Theorem dalem18 33683

Description: Lemma for dath 33738. 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 33625	. . 3 |- ( ph -> K e. HL )
3	1	dalempea 33628	. . 3 |- ( ph -> P e. A )
4	1	dalemqea 33629	. . 3 |- ( ph -> Q e. A )
5	1	dalemrea 33630	. . 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 33471	. . 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 1221	. 2 |- ( ph -> E. c e. A -. c .<_ ( ( P .\/ Q ) .\/ R ) )
11		dalem18.y	. . . . 5 |- Y = ( ( P .\/ Q ) .\/ R )
12	11	breq2i 4411	. . . 4 |- ( c .<_ Y <-> c .<_ ( ( P .\/ Q ) .\/ R ) )
13	12	notbii 296	. . 3 |- ( -. c .<_ Y <-> -. c .<_ ( ( P .\/ Q ) .\/ R ) )
14	13	rexbii 2862	. 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 369   /\ w3a 965   = wceq 1370   e. wcel 1758   E.wrex 2800   class class class wbr 4403   ` cfv 5529  (class class class)co 6203   Basecbs 14295   lecple 14367   joincjn 15236   Atomscatm 33266   HLchlt 33353

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 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485

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 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-poset 15238  df-plt 15250  df-lub 15266  df-glb 15267  df-join 15268  df-meet 15269  df-p0 15331  df-p1 15332  df-lat 15338  df-clat 15400  df-oposet 33179  df-ol 33181  df-oml 33182  df-covers 33269  df-ats 33270  df-atl 33301  df-cvlat 33325  df-hlat 33354

This theorem is referenced by:  dalem20  33695