Theorem dalem24 34710

Description: Lemma for dath 34749. Show that auxiliary atom G is outside of plane Y. (Contributed by NM, 2-Aug-2012.)

Hypotheses

Ref	Expression
dalem.ph	|- ( ph <-> ( ( ( K e. HL /\ C e. ( Base ` K ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( Y e. O /\ Z e. O ) /\ ( ( -. C .<_ ( P .\/ Q ) /\ -. C .<_ ( Q .\/ R ) /\ -. C .<_ ( R .\/ P ) ) /\ ( -. C .<_ ( S .\/ T ) /\ -. C .<_ ( T .\/ U ) /\ -. C .<_ ( U .\/ S ) ) /\ ( C .<_ ( P .\/ S ) /\ C .<_ ( Q .\/ T ) /\ C .<_ ( R .\/ U ) ) ) ) )
dalem.l	|- .<_ = ( le ` K )
dalem.j	|- .\/ = ( join ` K )
dalem.a	|- A = ( Atoms ` K )
dalem.ps	|- ( ps <-> ( ( c e. A /\ d e. A ) /\ -. c .<_ Y /\ ( d =/= c /\ -. d .<_ Y /\ C .<_ ( c .\/ d ) ) ) )
dalem23.m	|- ./\ = ( meet ` K )
dalem23.o	|- O = ( LPlanes ` K )
dalem23.y	|- Y = ( ( P .\/ Q ) .\/ R )
dalem23.z	|- Z = ( ( S .\/ T ) .\/ U )
dalem23.g	|- G = ( ( c .\/ P ) ./\ ( d .\/ S ) )

Assertion

Ref	Expression
dalem24	|- ( ( ph /\ Y = Z /\ ps ) -> -. G .<_ Y )

Proof of Theorem dalem24

Step	Hyp	Ref	Expression
1		dalem23.g	. . . . 5 |- G = ( ( c .\/ P ) ./\ ( d .\/ S ) )
2	1	oveq1i 6295	. . . 4 |- ( G ./\ Y ) = ( ( ( c .\/ P ) ./\ ( d .\/ S ) ) ./\ Y )
3		dalem.ph	. . . . . . . 8 |- ( ph <-> ( ( ( K e. HL /\ C e. ( Base ` K ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( Y e. O /\ Z e. O ) /\ ( ( -. C .<_ ( P .\/ Q ) /\ -. C .<_ ( Q .\/ R ) /\ -. C .<_ ( R .\/ P ) ) /\ ( -. C .<_ ( S .\/ T ) /\ -. C .<_ ( T .\/ U ) /\ -. C .<_ ( U .\/ S ) ) /\ ( C .<_ ( P .\/ S ) /\ C .<_ ( Q .\/ T ) /\ C .<_ ( R .\/ U ) ) ) ) )
4	3	dalemkehl 34636	. . . . . . 7 |- ( ph -> K e. HL )
5		hlol 34375	. . . . . . 7 |- ( K e. HL -> K e. OL )
6	4, 5	syl 16	. . . . . 6 |- ( ph -> K e. OL )
7	6	3ad2ant1 1017	. . . . 5 |- ( ( ph /\ Y = Z /\ ps ) -> K e. OL )
8	4	3ad2ant1 1017	. . . . . 6 |- ( ( ph /\ Y = Z /\ ps ) -> K e. HL )
9		dalem.ps	. . . . . . . 8 |- ( ps <-> ( ( c e. A /\ d e. A ) /\ -. c .<_ Y /\ ( d =/= c /\ -. d .<_ Y /\ C .<_ ( c .\/ d ) ) ) )
10	9	dalemccea 34696	. . . . . . 7 |- ( ps -> c e. A )
11	10	3ad2ant3 1019	. . . . . 6 |- ( ( ph /\ Y = Z /\ ps ) -> c e. A )
12	3	dalempea 34639	. . . . . . 7 |- ( ph -> P e. A )
13	12	3ad2ant1 1017	. . . . . 6 |- ( ( ph /\ Y = Z /\ ps ) -> P e. A )
14		eqid 2467	. . . . . . 7 |- ( Base ` K ) = ( Base ` K )
15		dalem.j	. . . . . . 7 |- .\/ = ( join ` K )
16		dalem.a	. . . . . . 7 |- A = ( Atoms ` K )
17	14, 15, 16	hlatjcl 34380	. . . . . 6 |- ( ( K e. HL /\ c e. A /\ P e. A ) -> ( c .\/ P ) e. ( Base ` K ) )
18	8, 11, 13, 17	syl3anc 1228	. . . . 5 |- ( ( ph /\ Y = Z /\ ps ) -> ( c .\/ P ) e. ( Base ` K ) )
19	9	dalemddea 34697	. . . . . . 7 |- ( ps -> d e. A )
20	19	3ad2ant3 1019	. . . . . 6 |- ( ( ph /\ Y = Z /\ ps ) -> d e. A )
21	3	dalemsea 34642	. . . . . . 7 |- ( ph -> S e. A )
22	21	3ad2ant1 1017	. . . . . 6 |- ( ( ph /\ Y = Z /\ ps ) -> S e. A )
23	14, 15, 16	hlatjcl 34380	. . . . . 6 |- ( ( K e. HL /\ d e. A /\ S e. A ) -> ( d .\/ S ) e. ( Base ` K ) )
24	8, 20, 22, 23	syl3anc 1228	. . . . 5 |- ( ( ph /\ Y = Z /\ ps ) -> ( d .\/ S ) e. ( Base ` K ) )
25		dalem23.o	. . . . . . 7 |- O = ( LPlanes ` K )
26	3, 25	dalemyeb 34662	. . . . . 6 |- ( ph -> Y e. ( Base ` K ) )
27	26	3ad2ant1 1017	. . . . 5 |- ( ( ph /\ Y = Z /\ ps ) -> Y e. ( Base ` K ) )
28		dalem23.m	. . . . . 6 |- ./\ = ( meet ` K )
29	14, 28	latmmdir 34249	. . . . 5 |- ( ( K e. OL /\ ( ( c .\/ P ) e. ( Base ` K ) /\ ( d .\/ S ) e. ( Base ` K ) /\ Y e. ( Base ` K ) ) ) -> ( ( ( c .\/ P ) ./\ ( d .\/ S ) ) ./\ Y ) = ( ( ( c .\/ P ) ./\ Y ) ./\ ( ( d .\/ S ) ./\ Y ) ) )
30	7, 18, 24, 27, 29	syl13anc 1230	. . . 4 |- ( ( ph /\ Y = Z /\ ps ) -> ( ( ( c .\/ P ) ./\ ( d .\/ S ) ) ./\ Y ) = ( ( ( c .\/ P ) ./\ Y ) ./\ ( ( d .\/ S ) ./\ Y ) ) )
31	2, 30	syl5eq 2520	. . 3 |- ( ( ph /\ Y = Z /\ ps ) -> ( G ./\ Y ) = ( ( ( c .\/ P ) ./\ Y ) ./\ ( ( d .\/ S ) ./\ Y ) ) )
32	15, 16	hlatjcom 34381	. . . . . . 7 |- ( ( K e. HL /\ c e. A /\ P e. A ) -> ( c .\/ P ) = ( P .\/ c ) )
33	8, 11, 13, 32	syl3anc 1228	. . . . . 6 |- ( ( ph /\ Y = Z /\ ps ) -> ( c .\/ P ) = ( P .\/ c ) )
34	33	oveq1d 6300	. . . . 5 |- ( ( ph /\ Y = Z /\ ps ) -> ( ( c .\/ P ) ./\ Y ) = ( ( P .\/ c ) ./\ Y ) )
35		dalem.l	. . . . . . . 8 |- .<_ = ( le ` K )
36		dalem23.y	. . . . . . . 8 |- Y = ( ( P .\/ Q ) .\/ R )
37	3, 35, 15, 16, 25, 36	dalemply 34667	. . . . . . 7 |- ( ph -> P .<_ Y )
38	37	3ad2ant1 1017	. . . . . 6 |- ( ( ph /\ Y = Z /\ ps ) -> P .<_ Y )
39	9	dalem-ccly 34698	. . . . . . 7 |- ( ps -> -. c .<_ Y )
40	39	3ad2ant3 1019	. . . . . 6 |- ( ( ph /\ Y = Z /\ ps ) -> -. c .<_ Y )
41	14, 35, 15, 28, 16	2atjm 34458	. . . . . 6 |- ( ( K e. HL /\ ( P e. A /\ c e. A /\ Y e. ( Base ` K ) ) /\ ( P .<_ Y /\ -. c .<_ Y ) ) -> ( ( P .\/ c ) ./\ Y ) = P )
42	8, 13, 11, 27, 38, 40, 41	syl132anc 1246	. . . . 5 |- ( ( ph /\ Y = Z /\ ps ) -> ( ( P .\/ c ) ./\ Y ) = P )
43	34, 42	eqtrd 2508	. . . 4 |- ( ( ph /\ Y = Z /\ ps ) -> ( ( c .\/ P ) ./\ Y ) = P )
44	15, 16	hlatjcom 34381	. . . . . . 7 |- ( ( K e. HL /\ d e. A /\ S e. A ) -> ( d .\/ S ) = ( S .\/ d ) )
45	8, 20, 22, 44	syl3anc 1228	. . . . . 6 |- ( ( ph /\ Y = Z /\ ps ) -> ( d .\/ S ) = ( S .\/ d ) )
46	45	oveq1d 6300	. . . . 5 |- ( ( ph /\ Y = Z /\ ps ) -> ( ( d .\/ S ) ./\ Y ) = ( ( S .\/ d ) ./\ Y ) )
47		dalem23.z	. . . . . . . 8 |- Z = ( ( S .\/ T ) .\/ U )
48	3, 35, 15, 16, 47	dalemsly 34668	. . . . . . 7 |- ( ( ph /\ Y = Z ) -> S .<_ Y )
49	48	3adant3 1016	. . . . . 6 |- ( ( ph /\ Y = Z /\ ps ) -> S .<_ Y )
50	9	dalem-ddly 34699	. . . . . . 7 |- ( ps -> -. d .<_ Y )
51	50	3ad2ant3 1019	. . . . . 6 |- ( ( ph /\ Y = Z /\ ps ) -> -. d .<_ Y )
52	14, 35, 15, 28, 16	2atjm 34458	. . . . . 6 |- ( ( K e. HL /\ ( S e. A /\ d e. A /\ Y e. ( Base ` K ) ) /\ ( S .<_ Y /\ -. d .<_ Y ) ) -> ( ( S .\/ d ) ./\ Y ) = S )
53	8, 22, 20, 27, 49, 51, 52	syl132anc 1246	. . . . 5 |- ( ( ph /\ Y = Z /\ ps ) -> ( ( S .\/ d ) ./\ Y ) = S )
54	46, 53	eqtrd 2508	. . . 4 |- ( ( ph /\ Y = Z /\ ps ) -> ( ( d .\/ S ) ./\ Y ) = S )
55	43, 54	oveq12d 6303	. . 3 |- ( ( ph /\ Y = Z /\ ps ) -> ( ( ( c .\/ P ) ./\ Y ) ./\ ( ( d .\/ S ) ./\ Y ) ) = ( P ./\ S ) )
56	3, 35, 15, 16, 25, 36	dalempnes 34664	. . . . 5 |- ( ph -> P =/= S )
57		hlatl 34374	. . . . . . 7 |- ( K e. HL -> K e. AtLat )
58	4, 57	syl 16	. . . . . 6 |- ( ph -> K e. AtLat )
59		eqid 2467	. . . . . . 7 |- ( 0. ` K ) = ( 0. ` K )
60	28, 59, 16	atnem0 34332	. . . . . 6 |- ( ( K e. AtLat /\ P e. A /\ S e. A ) -> ( P =/= S <-> ( P ./\ S ) = ( 0. ` K ) ) )
61	58, 12, 21, 60	syl3anc 1228	. . . . 5 |- ( ph -> ( P =/= S <-> ( P ./\ S ) = ( 0. ` K ) ) )
62	56, 61	mpbid 210	. . . 4 |- ( ph -> ( P ./\ S ) = ( 0. ` K ) )
63	62	3ad2ant1 1017	. . 3 |- ( ( ph /\ Y = Z /\ ps ) -> ( P ./\ S ) = ( 0. ` K ) )
64	31, 55, 63	3eqtrd 2512	. 2 |- ( ( ph /\ Y = Z /\ ps ) -> ( G ./\ Y ) = ( 0. ` K ) )
65	58	3ad2ant1 1017	. . 3 |- ( ( ph /\ Y = Z /\ ps ) -> K e. AtLat )
66	3, 35, 15, 16, 9, 28, 25, 36, 47, 1	dalem23 34709	. . 3 |- ( ( ph /\ Y = Z /\ ps ) -> G e. A )
67	14, 35, 28, 59, 16	atnle 34331	. . 3 |- ( ( K e. AtLat /\ G e. A /\ Y e. ( Base ` K ) ) -> ( -. G .<_ Y <-> ( G ./\ Y ) = ( 0. ` K ) ) )
68	65, 66, 27, 67	syl3anc 1228	. 2 |- ( ( ph /\ Y = Z /\ ps ) -> ( -. G .<_ Y <-> ( G ./\ Y ) = ( 0. ` K ) ) )
69	64, 68	mpbird 232	1 |- ( ( ph /\ Y = Z /\ ps ) -> -. G .<_ Y )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 973   = wceq 1379   e. wcel 1767   =/= wne 2662   class class class wbr 4447   ` cfv 5588  (class class class)co 6285   Basecbs 14493   lecple 14565   joincjn 15434   meetcmee 15435   0.cp0 15527   OLcol 34188   Atomscatm 34277   AtLatcal 34278   HLchlt 34364   LPlanesclpl 34505

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-poset 15436  df-plt 15448  df-lub 15464  df-glb 15465  df-join 15466  df-meet 15467  df-p0 15529  df-lat 15536  df-clat 15598  df-oposet 34190  df-ol 34192  df-oml 34193  df-covers 34280  df-ats 34281  df-atl 34312  df-cvlat 34336  df-hlat 34365  df-llines 34511  df-lplanes 34512

This theorem is referenced by:  dalem27  34712  dalem30  34715  dalem54  34739