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Theorem latjlej12 14498
Description: Add join to both sides of a lattice ordering. (chlej12i 22979 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 958 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  ->  K  e.  Lat )
2 simp2l 984 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  ->  X  e.  B )
3 simp2r 985 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  ->  Y  e.  B )
4 simp3l 986 . . 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 14496 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  .<_  Y  ->  ( X  .\/  Z )  .<_  ( Y  .\/  Z ) ) )
91, 2, 3, 4, 8syl13anc 1187 . 2  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  -> 
( X  .<_  Y  -> 
( X  .\/  Z
)  .<_  ( Y  .\/  Z ) ) )
10 simp3r 987 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  ->  W  e.  B )
115, 6, 7latjlej2 14497 . . 3  |-  ( ( K  e.  Lat  /\  ( Z  e.  B  /\  W  e.  B  /\  Y  e.  B
) )  ->  ( Z  .<_  W  ->  ( Y  .\/  Z )  .<_  ( Y  .\/  W ) ) )
121, 4, 10, 3, 11syl13anc 1187 . 2  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  -> 
( Z  .<_  W  -> 
( Y  .\/  Z
)  .<_  ( Y  .\/  W ) ) )
135, 7latjcl 14481 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Z  e.  B )  ->  ( X  .\/  Z
)  e.  B )
141, 2, 4, 13syl3anc 1185 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  -> 
( X  .\/  Z
)  e.  B )
155, 7latjcl 14481 . . . 4  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Y  .\/  Z
)  e.  B )
161, 3, 4, 15syl3anc 1185 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  -> 
( Y  .\/  Z
)  e.  B )
175, 7latjcl 14481 . . . 4  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  W  e.  B )  ->  ( Y  .\/  W
)  e.  B )
181, 3, 10, 17syl3anc 1185 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  -> 
( Y  .\/  W
)  e.  B )
195, 6lattr 14487 . . 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 1187 . 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 471 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 set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   class class class wbr 4214   ` cfv 5456  (class class class)co 6083   Basecbs 13471   lecple 13538   joincjn 14403   Latclat 14476
This theorem is referenced by:  latledi  14520  dalem-cly  30530  dalem38  30569  dalem44  30575  cdlema1N  30650  pmapjoin  30711  4atexlemc  30928  cdlemg33a  31565
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-undef 6545  df-riota 6551  df-poset 14405  df-lub 14433  df-join 14435  df-lat 14477
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