = ( Q .\/ s ) ) ) ) -> ( ( Q .\/ s ) .\/ r ) = ( ( r .\/ s ) .\/ Q ) )
47	35, 36, 46	3eqtrd 2660	. . . . . . . . . 10 |- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) /\ ( s e. A /\ ( Q =/= s /\ z = ( Q .\/ s ) ) ) ) -> X = ( ( r .\/ s ) .\/ Q ) )
48		breq2 4657	. . . . . . . . . . . . 13 |- ( y = ( r .\/ s ) -> ( Q .<_ y <-> Q .<_ ( r .\/ s ) ) )
49	48	notbid 308	. . . . . . . . . . . 12 |- ( y = ( r .\/ s ) -> ( -. Q .<_ y <-> -. Q .<_ ( r .\/ s ) ) )
50		oveq1 6657	. . . . . . . . . . . . 13 |- ( y = ( r .\/ s ) -> ( y .\/ Q ) = ( ( r .\/ s ) .\/ Q ) )
51	50	eqeq2d 2632	. . . . . . . . . . . 12 |- ( y = ( r .\/ s ) -> ( X = ( y .\/ Q ) <-> X = ( ( r .\/ s ) .\/ Q ) ) )
52	49, 51	anbi12d 747	. . . . . . . . . . 11 |- ( y = ( r .\/ s ) -> ( ( -. Q .<_ y /\ X = ( y .\/ Q ) ) <-> ( -. Q .<_ ( r .\/ s ) /\ X = ( ( r .\/ s ) .\/ Q ) ) ) )
53	52	rspcev 3309	. . . . . . . . . 10 |- ( ( ( r .\/ s ) e. N /\ ( -. Q .<_ ( r .\/ s ) /\ X = ( ( r .\/ s ) .\/ Q ) ) ) -> E. y e. N ( -. Q .<_ y /\ X = ( y .\/ Q ) ) )
54	31, 34, 47, 53	syl12anc 1324	. . . . . . . . 9 |- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) /\ ( s e. A /\ ( Q =/= s /\ z = ( Q .\/ s ) ) ) ) -> E. y e. N ( -. Q .<_ y /\ X = ( y .\/ Q ) ) )
55	54	3expia 1267	. . . . . . . 8 |- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) ) -> ( ( s e. A /\ ( Q =/= s /\ z = ( Q .\/ s ) ) ) -> E. y e. N ( -. Q .<_ y /\ X = ( y .\/ Q ) ) ) )
56	55	expd 452	. . . . . . 7 |- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) ) -> ( s e. A -> ( ( Q =/= s /\ z = ( Q .\/ s ) ) -> E. y e. N ( -. Q .<_ y /\ X = ( y .\/ Q ) ) ) ) )
57	56	rexlimdv 3030	. . . . . 6 |- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) ) -> ( E. s e. A ( Q =/= s /\ z = ( Q .\/ s ) ) -> E. y e. N ( -. Q .<_ y /\ X = ( y .\/ Q ) ) ) )
58	19, 57	mpd 15	. . . . 5 |- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) ) -> E. y e. N ( -. Q .<_ y /\ X = ( y .\/ Q ) ) )
59	58	3exp2 1285	. . . 4 |- ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) -> ( Q .<_ z -> ( ( z e. N /\ r e. A ) -> ( ( -. r .<_ z /\ X = ( z .\/ r ) ) -> E. y e. N ( -. Q .<_ y /\ X = ( y .\/ Q ) ) ) ) ) )
60		simpr2l 1120	. . . . . 6 |- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( -. Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) ) -> z e. N )
61		simpr1 1067	. . . . . 6 |- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( -. Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) ) -> -. Q .<_ z )
62		simpll1 1100	. . . . . . . . . . . 12 |- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( -. Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) ) -> K e. HL )
63	62, 37	syl 17	. . . . . . . . . . 11 |- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( -. Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) ) -> K e. Lat )
64	3, 10	llnbase 34795	. . . . . . . . . . . 12 |- ( z e. N -> z e. ( Base ` K ) )
65	60, 64	syl 17	. . . . . . . . . . 11 |- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( -. Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) ) -> z e. ( Base ` K ) )
66		simpr2r 1121	. . . . . . . . . . . 12 |- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( -. Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) ) -> r e. A )
67	66, 43	syl 17	. . . . . . . . . . 11 |- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( -. Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) ) -> r e. ( Base ` K ) )
68	3, 7, 8	latlej1 17060	. . . . . . . . . . 11 |- ( ( K e. Lat /\ z e. ( Base ` K ) /\ r e. ( Base ` K ) ) -> z .<_ ( z .\/ r ) )
69	63, 65, 67, 68	syl3anc 1326	. . . . . . . . . 10 |- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( -. Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) ) -> z .<_ ( z .\/ r ) )
70		simpr3r 1123	. . . . . . . . . 10 |- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( -. Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) ) -> X = ( z .\/ r ) )
71	69, 70	breqtrrd 4681	. . . . . . . . 9 |- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( -. Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) ) -> z .<_ X )
72		simplr 792	. . . . . . . . 9 |- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( -. Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) ) -> Q .<_ X )
73		simpll3 1102	. . . . . . . . . . 11 |- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( -. Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) ) -> Q e. A )
74	73, 39	syl 17	. . . . . . . . . 10 |- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( -. Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) ) -> Q e. ( Base ` K ) )
75		simpll2 1101	. . . . . . . . . . 11 |- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( -. Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) ) -> X e. P )
76	75, 5	syl 17	. . . . . . . . . 10 |- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( -. Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) ) -> X e. ( Base ` K ) )
77	3, 7, 8	latjle12 17062	. . . . . . . . . 10 |- ( ( K e. Lat /\ ( z e. ( Base ` K ) /\ Q e. ( Base ` K ) /\ X e. ( Base ` K ) ) ) -> ( ( z .<_ X /\ Q .<_ X ) <-> ( z .\/ Q ) .<_ X ) )
78	63, 65, 74, 76, 77	syl13anc 1328	. . . . . . . . 9 |- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( -. Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) ) -> ( ( z .<_ X /\ Q .<_ X ) <-> ( z .\/ Q ) .<_ X ) )
79	71, 72, 78	mpbi2and 956	. . . . . . . 8 |- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( -. Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) ) -> ( z .\/ Q ) .<_ X )
80	3, 8	latjcl 17051	. . . . . . . . . . 11 |- ( ( K e. Lat /\ z e. ( Base ` K ) /\ Q e. ( Base ` K ) ) -> ( z .\/ Q ) e. ( Base ` K ) )
81	63, 65, 74, 80	syl3anc 1326	. . . . . . . . . 10 |- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( -. Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) ) -> ( z .\/ Q ) e. ( Base ` K ) )
82		eqid 2622	. . . . . . . . . . . . 13 |- ( <o ` K ) = ( <o ` K )
83	3, 7, 8, 82, 9	cvr1 34696	. . . . . . . . . . . 12 |- ( ( K e. HL /\ z e. ( Base ` K ) /\ Q e. A ) -> ( -. Q .<_ z <-> z ( <o ` K ) ( z .\/ Q ) ) )
84	62, 65, 73, 83	syl3anc 1326	. . . . . . . . . . 11 |- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( -. Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) ) -> ( -. Q .<_ z <-> z ( <o ` K ) ( z .\/ Q ) ) )
85	61, 84	mpbid 222	. . . . . . . . . 10 |- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( -. Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) ) -> z ( <o ` K ) ( z .\/ Q ) )
86	3, 82, 10, 4	lplni 34818	. . . . . . . . . 10 |- ( ( ( K e. HL /\ ( z .\/ Q ) e. ( Base ` K ) /\ z e. N ) /\ z ( <o ` K ) ( z .\/ Q ) ) -> ( z .\/ Q ) e. P )
87	62, 81, 60, 85, 86	syl31anc 1329	. . . . . . . . 9 |- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( -. Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) ) -> ( z .\/ Q ) e. P )
88	7, 4	lplncmp 34848	. . . . . . . . 9 |- ( ( K e. HL /\ ( z .\/ Q ) e. P /\ X e. P ) -> ( ( z .\/ Q ) .<_ X <-> ( z .\/ Q ) = X ) )
89	62, 87, 75, 88	syl3anc 1326	. . . . . . . 8 |- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( -. Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) ) -> ( ( z .\/ Q ) .<_ X <-> ( z .\/ Q ) = X ) )
90	79, 89	mpbid 222	. . . . . . 7 |- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( -. Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) ) -> ( z .\/ Q ) = X )
91	90	eqcomd 2628	. . . . . 6 |- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( -. Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) ) -> X = ( z .\/ Q ) )
92		breq2 4657	. . . . . . . . 9 |- ( y = z -> ( Q .<_ y <-> Q .<_ z ) )
93	92	notbid 308	. . . . . . . 8 |- ( y = z -> ( -. Q .<_ y <-> -. Q .<_ z ) )
94		oveq1 6657	. . . . . . . . 9 |- ( y = z -> ( y .\/ Q ) = ( z .\/ Q ) )
95	94	eqeq2d 2632	. . . . . . . 8 |- ( y = z -> ( X = ( y .\/ Q ) <-> X = ( z .\/ Q ) ) )
96	93, 95	anbi12d 747	. . . . . . 7 |- ( y = z -> ( ( -. Q .<_ y /\ X = ( y .\/ Q ) ) <-> ( -. Q .<_ z /\ X = ( z .\/ Q ) ) ) )
97	96	rspcev 3309	. . . . . 6 |- ( ( z e. N /\ ( -. Q .<_ z /\ X = ( z .\/ Q ) ) ) -> E. y e. N ( -. Q .<_ y /\ X = ( y .\/ Q ) ) )
98	60, 61, 91, 97	syl12anc 1324	. . . . 5 |- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( -. Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) ) -> E. y e. N ( -. Q .<_ y /\ X = ( y .\/ Q ) ) )
99	98	3exp2 1285	. . . 4 |- ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) -> ( -. Q .<_ z -> ( ( z e. N /\ r e. A ) -> ( ( -. r .<_ z /\ X = ( z .\/ r ) ) -> E. y e. N ( -. Q .<_ y /\ X = ( y .\/ Q ) ) ) ) ) )
100	59, 99	pm2.61d 170	. . 3 |- ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) -> ( ( z e. N /\ r e. A ) -> ( ( -. r .<_ z /\ X = ( z .\/ r ) ) -> E. y e. N ( -. Q .<_ y /\ X = ( y .\/ Q ) ) ) ) )
101	100	rexlimdvv 3037	. 2 |- ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) -> ( E. z e. N E. r e. A ( -. r .<_ z /\ X = ( z .\/ r ) ) -> E. y e. N ( -. Q .<_ y /\ X = ( y .\/ Q ) ) ) )
102	13, 101	mpd 15	1 |- ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) -> E. y e. N ( -. Q .<_ y /\ X = ( y .\/ 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   Latclat 17045   <o ccvr 34549   Atomscatm 34550   HLchlt 34637   LLinesclln 34777   LPlanesclpl 34778

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  df-lplanes 34785

This theorem is referenced by: (None)