MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mod2ile Structured version   Unicode version

Theorem mod2ile 14535
Description: The weak direction of the modular law (e.g. pmod2iN 30646) 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
mod2ile  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( Z  .<_  X  ->  (
( X  ./\  Y
)  .\/  Z )  .<_  ( X  ./\  ( Y  .\/  Z ) ) ) )

Proof of Theorem mod2ile
StepHypRef Expression
1 simpll 731 . . . . 5  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  K  e.  Lat )
2 simplr3 1001 . . . . . 6  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  Z  e.  B )
3 simplr2 1000 . . . . . 6  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  Y  e.  B )
4 simplr1 999 . . . . . 6  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  X  e.  B )
52, 3, 43jca 1134 . . . . 5  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  ( Z  e.  B  /\  Y  e.  B  /\  X  e.  B )
)
61, 5jca 519 . . . 4  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  ( K  e.  Lat  /\  ( Z  e.  B  /\  Y  e.  B  /\  X  e.  B )
) )
7 simpr 448 . . . 4  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  Z  .<_  X )
8 modle.b . . . . 5  |-  B  =  ( Base `  K
)
9 modle.l . . . . 5  |-  .<_  =  ( le `  K )
10 modle.j . . . . 5  |-  .\/  =  ( join `  K )
11 modle.m . . . . 5  |-  ./\  =  ( meet `  K )
128, 9, 10, 11mod1ile 14534 . . . 4  |-  ( ( K  e.  Lat  /\  ( Z  e.  B  /\  Y  e.  B  /\  X  e.  B
) )  ->  ( Z  .<_  X  ->  ( Z  .\/  ( Y  ./\  X ) )  .<_  ( ( Z  .\/  Y ) 
./\  X ) ) )
136, 7, 12sylc 58 . . 3  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  ( Z  .\/  ( Y  ./\  X ) )  .<_  ( ( Z  .\/  Y ) 
./\  X ) )
148, 11latmcom 14504 . . . . . 6  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  =  ( Y 
./\  X ) )
151, 4, 3, 14syl3anc 1184 . . . . 5  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  ( X  ./\  Y )  =  ( Y  ./\  X
) )
1615oveq1d 6096 . . . 4  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  (
( X  ./\  Y
)  .\/  Z )  =  ( ( Y 
./\  X )  .\/  Z ) )
178, 11latmcl 14480 . . . . . 6  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  X  e.  B )  ->  ( Y  ./\  X
)  e.  B )
181, 3, 4, 17syl3anc 1184 . . . . 5  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  ( Y  ./\  X )  e.  B )
198, 10latjcom 14488 . . . . 5  |-  ( ( K  e.  Lat  /\  ( Y  ./\  X )  e.  B  /\  Z  e.  B )  ->  (
( Y  ./\  X
)  .\/  Z )  =  ( Z  .\/  ( Y  ./\  X ) ) )
201, 18, 2, 19syl3anc 1184 . . . 4  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  (
( Y  ./\  X
)  .\/  Z )  =  ( Z  .\/  ( Y  ./\  X ) ) )
2116, 20eqtrd 2468 . . 3  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  (
( X  ./\  Y
)  .\/  Z )  =  ( Z  .\/  ( Y  ./\  X ) ) )
228, 10latjcom 14488 . . . . . 6  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Y  .\/  Z
)  =  ( Z 
.\/  Y ) )
231, 3, 2, 22syl3anc 1184 . . . . 5  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  ( Y  .\/  Z )  =  ( Z  .\/  Y
) )
2423oveq2d 6097 . . . 4  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  ( X  ./\  ( Y  .\/  Z ) )  =  ( X  ./\  ( Z  .\/  Y ) ) )
258, 10latjcl 14479 . . . . . 6  |-  ( ( K  e.  Lat  /\  Z  e.  B  /\  Y  e.  B )  ->  ( Z  .\/  Y
)  e.  B )
261, 2, 3, 25syl3anc 1184 . . . . 5  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  ( Z  .\/  Y )  e.  B )
278, 11latmcom 14504 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  ( Z  .\/  Y )  e.  B )  -> 
( X  ./\  ( Z  .\/  Y ) )  =  ( ( Z 
.\/  Y )  ./\  X ) )
281, 4, 26, 27syl3anc 1184 . . . 4  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  ( X  ./\  ( Z  .\/  Y ) )  =  ( ( Z  .\/  Y
)  ./\  X )
)
2924, 28eqtrd 2468 . . 3  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  ( X  ./\  ( Y  .\/  Z ) )  =  ( ( Z  .\/  Y
)  ./\  X )
)
3013, 21, 293brtr4d 4242 . 2  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  (
( X  ./\  Y
)  .\/  Z )  .<_  ( X  ./\  ( Y  .\/  Z ) ) )
3130ex 424 1  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( Z  .<_  X  ->  (
( X  ./\  Y
)  .\/  Z )  .<_  ( X  ./\  ( Y  .\/  Z ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   Basecbs 13469   lecple 13536   joincjn 14401   meetcmee 14402   Latclat 14474
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-undef 6543  df-riota 6549  df-poset 14403  df-lub 14431  df-glb 14432  df-join 14433  df-meet 14434  df-lat 14475
  Copyright terms: Public domain W3C validator