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Theorem latjlej1 15541
Description: Add join to both sides of a lattice ordering. (chlej1i 26053 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 15536 . . . . 5  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  Z  e.  B )  ->  Y  .<_  ( Y  .\/  Z ) )
543adant3r1 1200 . . . 4  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Y  .<_  ( Y  .\/  Z
) )
6 simpl 457 . . . . 5  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  K  e.  Lat )
7 simpr1 997 . . . . 5  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  X  e.  B )
8 simpr2 998 . . . . 5  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Y  e.  B )
91, 3latjcl 15527 . . . . . 6  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Y  .\/  Z
)  e.  B )
1093adant3r1 1200 . . . . 5  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( Y  .\/  Z )  e.  B )
111, 2lattr 15532 . . . . 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 1225 . . . 4  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  .<_  Y  /\  Y  .<_  ( Y  .\/  Z ) )  ->  X  .<_  ( Y  .\/  Z
) ) )
135, 12mpan2d 674 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  .<_  Y  ->  X  .<_  ( Y  .\/  Z
) ) )
141, 2, 3latlej2 15537 . . . 4  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  Z  e.  B )  ->  Z  .<_  ( Y  .\/  Z ) )
15143adant3r1 1200 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Z  .<_  ( Y  .\/  Z
) )
1613, 15jctird 544 . 2  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  .<_  Y  ->  ( X  .<_  ( Y  .\/  Z )  /\  Z  .<_  ( Y  .\/  Z ) ) ) )
17 simpr3 999 . . . 4  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Z  e.  B )
187, 17, 103jca 1171 . . 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 15538 . . 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 470 . 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 214 1  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  .<_  Y  ->  ( X  .\/  Z )  .<_  ( Y  .\/  Z ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   class class class wbr 4440   ` cfv 5579  (class class class)co 6275   Basecbs 14479   lecple 14551   joincjn 15420   Latclat 15521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-poset 15422  df-lub 15450  df-glb 15451  df-join 15452  df-meet 15453  df-lat 15522
This theorem is referenced by:  latjlej2  15542  latjlej12  15543  ps-2  34149  dalem5  34338  cdlema1N  34462  dalawlem3  34544  dalawlem6  34547  dalawlem7  34548  dalawlem11  34552  dalawlem12  34553  cdleme20d  34983  trlcolem  35397  cdlemh1  35486
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