Theorem dalem24 35818

Description: Lemma for dath 35857. 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 6280	. . . 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 35744	. . . . . . 7 |- ( ph -> K e. HL )
5		hlol 35483	. . . . . . 7 |- ( K e. HL -> K e. OL )
6	4, 5	syl 16	. . . . . 6 |- ( ph -> K e. OL )
7	6	3ad2ant1 1015	. . . . 5 |- ( ( ph /\ Y = Z /\ ps ) -> K e. OL )
8	4	3ad2ant1 1015	. . . . . 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 35804	. . . . . . 7 |- ( ps -> c e. A )
11	10	3ad2ant3 1017	. . . . . 6 |- ( ( ph /\ Y = Z /\ ps ) -> c e. A )
12	3	dalempea 35747	. . . . . . 7 |- ( ph -> P e. A )
13	12	3ad2ant1 1015	. . . . . 6 |- ( ( ph /\ Y = Z /\ ps ) -> P e. A )
14		eqid 2454	. . . . . . 7 |- ( Base ` K ) = ( Base ` K )
15		dalem.j	. . . . . . 7 |- .\/ = ( join ` K )
16		dalem.a	. . . . . . 7 |- A = ( Atoms ` K )
17	14, 15, 16	hlatjcl 35488	. . . . . 6 |- ( ( K e. HL /\ c e. A /\ P e. A ) -> ( c .\/ P ) e. ( Base ` K ) )
18	8, 11, 13, 17	syl3anc 1226	. . . . 5 |- ( ( ph /\ Y = Z /\ ps ) -> ( c .\/ P ) e. ( Base ` K ) )
19	9	dalemddea 35805	. . . . . . 7 |- ( ps -> d e. A )
20	19	3ad2ant3 1017	. . . . . 6 |- ( ( ph /\ Y = Z /\ ps ) -> d e. A )
21	3	dalemsea 35750	. . . . . . 7 |- ( ph -> S e. A )
22	21	3ad2ant1 1015	. . . . . 6 |- ( ( ph /\ Y = Z /\ ps ) -> S e. A )
23	14, 15, 16	hlatjcl 35488	. . . . . 6 |- ( ( K e. HL /\ d e. A /\ S e. A ) -> ( d .\/ S ) e. ( Base ` K ) )
24	8, 20, 22, 23	syl3anc 1226	. . . . 5 |- ( ( ph /\ Y = Z /\ ps ) -> ( d .\/ S ) e. ( Base ` K ) )
25		dalem23.o	. . . . . . 7 |- O = ( LPlanes ` K )
26	3, 25	dalemyeb 35770	. . . . . 6 |- ( ph -> Y e. ( Base ` K ) )
27	26	3ad2ant1 1015	. . . . 5 |- ( ( ph /\ Y = Z /\ ps ) -> Y e. ( Base ` K ) )
28		dalem23.m	. . . . . 6 |- ./\ = ( meet ` K )
29	14, 28	latmmdir 35357	. . . . 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 1228	. . . 4 |- ( ( ph /\ Y = Z /\ ps ) -> ( ( ( c .\/ P ) ./\ ( d .\/ S ) ) ./\ Y ) = ( ( ( c .\/ P ) ./\ Y ) ./\ ( ( d .\/ S ) ./\ Y ) ) )
31	2, 30	syl5eq 2507	. . 3 |- ( ( ph /\ Y = Z /\ ps ) -> ( G ./\ Y ) = ( ( ( c .\/ P ) ./\ Y ) ./\ ( ( d .\/ S ) ./\ Y ) ) )
32	15, 16	hlatjcom 35489	. . . . . . 7 |- ( ( K e. HL /\ c e. A /\ P e. A ) -> ( c .\/ P ) = ( P .\/ c ) )
33	8, 11, 13, 32	syl3anc 1226	. . . . . 6 |- ( ( ph /\ Y = Z /\ ps ) -> ( c .\/ P ) = ( P .\/ c ) )
34	33	oveq1d 6285	. . . . 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 35775	. . . . . . 7 |- ( ph -> P .<_ Y )
38	37	3ad2ant1 1015	. . . . . 6 |- ( ( ph /\ Y = Z /\ ps ) -> P .<_ Y )
39	9	dalem-ccly 35806	. . . . . . 7 |- ( ps -> -. c .<_ Y )
40	39	3ad2ant3 1017	. . . . . 6 |- ( ( ph /\ Y = Z /\ ps ) -> -. c .<_ Y )
41	14, 35, 15, 28, 16	2atjm 35566	. . . . . 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 1244	. . . . 5 |- ( ( ph /\ Y = Z /\ ps ) -> ( ( P .\/ c ) ./\ Y ) = P )
43	34, 42	eqtrd 2495	. . . 4 |- ( ( ph /\ Y = Z /\ ps ) -> ( ( c .\/ P ) ./\ Y ) = P )
44	15, 16	hlatjcom 35489	. . . . . . 7 |- ( ( K e. HL /\ d e. A /\ S e. A ) -> ( d .\/ S ) = ( S .\/ d ) )
45	8, 20, 22, 44	syl3anc 1226	. . . . . 6 |- ( ( ph /\ Y = Z /\ ps ) -> ( d .\/ S ) = ( S .\/ d ) )
46	45	oveq1d 6285	. . . . 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 35776	. . . . . . 7 |- ( ( ph /\ Y = Z ) -> S .<_ Y )
49	48	3adant3 1014	. . . . . 6 |- ( ( ph /\ Y = Z /\ ps ) -> S .<_ Y )
50	9	dalem-ddly 35807	. . . . . . 7 |- ( ps -> -. d .<_ Y )
51	50	3ad2ant3 1017	. . . . . 6 |- ( ( ph /\ Y = Z /\ ps ) -> -. d .<_ Y )
52	14, 35, 15, 28, 16	2atjm 35566	. . . . . 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 1244	. . . . 5 |- ( ( ph /\ Y = Z /\ ps ) -> ( ( S .\/ d ) ./\ Y ) = S )
54	46, 53	eqtrd 2495	. . . 4 |- ( ( ph /\ Y = Z /\ ps ) -> ( ( d .\/ S ) ./\ Y ) = S )
55	43, 54	oveq12d 6288	. . 3 |- ( ( ph /\ Y = Z /\ ps ) -> ( ( ( c .\/ P ) ./\ Y ) ./\ ( ( d .\/ S ) ./\ Y ) ) = ( P ./\ S ) )
56	3, 35, 15, 16, 25, 36	dalempnes 35772	. . . . 5 |- ( ph -> P =/= S )
57		hlatl 35482	. . . . . . 7 |- ( K e. HL -> K e. AtLat )
58	4, 57	syl 16	. . . . . 6 |- ( ph -> K e. AtLat )
59		eqid 2454	. . . . . . 7 |- ( 0. ` K ) = ( 0. ` K )
60	28, 59, 16	atnem0 35440	. . . . . 6 |- ( ( K e. AtLat /\ P e. A /\ S e. A ) -> ( P =/= S <-> ( P ./\ S ) = ( 0. ` K ) ) )
61	58, 12, 21, 60	syl3anc 1226	. . . . 5 |- ( ph -> ( P =/= S <-> ( P ./\ S ) = ( 0. ` K ) ) )
62	56, 61	mpbid 210	. . . 4 |- ( ph -> ( P ./\ S ) = ( 0. ` K ) )
63	62	3ad2ant1 1015	. . 3 |- ( ( ph /\ Y = Z /\ ps ) -> ( P ./\ S ) = ( 0. ` K ) )
64	31, 55, 63	3eqtrd 2499	. 2 |- ( ( ph /\ Y = Z /\ ps ) -> ( G ./\ Y ) = ( 0. ` K ) )
65	58	3ad2ant1 1015	. . 3 |- ( ( ph /\ Y = Z /\ ps ) -> K e. AtLat )
66	3, 35, 15, 16, 9, 28, 25, 36, 47, 1	dalem23 35817	. . 3 |- ( ( ph /\ Y = Z /\ ps ) -> G e. A )
67	14, 35, 28, 59, 16	atnle 35439	. . 3 |- ( ( K e. AtLat /\ G e. A /\ Y e. ( Base ` K ) ) -> ( -. G .<_ Y <-> ( G ./\ Y ) = ( 0. ` K ) ) )
68	65, 66, 27, 67	syl3anc 1226	. 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 367   /\ w3a 971   = wceq 1398   e. wcel 1823   =/= wne 2649   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   Basecbs 14716   lecple 14791   joincjn 15772   meetcmee 15773   0.cp0 15866   OLcol 35296   Atomscatm 35385   AtLatcal 35386   HLchlt 35472   LPlanesclpl 35613

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-preset 15756  df-poset 15774  df-plt 15787  df-lub 15803  df-glb 15804  df-join 15805  df-meet 15806  df-p0 15868  df-lat 15875  df-clat 15937  df-oposet 35298  df-ol 35300  df-oml 35301  df-covers 35388  df-ats 35389  df-atl 35420  df-cvlat 35444  df-hlat 35473  df-llines 35619  df-lplanes 35620

This theorem is referenced by:  dalem27  35820  dalem30  35823  dalem54  35847