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Theorem dalemkehl 35087
Description: Lemma for dath 35200. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)
Hypothesis
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 ) ) ) ) )
Assertion
Ref Expression
dalemkehl  |-  ( ph  ->  K  e.  HL )

Proof of Theorem dalemkehl
StepHypRef Expression
1 dalema.ph . 2  |-  ( 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 simp11l 1108 . 2  |-  ( ( ( ( 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 ) ) ) )  ->  K  e.  HL )
31, 2sylbi 195 1  |-  ( ph  ->  K  e.  HL )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 974    e. wcel 1804   class class class wbr 4437   ` cfv 5578  (class class class)co 6281   Basecbs 14613   HLchlt 34815
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 976
This theorem is referenced by:  dalemkelat  35088  dalemkeop  35089  dalempjqeb  35109  dalemsjteb  35110  dalemtjueb  35111  dalemqrprot  35112  dalempnes  35115  dalemqnet  35116  dalempjsen  35117  dalemply  35118  dalemsly  35119  dalemswapyz  35120  dalemrot  35121  dalemrotyz  35122  dalem1  35123  dalemcea  35124  dalem2  35125  dalemdea  35126  dalem3  35128  dalem4  35129  dalem5  35131  dalem-cly  35135  dalem9  35136  dalem11  35138  dalem12  35139  dalem13  35140  dalem15  35142  dalem16  35143  dalem17  35144  dalem18  35145  dalem19  35146  dalemswapyzps  35154  dalemcjden  35156  dalem21  35158  dalem22  35159  dalem23  35160  dalem24  35161  dalem25  35162  dalem27  35163  dalem28  35164  dalem38  35174  dalem39  35175  dalem41  35177  dalem42  35178  dalem43  35179  dalem44  35180  dalem45  35181  dalem51  35187  dalem52  35188  dalem54  35190  dalem55  35191  dalem56  35192  dalem57  35193  dalem58  35194  dalem59  35195  dalem60  35196
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