Theorem dalem24 33263

Description: Lemma for dath 33302. 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 6285	. . . 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 33189	. . . . . . 7 |- ( ph -> K e. HL )
5		hlol 32928	. . . . . . 7 |- ( K e. HL -> K e. OL )
6	4, 5	syl 17	. . . . . 6 |- ( ph -> K e. OL )
7	6	3ad2ant1 1030	. . . . 5 |- ( ( ph /\ Y = Z /\ ps ) -> K e. OL )
8	4	3ad2ant1 1030	. . . . . 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 33249	. . . . . . 7 |- ( ps -> c e. A )
11	10	3ad2ant3 1032	. . . . . 6 |- ( ( ph /\ Y = Z /\ ps ) -> c e. A )
12	3	dalempea 33192	. . . . . . 7 |- ( ph -> P e. A )
13	12	3ad2ant1 1030	. . . . . 6 |- ( ( ph /\ Y = Z /\ ps ) -> P e. A )
14		eqid 2451	. . . . . . 7 |- ( Base ` K ) = ( Base ` K )
15		dalem.j	. . . . . . 7 |- .\/ = ( join ` K )
16		dalem.a	. . . . . . 7 |- A = ( Atoms ` K )
17	14, 15, 16	hlatjcl 32933	. . . . . 6 |- ( ( K e. HL /\ c e. A /\ P e. A ) -> ( c .\/ P ) e. ( Base ` K ) )
18	8, 11, 13, 17	syl3anc 1271	. . . . 5 |- ( ( ph /\ Y = Z /\ ps ) -> ( c .\/ P ) e. ( Base ` K ) )
19	9	dalemddea 33250	. . . . . . 7 |- ( ps -> d e. A )
20	19	3ad2ant3 1032	. . . . . 6 |- ( ( ph /\ Y = Z /\ ps ) -> d e. A )
21	3	dalemsea 33195	. . . . . . 7 |- ( ph -> S e. A )
22	21	3ad2ant1 1030	. . . . . 6 |- ( ( ph /\ Y = Z /\ ps ) -> S e. A )
23	14, 15, 16	hlatjcl 32933	. . . . . 6 |- ( ( K e. HL /\ d e. A /\ S e. A ) -> ( d .\/ S ) e. ( Base ` K ) )
24	8, 20, 22, 23	syl3anc 1271	. . . . 5 |- ( ( ph /\ Y = Z /\ ps ) -> ( d .\/ S ) e. ( Base ` K ) )
25		dalem23.o	. . . . . . 7 |- O = ( LPlanes ` K )
26	3, 25	dalemyeb 33215	. . . . . 6 |- ( ph -> Y e. ( Base ` K ) )
27	26	3ad2ant1 1030	. . . . 5 |- ( ( ph /\ Y = Z /\ ps ) -> Y e. ( Base ` K ) )
28		dalem23.m	. . . . . 6 |- ./\ = ( meet ` K )
29	14, 28	latmmdir 32802	. . . . 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 1273	. . . 4 |- ( ( ph /\ Y = Z /\ ps ) -> ( ( ( c .\/ P ) ./\ ( d .\/ S ) ) ./\ Y ) = ( ( ( c .\/ P ) ./\ Y ) ./\ ( ( d .\/ S ) ./\ Y ) ) )
31	2, 30	syl5eq 2497	. . 3 |- ( ( ph /\ Y = Z /\ ps ) -> ( G ./\ Y ) = ( ( ( c .\/ P ) ./\ Y ) ./\ ( ( d .\/ S ) ./\ Y ) ) )
32	15, 16	hlatjcom 32934	. . . . . . 7 |- ( ( K e. HL /\ c e. A /\ P e. A ) -> ( c .\/ P ) = ( P .\/ c ) )
33	8, 11, 13, 32	syl3anc 1271	. . . . . 6 |- ( ( ph /\ Y = Z /\ ps ) -> ( c .\/ P ) = ( P .\/ c ) )
34	33	oveq1d 6290	. . . . 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 33220	. . . . . . 7 |- ( ph -> P .<_ Y )
38	37	3ad2ant1 1030	. . . . . 6 |- ( ( ph /\ Y = Z /\ ps ) -> P .<_ Y )
39	9	dalem-ccly 33251	. . . . . . 7 |- ( ps -> -. c .<_ Y )
40	39	3ad2ant3 1032	. . . . . 6 |- ( ( ph /\ Y = Z /\ ps ) -> -. c .<_ Y )
41	14, 35, 15, 28, 16	2atjm 33011	. . . . . 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 1289	. . . . 5 |- ( ( ph /\ Y = Z /\ ps ) -> ( ( P .\/ c ) ./\ Y ) = P )
43	34, 42	eqtrd 2485	. . . 4 |- ( ( ph /\ Y = Z /\ ps ) -> ( ( c .\/ P ) ./\ Y ) = P )
44	15, 16	hlatjcom 32934	. . . . . . 7 |- ( ( K e. HL /\ d e. A /\ S e. A ) -> ( d .\/ S ) = ( S .\/ d ) )
45	8, 20, 22, 44	syl3anc 1271	. . . . . 6 |- ( ( ph /\ Y = Z /\ ps ) -> ( d .\/ S ) = ( S .\/ d ) )
46	45	oveq1d 6290	. . . . 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 33221	. . . . . . 7 |- ( ( ph /\ Y = Z ) -> S .<_ Y )
49	48	3adant3 1029	. . . . . 6 |- ( ( ph /\ Y = Z /\ ps ) -> S .<_ Y )
50	9	dalem-ddly 33252	. . . . . . 7 |- ( ps -> -. d .<_ Y )
51	50	3ad2ant3 1032	. . . . . 6 |- ( ( ph /\ Y = Z /\ ps ) -> -. d .<_ Y )
52	14, 35, 15, 28, 16	2atjm 33011	. . . . . 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 1289	. . . . 5 |- ( ( ph /\ Y = Z /\ ps ) -> ( ( S .\/ d ) ./\ Y ) = S )
54	46, 53	eqtrd 2485	. . . 4 |- ( ( ph /\ Y = Z /\ ps ) -> ( ( d .\/ S ) ./\ Y ) = S )
55	43, 54	oveq12d 6293	. . 3 |- ( ( ph /\ Y = Z /\ ps ) -> ( ( ( c .\/ P ) ./\ Y ) ./\ ( ( d .\/ S ) ./\ Y ) ) = ( P ./\ S ) )
56	3, 35, 15, 16, 25, 36	dalempnes 33217	. . . . 5 |- ( ph -> P =/= S )
57		hlatl 32927	. . . . . . 7 |- ( K e. HL -> K e. AtLat )
58	4, 57	syl 17	. . . . . 6 |- ( ph -> K e. AtLat )
59		eqid 2451	. . . . . . 7 |- ( 0. ` K ) = ( 0. ` K )
60	28, 59, 16	atnem0 32885	. . . . . 6 |- ( ( K e. AtLat /\ P e. A /\ S e. A ) -> ( P =/= S <-> ( P ./\ S ) = ( 0. ` K ) ) )
61	58, 12, 21, 60	syl3anc 1271	. . . . 5 |- ( ph -> ( P =/= S <-> ( P ./\ S ) = ( 0. ` K ) ) )
62	56, 61	mpbid 215	. . . 4 |- ( ph -> ( P ./\ S ) = ( 0. ` K ) )
63	62	3ad2ant1 1030	. . 3 |- ( ( ph /\ Y = Z /\ ps ) -> ( P ./\ S ) = ( 0. ` K ) )
64	31, 55, 63	3eqtrd 2489	. 2 |- ( ( ph /\ Y = Z /\ ps ) -> ( G ./\ Y ) = ( 0. ` K ) )
65	58	3ad2ant1 1030	. . 3 |- ( ( ph /\ Y = Z /\ ps ) -> K e. AtLat )
66	3, 35, 15, 16, 9, 28, 25, 36, 47, 1	dalem23 33262	. . 3 |- ( ( ph /\ Y = Z /\ ps ) -> G e. A )
67	14, 35, 28, 59, 16	atnle 32884	. . 3 |- ( ( K e. AtLat /\ G e. A /\ Y e. ( Base ` K ) ) -> ( -. G .<_ Y <-> ( G ./\ Y ) = ( 0. ` K ) ) )
68	65, 66, 27, 67	syl3anc 1271	. 2 |- ( ( ph /\ Y = Z /\ ps ) -> ( -. G .<_ Y <-> ( G ./\ Y ) = ( 0. ` K ) ) )
69	64, 68	mpbird 240	1 |- ( ( ph /\ Y = Z /\ ps ) -> -. G .<_ Y )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 189   /\ wa 375   /\ w3a 986   = wceq 1447   e. wcel 1890   =/= wne 2621   class class class wbr 4373   ` cfv 5560  (class class class)co 6275   Basecbs 15131   lecple 15207   joincjn 16199   meetcmee 16200   0.cp0 16293   OLcol 32741   Atomscatm 32830   AtLatcal 32831   HLchlt 32917   LPlanesclpl 33058

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1672  ax-4 1685  ax-5 1761  ax-6 1808  ax-7 1854  ax-8 1892  ax-9 1899  ax-10 1918  ax-11 1923  ax-12 1936  ax-13 2091  ax-ext 2431  ax-rep 4486  ax-sep 4496  ax-nul 4505  ax-pow 4553  ax-pr 4611  ax-un 6570

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 988  df-tru 1450  df-ex 1667  df-nf 1671  df-sb 1801  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 3014  df-sbc 3235  df-csb 3331  df-dif 3374  df-un 3376  df-in 3378  df-ss 3385  df-nul 3699  df-if 3849  df-pw 3920  df-sn 3936  df-pr 3938  df-op 3942  df-uni 4168  df-iun 4249  df-br 4374  df-opab 4433  df-mpt 4434  df-id 4726  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5524  df-fun 5562  df-fn 5563  df-f 5564  df-f1 5565  df-fo 5566  df-f1o 5567  df-fv 5568  df-riota 6237  df-ov 6278  df-oprab 6279  df-preset 16183  df-poset 16201  df-plt 16214  df-lub 16230  df-glb 16231  df-join 16232  df-meet 16233  df-p0 16295  df-lat 16302  df-clat 16364  df-oposet 32743  df-ol 32745  df-oml 32746  df-covers 32833  df-ats 32834  df-atl 32865  df-cvlat 32889  df-hlat 32918  df-llines 33064  df-lplanes 33065

This theorem is referenced by:  dalem27  33265  dalem30  33268  dalem54  33292