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Theorem latledi 14211
Description: An ortholattice is distributive in one ordering direction. (ledi 22135 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 14198 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  .<_  X )
543adant3r3 1162 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  ./\  Y )  .<_  X )
61, 2, 3latmle1 14198 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Z  e.  B )  ->  ( X  ./\  Z
)  .<_  X )
763adant3r2 1161 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  ./\  Z )  .<_  X )
81, 3latmcl 14173 . . . . . 6  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  e.  B )
983adant3r3 1162 . . . . 5  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  ./\  Y )  e.  B )
101, 3latmcl 14173 . . . . . 6  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Z  e.  B )  ->  ( X  ./\  Z
)  e.  B )
11103adant3r2 1161 . . . . 5  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  ./\  Z )  e.  B )
12 simpr1 961 . . . . 5  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  X  e.  B )
139, 11, 123jca 1132 . . . 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 14184 . . . 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 456 . . 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 887 . 2  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  ./\  Y
)  .\/  ( X  ./\ 
Z ) )  .<_  X )
181, 2, 3latmle2 14199 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  .<_  Y )
19183adant3r3 1162 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  ./\  Y )  .<_  Y )
201, 2, 3latmle2 14199 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Z  e.  B )  ->  ( X  ./\  Z
)  .<_  Z )
21203adant3r2 1161 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  ./\  Z )  .<_  Z )
22 simpl 443 . . . 4  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  K  e.  Lat )
23 simpr2 962 . . . 4  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Y  e.  B )
24 simpr3 963 . . . 4  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Z  e.  B )
251, 2, 14latjlej12 14189 . . . 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 1191 . . 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 660 . 2  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  ./\  Y
)  .\/  ( X  ./\ 
Z ) )  .<_  ( Y  .\/  Z ) )
281, 14latjcl 14172 . . . 4  |-  ( ( K  e.  Lat  /\  ( X  ./\  Y )  e.  B  /\  ( X  ./\  Z )  e.  B )  ->  (
( X  ./\  Y
)  .\/  ( X  ./\ 
Z ) )  e.  B )
2922, 9, 11, 28syl3anc 1182 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  ./\  Y
)  .\/  ( X  ./\ 
Z ) )  e.  B )
301, 14latjcl 14172 . . . 4  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Y  .\/  Z
)  e.  B )
31303adant3r1 1160 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( Y  .\/  Z )  e.  B )
321, 2, 3latlem12 14200 . . 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 1184 . 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 887 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 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   meetcmee 14095   Latclat 14167
This theorem is referenced by:  omlfh1N  30070
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-glb 14125  df-join 14126  df-meet 14127  df-lat 14168
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