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

Theorem mod2ile 15272
Description: The weak direction of the modular law (e.g. pmod2iN 33215) 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 748 . . . . 5  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  K  e.  Lat )
2 simplr3 1027 . . . . . 6  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  Z  e.  B )
3 simplr2 1026 . . . . . 6  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  Y  e.  B )
4 simplr1 1025 . . . . . 6  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  X  e.  B )
52, 3, 43jca 1163 . . . . 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 529 . . . 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 458 . . . 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 15271 . . . 4  |-  ( ( K  e.  Lat  /\  ( Z  e.  B  /\  Y  e.  B  /\  X  e.  B
) )  ->  ( Z  .<_  X  ->  ( Z  .\/  ( Y  ./\  X ) )  .<_  ( ( Z  .\/  Y ) 
./\  X ) ) )
136, 7, 12sylc 60 . . 3  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  ( Z  .\/  ( Y  ./\  X ) )  .<_  ( ( Z  .\/  Y ) 
./\  X ) )
148, 11latmcom 15241 . . . . . 6  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  =  ( Y 
./\  X ) )
151, 4, 3, 14syl3anc 1213 . . . . 5  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  ( X  ./\  Y )  =  ( Y  ./\  X
) )
1615oveq1d 6105 . . . 4  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  (
( X  ./\  Y
)  .\/  Z )  =  ( ( Y 
./\  X )  .\/  Z ) )
178, 11latmcl 15218 . . . . . 6  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  X  e.  B )  ->  ( Y  ./\  X
)  e.  B )
181, 3, 4, 17syl3anc 1213 . . . . 5  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  ( Y  ./\  X )  e.  B )
198, 10latjcom 15225 . . . . 5  |-  ( ( K  e.  Lat  /\  ( Y  ./\  X )  e.  B  /\  Z  e.  B )  ->  (
( Y  ./\  X
)  .\/  Z )  =  ( Z  .\/  ( Y  ./\  X ) ) )
201, 18, 2, 19syl3anc 1213 . . . 4  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  (
( Y  ./\  X
)  .\/  Z )  =  ( Z  .\/  ( Y  ./\  X ) ) )
2116, 20eqtrd 2473 . . 3  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  (
( X  ./\  Y
)  .\/  Z )  =  ( Z  .\/  ( Y  ./\  X ) ) )
228, 10latjcom 15225 . . . . . 6  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Y  .\/  Z
)  =  ( Z 
.\/  Y ) )
231, 3, 2, 22syl3anc 1213 . . . . 5  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  ( Y  .\/  Z )  =  ( Z  .\/  Y
) )
2423oveq2d 6106 . . . 4  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  ( X  ./\  ( Y  .\/  Z ) )  =  ( X  ./\  ( Z  .\/  Y ) ) )
258, 10latjcl 15217 . . . . . 6  |-  ( ( K  e.  Lat  /\  Z  e.  B  /\  Y  e.  B )  ->  ( Z  .\/  Y
)  e.  B )
261, 2, 3, 25syl3anc 1213 . . . . 5  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  ( Z  .\/  Y )  e.  B )
278, 11latmcom 15241 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  ( Z  .\/  Y )  e.  B )  -> 
( X  ./\  ( Z  .\/  Y ) )  =  ( ( Z 
.\/  Y )  ./\  X ) )
281, 4, 26, 27syl3anc 1213 . . . 4  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  ( X  ./\  ( Z  .\/  Y ) )  =  ( ( Z  .\/  Y
)  ./\  X )
)
2924, 28eqtrd 2473 . . 3  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  ( X  ./\  ( Y  .\/  Z ) )  =  ( ( Z  .\/  Y
)  ./\  X )
)
3013, 21, 293brtr4d 4319 . 2  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  (
( X  ./\  Y
)  .\/  Z )  .<_  ( X  ./\  ( Y  .\/  Z ) ) )
3130ex 434 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761   class class class wbr 4289   ` cfv 5415  (class class class)co 6090   Basecbs 14170   lecple 14241   joincjn 15110   meetcmee 15111   Latclat 15211
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-poset 15112  df-lub 15140  df-glb 15141  df-join 15142  df-meet 15143  df-lat 15212
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator