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Theorem dalemsly 35776
Description: Lemma for dath 35857. Frequently-used utility lemma. (Contributed by NM, 15-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 ) ) ) ) )
dalemc.l  |-  .<_  =  ( le `  K )
dalemc.j  |-  .\/  =  ( join `  K )
dalemc.a  |-  A  =  ( Atoms `  K )
dalemsly.z  |-  Z  =  ( ( S  .\/  T )  .\/  U )
Assertion
Ref Expression
dalemsly  |-  ( (
ph  /\  Y  =  Z )  ->  S  .<_  Y )

Proof of Theorem dalemsly
StepHypRef Expression
1 dalema.ph . . . . . . 7  |-  ( 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 ) ) ) ) )
21dalemkelat 35745 . . . . . 6  |-  ( ph  ->  K  e.  Lat )
3 dalemc.a . . . . . . 7  |-  A  =  ( Atoms `  K )
41, 3dalemseb 35763 . . . . . 6  |-  ( ph  ->  S  e.  ( Base `  K ) )
5 dalemc.j . . . . . . 7  |-  .\/  =  ( join `  K )
61, 5, 3dalemtjueb 35768 . . . . . 6  |-  ( ph  ->  ( T  .\/  U
)  e.  ( Base `  K ) )
7 eqid 2454 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
8 dalemc.l . . . . . . 7  |-  .<_  =  ( le `  K )
97, 8, 5latlej1 15889 . . . . . 6  |-  ( ( K  e.  Lat  /\  S  e.  ( Base `  K )  /\  ( T  .\/  U )  e.  ( Base `  K
) )  ->  S  .<_  ( S  .\/  ( T  .\/  U ) ) )
102, 4, 6, 9syl3anc 1226 . . . . 5  |-  ( ph  ->  S  .<_  ( S  .\/  ( T  .\/  U
) ) )
111dalemkehl 35744 . . . . . 6  |-  ( ph  ->  K  e.  HL )
121dalemsea 35750 . . . . . 6  |-  ( ph  ->  S  e.  A )
131dalemtea 35751 . . . . . 6  |-  ( ph  ->  T  e.  A )
141dalemuea 35752 . . . . . 6  |-  ( ph  ->  U  e.  A )
155, 3hlatjass 35491 . . . . . 6  |-  ( ( K  e.  HL  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A
) )  ->  (
( S  .\/  T
)  .\/  U )  =  ( S  .\/  ( T  .\/  U ) ) )
1611, 12, 13, 14, 15syl13anc 1228 . . . . 5  |-  ( ph  ->  ( ( S  .\/  T )  .\/  U )  =  ( S  .\/  ( T  .\/  U ) ) )
1710, 16breqtrrd 4465 . . . 4  |-  ( ph  ->  S  .<_  ( ( S  .\/  T )  .\/  U ) )
18 dalemsly.z . . . 4  |-  Z  =  ( ( S  .\/  T )  .\/  U )
1917, 18syl6breqr 4479 . . 3  |-  ( ph  ->  S  .<_  Z )
2019adantr 463 . 2  |-  ( (
ph  /\  Y  =  Z )  ->  S  .<_  Z )
21 simpr 459 . 2  |-  ( (
ph  /\  Y  =  Z )  ->  Y  =  Z )
2220, 21breqtrrd 4465 1  |-  ( (
ph  /\  Y  =  Z )  ->  S  .<_  Y )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   Basecbs 14716   lecple 14791   joincjn 15772   Latclat 15874   Atomscatm 35385   HLchlt 35472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-preset 15756  df-poset 15774  df-lub 15803  df-glb 15804  df-join 15805  df-meet 15806  df-lat 15875  df-ats 35389  df-atl 35420  df-cvlat 35444  df-hlat 35473
This theorem is referenced by:  dalem21  35815  dalem23  35817  dalem24  35818  dalem25  35819
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