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Theorem latjlej12 16291
Description: Add join to both sides of a lattice ordering. (chlej12i 27104 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
latjlej12  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  -> 
( ( X  .<_  Y  /\  Z  .<_  W )  ->  ( X  .\/  Z )  .<_  ( Y  .\/  W ) ) )

Proof of Theorem latjlej12
StepHypRef Expression
1 simp1 1005 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  ->  K  e.  Lat )
2 simp2l 1031 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  ->  X  e.  B )
3 simp2r 1032 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  ->  Y  e.  B )
4 simp3l 1033 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  ->  Z  e.  B )
5 latlej.b . . . 4  |-  B  =  ( Base `  K
)
6 latlej.l . . . 4  |-  .<_  =  ( le `  K )
7 latlej.j . . . 4  |-  .\/  =  ( join `  K )
85, 6, 7latjlej1 16289 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  .<_  Y  ->  ( X  .\/  Z )  .<_  ( Y  .\/  Z ) ) )
91, 2, 3, 4, 8syl13anc 1266 . 2  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  -> 
( X  .<_  Y  -> 
( X  .\/  Z
)  .<_  ( Y  .\/  Z ) ) )
10 simp3r 1034 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  ->  W  e.  B )
115, 6, 7latjlej2 16290 . . 3  |-  ( ( K  e.  Lat  /\  ( Z  e.  B  /\  W  e.  B  /\  Y  e.  B
) )  ->  ( Z  .<_  W  ->  ( Y  .\/  Z )  .<_  ( Y  .\/  W ) ) )
121, 4, 10, 3, 11syl13anc 1266 . 2  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  -> 
( Z  .<_  W  -> 
( Y  .\/  Z
)  .<_  ( Y  .\/  W ) ) )
135, 7latjcl 16275 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Z  e.  B )  ->  ( X  .\/  Z
)  e.  B )
141, 2, 4, 13syl3anc 1264 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  -> 
( X  .\/  Z
)  e.  B )
155, 7latjcl 16275 . . . 4  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Y  .\/  Z
)  e.  B )
161, 3, 4, 15syl3anc 1264 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  -> 
( Y  .\/  Z
)  e.  B )
175, 7latjcl 16275 . . . 4  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  W  e.  B )  ->  ( Y  .\/  W
)  e.  B )
181, 3, 10, 17syl3anc 1264 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  -> 
( Y  .\/  W
)  e.  B )
195, 6lattr 16280 . . 3  |-  ( ( K  e.  Lat  /\  ( ( X  .\/  Z )  e.  B  /\  ( Y  .\/  Z )  e.  B  /\  ( Y  .\/  W )  e.  B ) )  -> 
( ( ( X 
.\/  Z )  .<_  ( Y  .\/  Z )  /\  ( Y  .\/  Z )  .<_  ( Y  .\/  W ) )  -> 
( X  .\/  Z
)  .<_  ( Y  .\/  W ) ) )
201, 14, 16, 18, 19syl13anc 1266 . 2  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  -> 
( ( ( X 
.\/  Z )  .<_  ( Y  .\/  Z )  /\  ( Y  .\/  Z )  .<_  ( Y  .\/  W ) )  -> 
( X  .\/  Z
)  .<_  ( Y  .\/  W ) ) )
219, 12, 20syl2and 485 1  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  -> 
( ( X  .<_  Y  /\  Z  .<_  W )  ->  ( X  .\/  Z )  .<_  ( Y  .\/  W ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1867   class class class wbr 4417   ` cfv 5593  (class class class)co 6297   Basecbs 15099   lecple 15175   joincjn 16167   Latclat 16269
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4530  ax-sep 4540  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6589
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4477  df-mpt 4478  df-id 4761  df-xp 4852  df-rel 4853  df-cnv 4854  df-co 4855  df-dm 4856  df-rn 4857  df-res 4858  df-ima 4859  df-iota 5557  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-riota 6259  df-ov 6300  df-oprab 6301  df-poset 16169  df-lub 16198  df-glb 16199  df-join 16200  df-meet 16201  df-lat 16270
This theorem is referenced by:  latledi  16313  dalem-cly  33149  dalem38  33188  dalem44  33194  cdlema1N  33269  pmapjoin  33330  4atexlemc  33547  cdlemg33a  34186
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