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Theorem latjlej1 14487
Description: Add join to both sides of a lattice ordering. (chlej1i 22968 analog.) (Contributed by NM, 8-Nov-2011.)
Hypotheses
Ref Expression
latlej.b  |-  B  =  ( Base `  K
)
latlej.l  |-  .<_  =  ( le `  K )
latlej.j  |-  .\/  =  ( join `  K )
Assertion
Ref Expression
latjlej1  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  .<_  Y  ->  ( X  .\/  Z )  .<_  ( Y  .\/  Z ) ) )

Proof of Theorem latjlej1
StepHypRef Expression
1 latlej.b . . . . . 6  |-  B  =  ( Base `  K
)
2 latlej.l . . . . . 6  |-  .<_  =  ( le `  K )
3 latlej.j . . . . . 6  |-  .\/  =  ( join `  K )
41, 2, 3latlej1 14482 . . . . 5  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  Z  e.  B )  ->  Y  .<_  ( Y  .\/  Z ) )
543adant3r1 1162 . . . 4  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Y  .<_  ( Y  .\/  Z
) )
6 simpl 444 . . . . 5  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  K  e.  Lat )
7 simpr1 963 . . . . 5  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  X  e.  B )
8 simpr2 964 . . . . 5  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Y  e.  B )
91, 3latjcl 14472 . . . . . 6  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Y  .\/  Z
)  e.  B )
1093adant3r1 1162 . . . . 5  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( Y  .\/  Z )  e.  B )
111, 2lattr 14478 . . . . 5  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  ( Y  .\/  Z
)  e.  B ) )  ->  ( ( X  .<_  Y  /\  Y  .<_  ( Y  .\/  Z
) )  ->  X  .<_  ( Y  .\/  Z
) ) )
126, 7, 8, 10, 11syl13anc 1186 . . . 4  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  .<_  Y  /\  Y  .<_  ( Y  .\/  Z ) )  ->  X  .<_  ( Y  .\/  Z
) ) )
135, 12mpan2d 656 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  .<_  Y  ->  X  .<_  ( Y  .\/  Z
) ) )
141, 2, 3latlej2 14483 . . . 4  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  Z  e.  B )  ->  Z  .<_  ( Y  .\/  Z ) )
15143adant3r1 1162 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Z  .<_  ( Y  .\/  Z
) )
1613, 15jctird 529 . 2  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  .<_  Y  ->  ( X  .<_  ( Y  .\/  Z )  /\  Z  .<_  ( Y  .\/  Z ) ) ) )
17 simpr3 965 . . . 4  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Z  e.  B )
187, 17, 103jca 1134 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  e.  B  /\  Z  e.  B  /\  ( Y  .\/  Z )  e.  B ) )
191, 2, 3latjle12 14484 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Z  e.  B  /\  ( Y  .\/  Z
)  e.  B ) )  ->  ( ( X  .<_  ( Y  .\/  Z )  /\  Z  .<_  ( Y  .\/  Z ) )  <->  ( X  .\/  Z )  .<_  ( Y  .\/  Z ) ) )
2018, 19syldan 457 . 2  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  .<_  ( Y 
.\/  Z )  /\  Z  .<_  ( Y  .\/  Z ) )  <->  ( X  .\/  Z )  .<_  ( Y 
.\/  Z ) ) )
2116, 20sylibd 206 1  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  .<_  Y  ->  ( X  .\/  Z )  .<_  ( Y  .\/  Z ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   class class class wbr 4205   ` cfv 5447  (class class class)co 6074   Basecbs 13462   lecple 13529   joincjn 14394   Latclat 14467
This theorem is referenced by:  latjlej2  14488  latjlej12  14489  ps-2  30213  dalem5  30402  cdlema1N  30526  dalawlem3  30608  dalawlem6  30611  dalawlem7  30612  dalawlem11  30616  dalawlem12  30617  cdleme20d  31047  trlcolem  31461  cdlemh1  31550
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4313  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2703  df-rex 2704  df-reu 2705  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-iun 4088  df-br 4206  df-opab 4260  df-mpt 4261  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-1st 6342  df-2nd 6343  df-undef 6536  df-riota 6542  df-poset 14396  df-lub 14424  df-join 14426  df-lat 14468
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