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Theorem latledi 14473
Description: An ortholattice is distributive in one ordering direction. (ledi 22995 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 14460 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  .<_  X )
543adant3r3 1164 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  ./\  Y )  .<_  X )
61, 2, 3latmle1 14460 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Z  e.  B )  ->  ( X  ./\  Z
)  .<_  X )
763adant3r2 1163 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  ./\  Z )  .<_  X )
81, 3latmcl 14435 . . . . . 6  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  e.  B )
983adant3r3 1164 . . . . 5  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  ./\  Y )  e.  B )
101, 3latmcl 14435 . . . . . 6  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Z  e.  B )  ->  ( X  ./\  Z
)  e.  B )
11103adant3r2 1163 . . . . 5  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  ./\  Z )  e.  B )
12 simpr1 963 . . . . 5  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  X  e.  B )
139, 11, 123jca 1134 . . . 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 14446 . . . 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 457 . . 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 888 . 2  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  ./\  Y
)  .\/  ( X  ./\ 
Z ) )  .<_  X )
181, 2, 3latmle2 14461 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  .<_  Y )
19183adant3r3 1164 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  ./\  Y )  .<_  Y )
201, 2, 3latmle2 14461 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Z  e.  B )  ->  ( X  ./\  Z
)  .<_  Z )
21203adant3r2 1163 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  ./\  Z )  .<_  Z )
22 simpl 444 . . . 4  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  K  e.  Lat )
23 simpr2 964 . . . 4  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Y  e.  B )
24 simpr3 965 . . . 4  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Z  e.  B )
251, 2, 14latjlej12 14451 . . . 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 1193 . . 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 661 . 2  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  ./\  Y
)  .\/  ( X  ./\ 
Z ) )  .<_  ( Y  .\/  Z ) )
281, 14latjcl 14434 . . . 4  |-  ( ( K  e.  Lat  /\  ( X  ./\  Y )  e.  B  /\  ( X  ./\  Z )  e.  B )  ->  (
( X  ./\  Y
)  .\/  ( X  ./\ 
Z ) )  e.  B )
2922, 9, 11, 28syl3anc 1184 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  ./\  Y
)  .\/  ( X  ./\ 
Z ) )  e.  B )
301, 14latjcl 14434 . . . 4  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Y  .\/  Z
)  e.  B )
31303adant3r1 1162 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( Y  .\/  Z )  e.  B )
321, 2, 3latlem12 14462 . . 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 1186 . 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 888 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   Basecbs 13424   lecple 13491   joincjn 14356   meetcmee 14357   Latclat 14429
This theorem is referenced by:  omlfh1N  29741
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-poset 14358  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-lat 14430
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