Theorem dalem18 33292

Description: Lemma for dath 33347. 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 33234	. . 3 |- ( ph -> K e. HL )
3	1	dalempea 33237	. . 3 |- ( ph -> P e. A )
4	1	dalemqea 33238	. . 3 |- ( ph -> Q e. A )
5	1	dalemrea 33239	. . 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 33080	. . 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 1278	. 2 |- ( ph -> E. c e. A -. c .<_ ( ( P .\/ Q ) .\/ R ) )
11		dalem18.y	. . . . 5 |- Y = ( ( P .\/ Q ) .\/ R )
12	11	breq2i 4426	. . . 4 |- ( c .<_ Y <-> c .<_ ( ( P .\/ Q ) .\/ R ) )
13	12	notbii 302	. . 3 |- ( -. c .<_ Y <-> -. c .<_ ( ( P .\/ Q ) .\/ R ) )
14	13	rexbii 2901	. 2 |- ( E. c e. A -. c .<_ Y <-> E. c e. A -. c .<_ ( ( P .\/ Q ) .\/ R ) )
15	10, 14	sylibr 217	1 |- ( ph -> E. c e. A -. c .<_ Y )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 189   /\ wa 375   /\ w3a 991   = wceq 1455   e. wcel 1898   E.wrex 2750   class class class wbr 4418   ` cfv 5605  (class class class)co 6320   Basecbs 15176   lecple 15252   joincjn 16244   Atomscatm 32875   HLchlt 32962

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4531  ax-sep 4541  ax-nul 4550  ax-pow 4598  ax-pr 4656  ax-un 6615

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-reu 2756  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-iun 4294  df-br 4419  df-opab 4478  df-mpt 4479  df-id 4771  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-res 4868  df-ima 4869  df-iota 5569  df-fun 5607  df-fn 5608  df-f 5609  df-f1 5610  df-fo 5611  df-f1o 5612  df-fv 5613  df-riota 6282  df-ov 6323  df-oprab 6324  df-preset 16228  df-poset 16246  df-plt 16259  df-lub 16275  df-glb 16276  df-join 16277  df-meet 16278  df-p0 16340  df-p1 16341  df-lat 16347  df-clat 16409  df-oposet 32788  df-ol 32790  df-oml 32791  df-covers 32878  df-ats 32879  df-atl 32910  df-cvlat 32934  df-hlat 32963

This theorem is referenced by:  dalem20  33304