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Theorem mod1ile 15374
Description: The weak direction of the modular law (e.g. pmod1i 33795, atmod1i1 33804) 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 753 . . . . 5  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X  .<_  Z )  ->  K  e.  Lat )
2 simplr1 1030 . . . . 5  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X  .<_  Z )  ->  X  e.  B )
3 simplr2 1031 . . . . 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 15329 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  X  .<_  ( X  .\/  Y ) )
81, 2, 3, 7syl3anc 1219 . . . 4  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X  .<_  Z )  ->  X  .<_  ( X  .\/  Y
) )
9 simpr 461 . . . 4  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X  .<_  Z )  ->  X  .<_  Z )
104, 6latjcl 15320 . . . . . 6  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .\/  Y
)  e.  B )
111, 2, 3, 10syl3anc 1219 . . . . 5  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X  .<_  Z )  ->  ( X  .\/  Y )  e.  B )
12 simplr3 1032 . . . . 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 15347 . . . . 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 1221 . . . 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 912 . . 3  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X  .<_  Z )  ->  X  .<_  ( ( X  .\/  Y )  ./\  Z )
)
174, 5, 6, 13latmlej12 15360 . . . . 5  |-  ( ( K  e.  Lat  /\  ( Y  e.  B  /\  Z  e.  B  /\  X  e.  B
) )  ->  ( Y  ./\  Z )  .<_  ( X  .\/  Y ) )
181, 3, 12, 2, 17syl13anc 1221 . . . 4  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X  .<_  Z )  ->  ( Y  ./\  Z )  .<_  ( X  .\/  Y ) )
194, 5, 13latmle2 15346 . . . . 5  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Y  ./\  Z
)  .<_  Z )
201, 3, 12, 19syl3anc 1219 . . . 4  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X  .<_  Z )  ->  ( Y  ./\  Z )  .<_  Z )
214, 13latmcl 15321 . . . . . 6  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Y  ./\  Z
)  e.  B )
221, 3, 12, 21syl3anc 1219 . . . . 5  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X  .<_  Z )  ->  ( Y  ./\  Z )  e.  B )
234, 5, 13latlem12 15347 . . . . 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 1221 . . . 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 912 . . 3  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X  .<_  Z )  ->  ( Y  ./\  Z )  .<_  ( ( X  .\/  Y )  ./\  Z )
)
264, 13latmcl 15321 . . . . 5  |-  ( ( K  e.  Lat  /\  ( X  .\/  Y )  e.  B  /\  Z  e.  B )  ->  (
( X  .\/  Y
)  ./\  Z )  e.  B )
271, 11, 12, 26syl3anc 1219 . . . 4  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X  .<_  Z )  ->  (
( X  .\/  Y
)  ./\  Z )  e.  B )
284, 5, 6latjle12 15331 . . . 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 1221 . . 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 912 . 2  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X  .<_  Z )  ->  ( X  .\/  ( Y  ./\  Z ) )  .<_  ( ( X  .\/  Y ) 
./\  Z ) )
3130ex 434 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   class class class wbr 4387   ` cfv 5513  (class class class)co 6187   Basecbs 14273   lecple 14344   joincjn 15213   meetcmee 15214   Latclat 15314
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4498  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4187  df-iun 4268  df-br 4388  df-opab 4446  df-mpt 4447  df-id 4731  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-riota 6148  df-ov 6190  df-oprab 6191  df-poset 15215  df-lub 15243  df-glb 15244  df-join 15245  df-meet 15246  df-lat 15315
This theorem is referenced by:  mod2ile  15375  hlmod1i  33803
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