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Metamath Proof Explorer

Theorem dalem18

Description: Lemma for dath . 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. (Desargues's theorem does not always hold in 2 dimensions.) (Contributed by NM, 29-Jul-2012)

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

Proof

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