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Theorem dalemkelat 35088
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
dalemkelat  |-  ( ph  ->  K  e.  Lat )

Proof of Theorem dalemkelat
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 ) ) ) ) )
21dalemkehl 35087 . 2  |-  ( ph  ->  K  e.  HL )
3 hllat 34828 . 2  |-  ( K  e.  HL  ->  K  e.  Lat )
42, 3syl 16 1  |-  ( ph  ->  K  e.  Lat )
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 14509   Latclat 15549   HLchlt 34815
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-dm 4999  df-iota 5541  df-fv 5586  df-ov 6284  df-atl 34763  df-cvlat 34787  df-hlat 34816
This theorem is referenced by:  dalemcnes  35114  dalempnes  35115  dalemqnet  35116  dalemply  35118  dalemsly  35119  dalem1  35123  dalemcea  35124  dalem3  35128  dalem4  35129  dalem5  35131  dalem8  35134  dalem-cly  35135  dalem10  35137  dalem13  35140  dalem16  35143  dalem17  35144  dalem21  35158  dalem25  35162  dalem27  35163  dalem38  35174  dalem39  35175  dalem43  35179  dalem44  35180  dalem45  35181  dalem48  35184  dalem54  35190  dalem55  35191  dalem56  35192  dalem57  35193  dalem60  35196
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