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Theorem dalemsly 29820
Description: Lemma for dath 29901. 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 29789 . . . . . 6  |-  ( ph  ->  K  e.  Lat )
3 dalemc.a . . . . . . 7  |-  A  =  ( Atoms `  K )
41, 3dalemseb 29807 . . . . . 6  |-  ( ph  ->  S  e.  ( Base `  K ) )
5 dalemc.j . . . . . . 7  |-  .\/  =  ( join `  K )
61, 5, 3dalemtjueb 29812 . . . . . 6  |-  ( ph  ->  ( T  .\/  U
)  e.  ( Base `  K ) )
7 eqid 2380 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
8 dalemc.l . . . . . . 7  |-  .<_  =  ( le `  K )
97, 8, 5latlej1 14409 . . . . . 6  |-  ( ( K  e.  Lat  /\  S  e.  ( Base `  K )  /\  ( T  .\/  U )  e.  ( Base `  K
) )  ->  S  .<_  ( S  .\/  ( T  .\/  U ) ) )
102, 4, 6, 9syl3anc 1184 . . . . 5  |-  ( ph  ->  S  .<_  ( S  .\/  ( T  .\/  U
) ) )
111dalemkehl 29788 . . . . . 6  |-  ( ph  ->  K  e.  HL )
121dalemsea 29794 . . . . . 6  |-  ( ph  ->  S  e.  A )
131dalemtea 29795 . . . . . 6  |-  ( ph  ->  T  e.  A )
141dalemuea 29796 . . . . . 6  |-  ( ph  ->  U  e.  A )
155, 3hlatjass 29535 . . . . . 6  |-  ( ( K  e.  HL  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A
) )  ->  (
( S  .\/  T
)  .\/  U )  =  ( S  .\/  ( T  .\/  U ) ) )
1611, 12, 13, 14, 15syl13anc 1186 . . . . 5  |-  ( ph  ->  ( ( S  .\/  T )  .\/  U )  =  ( S  .\/  ( T  .\/  U ) ) )
1710, 16breqtrrd 4172 . . . 4  |-  ( ph  ->  S  .<_  ( ( S  .\/  T )  .\/  U ) )
18 dalemsly.z . . . 4  |-  Z  =  ( ( S  .\/  T )  .\/  U )
1917, 18syl6breqr 4186 . . 3  |-  ( ph  ->  S  .<_  Z )
2019adantr 452 . 2  |-  ( (
ph  /\  Y  =  Z )  ->  S  .<_  Z )
21 simpr 448 . 2  |-  ( (
ph  /\  Y  =  Z )  ->  Y  =  Z )
2220, 21breqtrrd 4172 1  |-  ( (
ph  /\  Y  =  Z )  ->  S  .<_  Y )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   class class class wbr 4146   ` cfv 5387  (class class class)co 6013   Basecbs 13389   lecple 13456   joincjn 14321   Latclat 14394   Atomscatm 29429   HLchlt 29516
This theorem is referenced by:  dalem21  29859  dalem23  29861  dalem24  29862  dalem25  29863
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-undef 6472  df-riota 6478  df-poset 14323  df-lub 14351  df-join 14353  df-lat 14395  df-ats 29433  df-atl 29464  df-cvlat 29488  df-hlat 29517
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