Theorem dalem60 35018

Description: Lemma for dath 35022. B is an axis of perspectivity (almost). (Contributed by NM, 11-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 ) ) ) )
dalem60.m	|- ./\ = ( meet ` K )
dalem60.o	|- O = ( LPlanes ` K )
dalem60.y	|- Y = ( ( P .\/ Q ) .\/ R )
dalem60.z	|- Z = ( ( S .\/ T ) .\/ U )
dalem60.d	|- D = ( ( P .\/ Q ) ./\ ( S .\/ T ) )
dalem60.e	|- E = ( ( Q .\/ R ) ./\ ( T .\/ U ) )
dalem60.g	|- G = ( ( c .\/ P ) ./\ ( d .\/ S ) )
dalem60.h	|- H = ( ( c .\/ Q ) ./\ ( d .\/ T ) )
dalem60.i	|- I = ( ( c .\/ R ) ./\ ( d .\/ U ) )
dalem60.b1	|- B = ( ( ( G .\/ H ) .\/ I ) ./\ Y )

Assertion

Ref	Expression
dalem60	|- ( ( ph /\ Y = Z /\ ps ) -> ( D .\/ E ) = B )

Proof of Theorem dalem60

Step	Hyp	Ref	Expression
1		dalem.ph	. . . 4 |- ( 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 ) ) ) ) )
2		dalem.l	. . . 4 |- .<_ = ( le ` K )
3		dalem.j	. . . 4 |- .\/ = ( join ` K )
4		dalem.a	. . . 4 |- A = ( Atoms ` K )
5		dalem.ps	. . . 4 |- ( ps <-> ( ( c e. A /\ d e. A ) /\ -. c .<_ Y /\ ( d =/= c /\ -. d .<_ Y /\ C .<_ ( c .\/ d ) ) ) )
6		dalem60.m	. . . 4 |- ./\ = ( meet ` K )
7		dalem60.o	. . . 4 |- O = ( LPlanes ` K )
8		dalem60.y	. . . 4 |- Y = ( ( P .\/ Q ) .\/ R )
9		dalem60.z	. . . 4 |- Z = ( ( S .\/ T ) .\/ U )
10		dalem60.d	. . . 4 |- D = ( ( P .\/ Q ) ./\ ( S .\/ T ) )
11		dalem60.g	. . . 4 |- G = ( ( c .\/ P ) ./\ ( d .\/ S ) )
12		dalem60.h	. . . 4 |- H = ( ( c .\/ Q ) ./\ ( d .\/ T ) )
13		dalem60.i	. . . 4 |- I = ( ( c .\/ R ) ./\ ( d .\/ U ) )
14		dalem60.b1	. . . 4 |- B = ( ( ( G .\/ H ) .\/ I ) ./\ Y )
15	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14	dalem57 35015	. . 3 |- ( ( ph /\ Y = Z /\ ps ) -> D .<_ B )
16		dalem60.e	. . . 4 |- E = ( ( Q .\/ R ) ./\ ( T .\/ U ) )
17	1, 2, 3, 4, 5, 6, 7, 8, 9, 16, 11, 12, 13, 14	dalem58 35016	. . 3 |- ( ( ph /\ Y = Z /\ ps ) -> E .<_ B )
18	1	dalemkelat 34910	. . . . 5 |- ( ph -> K e. Lat )
19	18	3ad2ant1 1082	. . . 4 |- ( ( ph /\ Y = Z /\ ps ) -> K e. Lat )
20	1, 2, 3, 4, 6, 7, 8, 9, 10	dalemdea 34948	. . . . . 6 |- ( ph -> D e. A )
21		eqid 2622	. . . . . . 7 |- ( Base ` K ) = ( Base ` K )
22	21, 4	atbase 34576	. . . . . 6 |- ( D e. A -> D e. ( Base ` K ) )
23	20, 22	syl 17	. . . . 5 |- ( ph -> D e. ( Base ` K ) )
24	23	3ad2ant1 1082	. . . 4 |- ( ( ph /\ Y = Z /\ ps ) -> D e. ( Base ` K ) )
25	1, 2, 3, 4, 6, 7, 8, 9, 16	dalemeea 34949	. . . . . 6 |- ( ph -> E e. A )
26	21, 4	atbase 34576	. . . . . 6 |- ( E e. A -> E e. ( Base ` K ) )
27	25, 26	syl 17	. . . . 5 |- ( ph -> E e. ( Base ` K ) )
28	27	3ad2ant1 1082	. . . 4 |- ( ( ph /\ Y = Z /\ ps ) -> E e. ( Base ` K ) )
29		eqid 2622	. . . . . 6 |- ( LLines ` K ) = ( LLines ` K )
30	1, 2, 3, 4, 5, 6, 29, 7, 8, 9, 11, 12, 13, 14	dalem53 35011	. . . . 5 |- ( ( ph /\ Y = Z /\ ps ) -> B e. ( LLines ` K ) )
31	21, 29	llnbase 34795	. . . . 5 |- ( B e. ( LLines ` K ) -> B e. ( Base ` K ) )
32	30, 31	syl 17	. . . 4 |- ( ( ph /\ Y = Z /\ ps ) -> B e. ( Base ` K ) )
33	21, 2, 3	latjle12 17062	. . . 4 |- ( ( K e. Lat /\ ( D e. ( Base ` K ) /\ E e. ( Base ` K ) /\ B e. ( Base ` K ) ) ) -> ( ( D .<_ B /\ E .<_ B ) <-> ( D .\/ E ) .<_ B ) )
34	19, 24, 28, 32, 33	syl13anc 1328	. . 3 |- ( ( ph /\ Y = Z /\ ps ) -> ( ( D .<_ B /\ E .<_ B ) <-> ( D .\/ E ) .<_ B ) )
35	15, 17, 34	mpbi2and 956	. 2 |- ( ( ph /\ Y = Z /\ ps ) -> ( D .\/ E ) .<_ B )
36	1	dalemkehl 34909	. . . 4 |- ( ph -> K e. HL )
37	36	3ad2ant1 1082	. . 3 |- ( ( ph /\ Y = Z /\ ps ) -> K e. HL )
38	1, 2, 3, 4, 6, 7, 8, 9, 10, 16	dalemdnee 34952	. . . . 5 |- ( ph -> D =/= E )
39	3, 4, 29	llni2 34798	. . . . 5 |- ( ( ( K e. HL /\ D e. A /\ E e. A ) /\ D =/= E ) -> ( D .\/ E ) e. ( LLines ` K ) )
40	36, 20, 25, 38, 39	syl31anc 1329	. . . 4 |- ( ph -> ( D .\/ E ) e. ( LLines ` K ) )
41	40	3ad2ant1 1082	. . 3 |- ( ( ph /\ Y = Z /\ ps ) -> ( D .\/ E ) e. ( LLines ` K ) )
42	2, 29	llncmp 34808	. . 3 |- ( ( K e. HL /\ ( D .\/ E ) e. ( LLines ` K ) /\ B e. ( LLines ` K ) ) -> ( ( D .\/ E ) .<_ B <-> ( D .\/ E ) = B ) )
43	37, 41, 30, 42	syl3anc 1326	. 2 |- ( ( ph /\ Y = Z /\ ps ) -> ( ( D .\/ E ) .<_ B <-> ( D .\/ E ) = B ) )
44	35, 43	mpbid 222	1 |- ( ( ph /\ Y = Z /\ ps ) -> ( D .\/ E ) = B )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   lecple 15948   joincjn 16944   meetcmee 16945   Latclat 17045   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-3or 1038  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  df-lvols 34786

This theorem is referenced by:  dalem61  35019