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Theorem latjlej1 15254
Description: Add join to both sides of a lattice ordering. (chlej1i 24895 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 15249 . . . . 5  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  Z  e.  B )  ->  Y  .<_  ( Y  .\/  Z ) )
543adant3r1 1196 . . . 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 994 . . . . 5  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  X  e.  B )
8 simpr2 995 . . . . 5  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Y  e.  B )
91, 3latjcl 15240 . . . . . 6  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Y  .\/  Z
)  e.  B )
1093adant3r1 1196 . . . . 5  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( Y  .\/  Z )  e.  B )
111, 2lattr 15245 . . . . 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 1220 . . . 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 15250 . . . 4  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  Z  e.  B )  ->  Z  .<_  ( Y  .\/  Z ) )
15143adant3r1 1196 . . 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 996 . . . 4  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Z  e.  B )
187, 17, 103jca 1168 . . 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 15251 . . 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 965    = wceq 1369    e. wcel 1756   class class class wbr 4311   ` cfv 5437  (class class class)co 6110   Basecbs 14193   lecple 14264   joincjn 15133   Latclat 15234
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4422  ax-sep 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550  ax-un 6391
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2739  df-rex 2740  df-reu 2741  df-rab 2743  df-v 2993  df-sbc 3206  df-csb 3308  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-nul 3657  df-if 3811  df-pw 3881  df-sn 3897  df-pr 3899  df-op 3903  df-uni 4111  df-iun 4192  df-br 4312  df-opab 4370  df-mpt 4371  df-id 4655  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5400  df-fun 5439  df-fn 5440  df-f 5441  df-f1 5442  df-fo 5443  df-f1o 5444  df-fv 5445  df-riota 6071  df-ov 6113  df-oprab 6114  df-poset 15135  df-lub 15163  df-glb 15164  df-join 15165  df-meet 15166  df-lat 15235
This theorem is referenced by:  latjlej2  15255  latjlej12  15256  ps-2  33145  dalem5  33334  cdlema1N  33458  dalawlem3  33540  dalawlem6  33543  dalawlem7  33544  dalawlem11  33548  dalawlem12  33549  cdleme20d  33979  trlcolem  34393  cdlemh1  34482
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