Theorem dalem18 32662

Description: Lemma for dath 32717. Show that a dummy atom c exists outside of the Y and Z planes (when those planes are equal). This requires that the projective space be 3-dimensional. (Desargue's theorem doesn't always hold in 2 dimensions.) (Contributed by NM, 29-Jul-2012.)

Hypotheses

Ref	Expression
dalema.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 ) ) ) ) )
dalemc.l	|- .<_ = ( le ` K )
dalemc.j	|- .\/ = ( join ` K )
dalemc.a	|- A = ( Atoms ` K )
dalem18.y	|- Y = ( ( P .\/ Q ) .\/ R )

Assertion

Ref	Expression
dalem18	|- ( ph -> E. c e. A -. c .<_ Y )

Distinct variable groups:   A, c   .\/ , c   .<_ , c   P, c   Q, c   R, c

Allowed substitution hints:   ph( c)   C( c)   S( c)   T( c)   U( c)   K( c)   O( c)   Y( c)   Z( c)

Proof of Theorem dalem18

Step	Hyp	Ref	Expression
1		dalema.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	1	dalemkehl 32604	. . 3 |- ( ph -> K e. HL )
3	1	dalempea 32607	. . 3 |- ( ph -> P e. A )
4	1	dalemqea 32608	. . 3 |- ( ph -> Q e. A )
5	1	dalemrea 32609	. . 3 |- ( ph -> R e. A )
6		dalemc.j	. . . 4 |- .\/ = ( join ` K )
7		dalemc.l	. . . 4 |- .<_ = ( le ` K )
8		dalemc.a	. . . 4 |- A = ( Atoms ` K )
9	6, 7, 8	3dim3 32450	. . 3 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> E. c e. A -. c .<_ ( ( P .\/ Q ) .\/ R ) )
10	2, 3, 4, 5, 9	syl13anc 1230	. 2 |- ( ph -> E. c e. A -. c .<_ ( ( P .\/ Q ) .\/ R ) )
11		dalem18.y	. . . . 5 |- Y = ( ( P .\/ Q ) .\/ R )
12	11	breq2i 4400	. . . 4 |- ( c .<_ Y <-> c .<_ ( ( P .\/ Q ) .\/ R ) )
13	12	notbii 294	. . 3 |- ( -. c .<_ Y <-> -. c .<_ ( ( P .\/ Q ) .\/ R ) )
14	13	rexbii 2903	. 2 |- ( E. c e. A -. c .<_ Y <-> E. c e. A -. c .<_ ( ( P .\/ Q ) .\/ R ) )
15	10, 14	sylibr 212	1 |- ( ph -> E. c e. A -. c .<_ Y )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 367   /\ w3a 972   = wceq 1403   e. wcel 1840   E.wrex 2752   class class class wbr 4392   ` cfv 5523  (class class class)co 6232   Basecbs 14731   lecple 14806   joincjn 15787   Atomscatm 32245   HLchlt 32332

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-reu 2758  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-riota 6194  df-ov 6235  df-oprab 6236  df-preset 15771  df-poset 15789  df-plt 15802  df-lub 15818  df-glb 15819  df-join 15820  df-meet 15821  df-p0 15883  df-p1 15884  df-lat 15890  df-clat 15952  df-oposet 32158  df-ol 32160  df-oml 32161  df-covers 32248  df-ats 32249  df-atl 32280  df-cvlat 32304  df-hlat 32333

This theorem is referenced by:  dalem20  32674