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Theorem mod1ile 14535
Description: The weak direction of the modular law (e.g. pmod1i 30646, atmod1i1 30655) that holds in any lattice. (Contributed by NM, 11-May-2012.)
Hypotheses
Ref Expression
modle.b  |-  B  =  ( Base `  K
)
modle.l  |-  .<_  =  ( le `  K )
modle.j  |-  .\/  =  ( join `  K )
modle.m  |-  ./\  =  ( meet `  K )
Assertion
Ref Expression
mod1ile  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  .<_  Z  ->  ( X  .\/  ( Y  ./\  Z ) )  .<_  ( ( X  .\/  Y ) 
./\  Z ) ) )

Proof of Theorem mod1ile
StepHypRef Expression
1 simpll 732 . . . . 5  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X  .<_  Z )  ->  K  e.  Lat )
2 simplr1 1000 . . . . 5  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X  .<_  Z )  ->  X  e.  B )
3 simplr2 1001 . . . . 5  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X  .<_  Z )  ->  Y  e.  B )
4 modle.b . . . . . 6  |-  B  =  ( Base `  K
)
5 modle.l . . . . . 6  |-  .<_  =  ( le `  K )
6 modle.j . . . . . 6  |-  .\/  =  ( join `  K )
74, 5, 6latlej1 14490 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  X  .<_  ( X  .\/  Y ) )
81, 2, 3, 7syl3anc 1185 . . . 4  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X  .<_  Z )  ->  X  .<_  ( X  .\/  Y
) )
9 simpr 449 . . . 4  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X  .<_  Z )  ->  X  .<_  Z )
104, 6latjcl 14480 . . . . . 6  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .\/  Y
)  e.  B )
111, 2, 3, 10syl3anc 1185 . . . . 5  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X  .<_  Z )  ->  ( X  .\/  Y )  e.  B )
12 simplr3 1002 . . . . 5  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X  .<_  Z )  ->  Z  e.  B )
13 modle.m . . . . . 6  |-  ./\  =  ( meet `  K )
144, 5, 13latlem12 14508 . . . . 5  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  ( X  .\/  Y
)  e.  B  /\  Z  e.  B )
)  ->  ( ( X  .<_  ( X  .\/  Y )  /\  X  .<_  Z )  <->  X  .<_  ( ( X  .\/  Y ) 
./\  Z ) ) )
151, 2, 11, 12, 14syl13anc 1187 . . . 4  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X  .<_  Z )  ->  (
( X  .<_  ( X 
.\/  Y )  /\  X  .<_  Z )  <->  X  .<_  ( ( X  .\/  Y
)  ./\  Z )
) )
168, 9, 15mpbi2and 889 . . 3  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X  .<_  Z )  ->  X  .<_  ( ( X  .\/  Y )  ./\  Z )
)
174, 5, 6, 13latmlej12 14521 . . . . 5  |-  ( ( K  e.  Lat  /\  ( Y  e.  B  /\  Z  e.  B  /\  X  e.  B
) )  ->  ( Y  ./\  Z )  .<_  ( X  .\/  Y ) )
181, 3, 12, 2, 17syl13anc 1187 . . . 4  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X  .<_  Z )  ->  ( Y  ./\  Z )  .<_  ( X  .\/  Y ) )
194, 5, 13latmle2 14507 . . . . 5  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Y  ./\  Z
)  .<_  Z )
201, 3, 12, 19syl3anc 1185 . . . 4  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X  .<_  Z )  ->  ( Y  ./\  Z )  .<_  Z )
214, 13latmcl 14481 . . . . . 6  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Y  ./\  Z
)  e.  B )
221, 3, 12, 21syl3anc 1185 . . . . 5  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X  .<_  Z )  ->  ( Y  ./\  Z )  e.  B )
234, 5, 13latlem12 14508 . . . . 5  |-  ( ( K  e.  Lat  /\  ( ( Y  ./\  Z )  e.  B  /\  ( X  .\/  Y )  e.  B  /\  Z  e.  B ) )  -> 
( ( ( Y 
./\  Z )  .<_  ( X  .\/  Y )  /\  ( Y  ./\  Z )  .<_  Z )  <->  ( Y  ./\  Z )  .<_  ( ( X  .\/  Y )  ./\  Z )
) )
241, 22, 11, 12, 23syl13anc 1187 . . . 4  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X  .<_  Z )  ->  (
( ( Y  ./\  Z )  .<_  ( X  .\/  Y )  /\  ( Y  ./\  Z )  .<_  Z )  <->  ( Y  ./\ 
Z )  .<_  ( ( X  .\/  Y ) 
./\  Z ) ) )
2518, 20, 24mpbi2and 889 . . 3  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X  .<_  Z )  ->  ( Y  ./\  Z )  .<_  ( ( X  .\/  Y )  ./\  Z )
)
264, 13latmcl 14481 . . . . 5  |-  ( ( K  e.  Lat  /\  ( X  .\/  Y )  e.  B  /\  Z  e.  B )  ->  (
( X  .\/  Y
)  ./\  Z )  e.  B )
271, 11, 12, 26syl3anc 1185 . . . 4  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X  .<_  Z )  ->  (
( X  .\/  Y
)  ./\  Z )  e.  B )
284, 5, 6latjle12 14492 . . . 4  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  ( Y  ./\  Z
)  e.  B  /\  ( ( X  .\/  Y )  ./\  Z )  e.  B ) )  -> 
( ( X  .<_  ( ( X  .\/  Y
)  ./\  Z )  /\  ( Y  ./\  Z
)  .<_  ( ( X 
.\/  Y )  ./\  Z ) )  <->  ( X  .\/  ( Y  ./\  Z
) )  .<_  ( ( X  .\/  Y ) 
./\  Z ) ) )
291, 2, 22, 27, 28syl13anc 1187 . . 3  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X  .<_  Z )  ->  (
( X  .<_  ( ( X  .\/  Y ) 
./\  Z )  /\  ( Y  ./\  Z ) 
.<_  ( ( X  .\/  Y )  ./\  Z )
)  <->  ( X  .\/  ( Y  ./\  Z ) )  .<_  ( ( X  .\/  Y )  ./\  Z ) ) )
3016, 25, 29mpbi2and 889 . 2  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X  .<_  Z )  ->  ( X  .\/  ( Y  ./\  Z ) )  .<_  ( ( X  .\/  Y ) 
./\  Z ) )
3130ex 425 1  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  .<_  Z  ->  ( X  .\/  ( Y  ./\  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 4213   ` cfv 5455  (class class class)co 6082   Basecbs 13470   lecple 13537   joincjn 14402   meetcmee 14403   Latclat 14475
This theorem is referenced by:  mod2ile  14536  hlmod1i  30654
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 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
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 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-undef 6544  df-riota 6550  df-poset 14404  df-lub 14432  df-glb 14433  df-join 14434  df-meet 14435  df-lat 14476
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