MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mod1ile Structured version   Unicode version

Theorem mod1ile 15604
Description: The weak direction of the modular law (e.g. pmod1i 35274, atmod1i1 35283) 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 1037 . . . . 5  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X  .<_  Z )  ->  X  e.  B )
3 simplr2 1038 . . . . 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 15559 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  X  .<_  ( X  .\/  Y ) )
81, 2, 3, 7syl3anc 1227 . . . 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 15550 . . . . . 6  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .\/  Y
)  e.  B )
111, 2, 3, 10syl3anc 1227 . . . . 5  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X  .<_  Z )  ->  ( X  .\/  Y )  e.  B )
12 simplr3 1039 . . . . 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 15577 . . . . 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 1229 . . . 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 919 . . 3  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X  .<_  Z )  ->  X  .<_  ( ( X  .\/  Y )  ./\  Z )
)
174, 5, 6, 13latmlej12 15590 . . . . 5  |-  ( ( K  e.  Lat  /\  ( Y  e.  B  /\  Z  e.  B  /\  X  e.  B
) )  ->  ( Y  ./\  Z )  .<_  ( X  .\/  Y ) )
181, 3, 12, 2, 17syl13anc 1229 . . . 4  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X  .<_  Z )  ->  ( Y  ./\  Z )  .<_  ( X  .\/  Y ) )
194, 5, 13latmle2 15576 . . . . 5  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Y  ./\  Z
)  .<_  Z )
201, 3, 12, 19syl3anc 1227 . . . 4  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X  .<_  Z )  ->  ( Y  ./\  Z )  .<_  Z )
214, 13latmcl 15551 . . . . . 6  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Y  ./\  Z
)  e.  B )
221, 3, 12, 21syl3anc 1227 . . . . 5  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X  .<_  Z )  ->  ( Y  ./\  Z )  e.  B )
234, 5, 13latlem12 15577 . . . . 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 1229 . . . 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 919 . . 3  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X  .<_  Z )  ->  ( Y  ./\  Z )  .<_  ( ( X  .\/  Y )  ./\  Z )
)
264, 13latmcl 15551 . . . . 5  |-  ( ( K  e.  Lat  /\  ( X  .\/  Y )  e.  B  /\  Z  e.  B )  ->  (
( X  .\/  Y
)  ./\  Z )  e.  B )
271, 11, 12, 26syl3anc 1227 . . . 4  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X  .<_  Z )  ->  (
( X  .\/  Y
)  ./\  Z )  e.  B )
284, 5, 6latjle12 15561 . . . 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 1229 . . 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 919 . 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 972    = wceq 1381    e. wcel 1802   class class class wbr 4433   ` cfv 5574  (class class class)co 6277   Basecbs 14504   lecple 14576   joincjn 15442   meetcmee 15443   Latclat 15544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-poset 15444  df-lub 15473  df-glb 15474  df-join 15475  df-meet 15476  df-lat 15545
This theorem is referenced by:  mod2ile  15605  hlmod1i  35282
  Copyright terms: Public domain W3C validator