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Theorem latledi 14520
Description: An ortholattice is distributive in one ordering direction. (ledi 23044 analog.) (Contributed by NM, 7-Nov-2011.)
Hypotheses
Ref Expression
latledi.b  |-  B  =  ( Base `  K
)
latledi.l  |-  .<_  =  ( le `  K )
latledi.j  |-  .\/  =  ( join `  K )
latledi.m  |-  ./\  =  ( meet `  K )
Assertion
Ref Expression
latledi  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  ./\  Y
)  .\/  ( X  ./\ 
Z ) )  .<_  ( X  ./\  ( Y 
.\/  Z ) ) )

Proof of Theorem latledi
StepHypRef Expression
1 latledi.b . . . . 5  |-  B  =  ( Base `  K
)
2 latledi.l . . . . 5  |-  .<_  =  ( le `  K )
3 latledi.m . . . . 5  |-  ./\  =  ( meet `  K )
41, 2, 3latmle1 14507 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  .<_  X )
543adant3r3 1165 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  ./\  Y )  .<_  X )
61, 2, 3latmle1 14507 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Z  e.  B )  ->  ( X  ./\  Z
)  .<_  X )
763adant3r2 1164 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  ./\  Z )  .<_  X )
81, 3latmcl 14482 . . . . . 6  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  e.  B )
983adant3r3 1165 . . . . 5  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  ./\  Y )  e.  B )
101, 3latmcl 14482 . . . . . 6  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Z  e.  B )  ->  ( X  ./\  Z
)  e.  B )
11103adant3r2 1164 . . . . 5  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  ./\  Z )  e.  B )
12 simpr1 964 . . . . 5  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  X  e.  B )
139, 11, 123jca 1135 . . . 4  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  ./\  Y
)  e.  B  /\  ( X  ./\  Z )  e.  B  /\  X  e.  B ) )
14 latledi.j . . . . 5  |-  .\/  =  ( join `  K )
151, 2, 14latjle12 14493 . . . 4  |-  ( ( K  e.  Lat  /\  ( ( X  ./\  Y )  e.  B  /\  ( X  ./\  Z )  e.  B  /\  X  e.  B ) )  -> 
( ( ( X 
./\  Y )  .<_  X  /\  ( X  ./\  Z )  .<_  X )  <->  ( ( X  ./\  Y
)  .\/  ( X  ./\ 
Z ) )  .<_  X ) )
1613, 15syldan 458 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( ( X  ./\  Y )  .<_  X  /\  ( X  ./\  Z ) 
.<_  X )  <->  ( ( X  ./\  Y )  .\/  ( X  ./\  Z ) )  .<_  X )
)
175, 7, 16mpbi2and 889 . 2  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  ./\  Y
)  .\/  ( X  ./\ 
Z ) )  .<_  X )
181, 2, 3latmle2 14508 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  .<_  Y )
19183adant3r3 1165 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  ./\  Y )  .<_  Y )
201, 2, 3latmle2 14508 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Z  e.  B )  ->  ( X  ./\  Z
)  .<_  Z )
21203adant3r2 1164 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  ./\  Z )  .<_  Z )
22 simpl 445 . . . 4  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  K  e.  Lat )
23 simpr2 965 . . . 4  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Y  e.  B )
24 simpr3 966 . . . 4  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Z  e.  B )
251, 2, 14latjlej12 14498 . . . 4  |-  ( ( K  e.  Lat  /\  ( ( X  ./\  Y )  e.  B  /\  Y  e.  B )  /\  ( ( X  ./\  Z )  e.  B  /\  Z  e.  B )
)  ->  ( (
( X  ./\  Y
)  .<_  Y  /\  ( X  ./\  Z )  .<_  Z )  ->  (
( X  ./\  Y
)  .\/  ( X  ./\ 
Z ) )  .<_  ( Y  .\/  Z ) ) )
2622, 9, 23, 11, 24, 25syl122anc 1194 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( ( X  ./\  Y )  .<_  Y  /\  ( X  ./\  Z ) 
.<_  Z )  ->  (
( X  ./\  Y
)  .\/  ( X  ./\ 
Z ) )  .<_  ( Y  .\/  Z ) ) )
2719, 21, 26mp2and 662 . 2  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  ./\  Y
)  .\/  ( X  ./\ 
Z ) )  .<_  ( Y  .\/  Z ) )
281, 14latjcl 14481 . . . 4  |-  ( ( K  e.  Lat  /\  ( X  ./\  Y )  e.  B  /\  ( X  ./\  Z )  e.  B )  ->  (
( X  ./\  Y
)  .\/  ( X  ./\ 
Z ) )  e.  B )
2922, 9, 11, 28syl3anc 1185 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  ./\  Y
)  .\/  ( X  ./\ 
Z ) )  e.  B )
301, 14latjcl 14481 . . . 4  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Y  .\/  Z
)  e.  B )
31303adant3r1 1163 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( Y  .\/  Z )  e.  B )
321, 2, 3latlem12 14509 . . 3  |-  ( ( K  e.  Lat  /\  ( ( ( X 
./\  Y )  .\/  ( X  ./\  Z ) )  e.  B  /\  X  e.  B  /\  ( Y  .\/  Z )  e.  B ) )  ->  ( ( ( ( X  ./\  Y
)  .\/  ( X  ./\ 
Z ) )  .<_  X  /\  ( ( X 
./\  Y )  .\/  ( X  ./\  Z ) )  .<_  ( Y  .\/  Z ) )  <->  ( ( X  ./\  Y )  .\/  ( X  ./\  Z ) )  .<_  ( X  ./\  ( Y  .\/  Z
) ) ) )
3322, 29, 12, 31, 32syl13anc 1187 . 2  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( ( ( X 
./\  Y )  .\/  ( X  ./\  Z ) )  .<_  X  /\  ( ( X  ./\  Y )  .\/  ( X 
./\  Z ) ) 
.<_  ( Y  .\/  Z
) )  <->  ( ( X  ./\  Y )  .\/  ( X  ./\  Z ) )  .<_  ( X  ./\  ( Y  .\/  Z
) ) ) )
3417, 27, 33mpbi2and 889 1  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  ./\  Y
)  .\/  ( X  ./\ 
Z ) )  .<_  ( X  ./\  ( Y 
.\/  Z ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ 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   meetcmee 14404   Latclat 14476
This theorem is referenced by:  omlfh1N  30118
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-glb 14434  df-join 14435  df-meet 14436  df-lat 14477
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