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Theorem latjlej1 14187
Description: Add join to both sides of a lattice ordering. (chlej1i 22068 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 14182 . . . . 5  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  Z  e.  B )  ->  Y  .<_  ( Y  .\/  Z ) )
543adant3r1 1160 . . . 4  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Y  .<_  ( Y  .\/  Z
) )
6 simpl 443 . . . . 5  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  K  e.  Lat )
7 simpr1 961 . . . . 5  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  X  e.  B )
8 simpr2 962 . . . . 5  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Y  e.  B )
91, 3latjcl 14172 . . . . . 6  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Y  .\/  Z
)  e.  B )
1093adant3r1 1160 . . . . 5  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( Y  .\/  Z )  e.  B )
111, 2lattr 14178 . . . . 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 1184 . . . 4  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  .<_  Y  /\  Y  .<_  ( Y  .\/  Z ) )  ->  X  .<_  ( Y  .\/  Z
) ) )
135, 12mpan2d 655 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  .<_  Y  ->  X  .<_  ( Y  .\/  Z
) ) )
141, 2, 3latlej2 14183 . . . 4  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  Z  e.  B )  ->  Z  .<_  ( Y  .\/  Z ) )
15143adant3r1 1160 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Z  .<_  ( Y  .\/  Z
) )
1613, 15jctird 528 . 2  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  .<_  Y  ->  ( X  .<_  ( Y  .\/  Z )  /\  Z  .<_  ( Y  .\/  Z ) ) ) )
17 simpr3 963 . . . 4  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Z  e.  B )
187, 17, 103jca 1132 . . 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 14184 . . 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 456 . 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 205 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 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   joincjn 14094   Latclat 14167
This theorem is referenced by:  latjlej2  14188  latjlej12  14189  ps-2  30289  dalem5  30478  cdlema1N  30602  dalawlem3  30684  dalawlem6  30687  dalawlem7  30688  dalawlem11  30692  dalawlem12  30693  cdleme20d  31123  trlcolem  31537  cdlemh1  31626
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-lub 14124  df-join 14126  df-lat 14168
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