Theorem dalem24 33680

Description: Lemma for dath 33719. 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 6211	. . . 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 33606	. . . . . . 7 |- ( ph -> K e. HL )
5		hlol 33345	. . . . . . 7 |- ( K e. HL -> K e. OL )
6	4, 5	syl 16	. . . . . 6 |- ( ph -> K e. OL )
7	6	3ad2ant1 1009	. . . . 5 |- ( ( ph /\ Y = Z /\ ps ) -> K e. OL )
8	4	3ad2ant1 1009	. . . . . 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 33666	. . . . . . 7 |- ( ps -> c e. A )
11	10	3ad2ant3 1011	. . . . . 6 |- ( ( ph /\ Y = Z /\ ps ) -> c e. A )
12	3	dalempea 33609	. . . . . . 7 |- ( ph -> P e. A )
13	12	3ad2ant1 1009	. . . . . 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 33350	. . . . . 6 |- ( ( K e. HL /\ c e. A /\ P e. A ) -> ( c .\/ P ) e. ( Base ` K ) )
18	8, 11, 13, 17	syl3anc 1219	. . . . 5 |- ( ( ph /\ Y = Z /\ ps ) -> ( c .\/ P ) e. ( Base ` K ) )
19	9	dalemddea 33667	. . . . . . 7 |- ( ps -> d e. A )
20	19	3ad2ant3 1011	. . . . . 6 |- ( ( ph /\ Y = Z /\ ps ) -> d e. A )
21	3	dalemsea 33612	. . . . . . 7 |- ( ph -> S e. A )
22	21	3ad2ant1 1009	. . . . . 6 |- ( ( ph /\ Y = Z /\ ps ) -> S e. A )
23	14, 15, 16	hlatjcl 33350	. . . . . 6 |- ( ( K e. HL /\ d e. A /\ S e. A ) -> ( d .\/ S ) e. ( Base ` K ) )
24	8, 20, 22, 23	syl3anc 1219	. . . . 5 |- ( ( ph /\ Y = Z /\ ps ) -> ( d .\/ S ) e. ( Base ` K ) )
25		dalem23.o	. . . . . . 7 |- O = ( LPlanes ` K )
26	3, 25	dalemyeb 33632	. . . . . 6 |- ( ph -> Y e. ( Base ` K ) )
27	26	3ad2ant1 1009	. . . . 5 |- ( ( ph /\ Y = Z /\ ps ) -> Y e. ( Base ` K ) )
28		dalem23.m	. . . . . 6 |- ./\ = ( meet ` K )
29	14, 28	latmmdir 33219	. . . . 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 1221	. . . 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 33351	. . . . . . 7 |- ( ( K e. HL /\ c e. A /\ P e. A ) -> ( c .\/ P ) = ( P .\/ c ) )
33	8, 11, 13, 32	syl3anc 1219	. . . . . 6 |- ( ( ph /\ Y = Z /\ ps ) -> ( c .\/ P ) = ( P .\/ c ) )
34	33	oveq1d 6216	. . . . 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 33637	. . . . . . 7 |- ( ph -> P .<_ Y )
38	37	3ad2ant1 1009	. . . . . 6 |- ( ( ph /\ Y = Z /\ ps ) -> P .<_ Y )
39	9	dalem-ccly 33668	. . . . . . 7 |- ( ps -> -. c .<_ Y )
40	39	3ad2ant3 1011	. . . . . 6 |- ( ( ph /\ Y = Z /\ ps ) -> -. c .<_ Y )
41	14, 35, 15, 28, 16	2atjm 33428	. . . . . 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 1237	. . . . 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 33351	. . . . . . 7 |- ( ( K e. HL /\ d e. A /\ S e. A ) -> ( d .\/ S ) = ( S .\/ d ) )
45	8, 20, 22, 44	syl3anc 1219	. . . . . 6 |- ( ( ph /\ Y = Z /\ ps ) -> ( d .\/ S ) = ( S .\/ d ) )
46	45	oveq1d 6216	. . . . 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 33638	. . . . . . 7 |- ( ( ph /\ Y = Z ) -> S .<_ Y )
49	48	3adant3 1008	. . . . . 6 |- ( ( ph /\ Y = Z /\ ps ) -> S .<_ Y )
50	9	dalem-ddly 33669	. . . . . . 7 |- ( ps -> -. d .<_ Y )
51	50	3ad2ant3 1011	. . . . . 6 |- ( ( ph /\ Y = Z /\ ps ) -> -. d .<_ Y )
52	14, 35, 15, 28, 16	2atjm 33428	. . . . . 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 1237	. . . . 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 6219	. . 3 |- ( ( ph /\ Y = Z /\ ps ) -> ( ( ( c .\/ P ) ./\ Y ) ./\ ( ( d .\/ S ) ./\ Y ) ) = ( P ./\ S ) )
56	3, 35, 15, 16, 25, 36	dalempnes 33634	. . . . 5 |- ( ph -> P =/= S )
57		hlatl 33344	. . . . . . 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 33302	. . . . . 6 |- ( ( K e. AtLat /\ P e. A /\ S e. A ) -> ( P =/= S <-> ( P ./\ S ) = ( 0. ` K ) ) )
61	58, 12, 21, 60	syl3anc 1219	. . . . 5 |- ( ph -> ( P =/= S <-> ( P ./\ S ) = ( 0. ` K ) ) )
62	56, 61	mpbid 210	. . . 4 |- ( ph -> ( P ./\ S ) = ( 0. ` K ) )
63	62	3ad2ant1 1009	. . 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 1009	. . 3 |- ( ( ph /\ Y = Z /\ ps ) -> K e. AtLat )
66	3, 35, 15, 16, 9, 28, 25, 36, 47, 1	dalem23 33679	. . 3 |- ( ( ph /\ Y = Z /\ ps ) -> G e. A )
67	14, 35, 28, 59, 16	atnle 33301	. . 3 |- ( ( K e. AtLat /\ G e. A /\ Y e. ( Base ` K ) ) -> ( -. G .<_ Y <-> ( G ./\ Y ) = ( 0. ` K ) ) )
68	65, 66, 27, 67	syl3anc 1219	. 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 965   = wceq 1370   e. wcel 1758   =/= wne 2648   class class class wbr 4401   ` cfv 5527  (class class class)co 6201   Basecbs 14293   lecple 14365   joincjn 15234   meetcmee 15235   0.cp0 15327   OLcol 33158   Atomscatm 33247   AtLatcal 33248   HLchlt 33334   LPlanesclpl 33475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-poset 15236  df-plt 15248  df-lub 15264  df-glb 15265  df-join 15266  df-meet 15267  df-p0 15329  df-lat 15336  df-clat 15398  df-oposet 33160  df-ol 33162  df-oml 33163  df-covers 33250  df-ats 33251  df-atl 33282  df-cvlat 33306  df-hlat 33335  df-llines 33481  df-lplanes 33482

This theorem is referenced by:  dalem27  33682  dalem30  33685  dalem54  33709