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Theorem dalempeb 32656
Description: Lemma for dath 32753. Frequently-used utility lemma. (Contributed by NM, 13-Aug-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 ) ) ) ) )
dalema.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
dalempeb  |-  ( ph  ->  P  e.  ( Base `  K ) )

Proof of Theorem dalempeb
StepHypRef Expression
1 dalema.ph . . 3  |-  ( 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 ) ) ) ) )
21dalempea 32643 . 2  |-  ( ph  ->  P  e.  A )
3 eqid 2402 . . 3  |-  ( Base `  K )  =  (
Base `  K )
4 dalema.a . . 3  |-  A  =  ( Atoms `  K )
53, 4atbase 32307 . 2  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
62, 5syl 17 1  |-  ( ph  ->  P  e.  ( Base `  K ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   class class class wbr 4395   ` cfv 5569  (class class class)co 6278   Basecbs 14841   Atomscatm 32281   HLchlt 32368
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-iota 5533  df-fun 5571  df-fv 5577  df-ats 32285
This theorem is referenced by:  dalemply  32671  dalem1  32676  dalem-cly  32688  dalem21  32711  dalem38  32727  dalem44  32733
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