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Theorem mod2ile 15378
Description: The weak direction of the modular law (e.g. pmod2iN 33799) 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 753 . . . . 5  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  K  e.  Lat )
2 simplr3 1032 . . . . . 6  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  Z  e.  B )
3 simplr2 1031 . . . . . 6  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  Y  e.  B )
4 simplr1 1030 . . . . . 6  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  X  e.  B )
52, 3, 43jca 1168 . . . . 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 532 . . . 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 461 . . . 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 15377 . . . 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 15347 . . . . . 6  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  =  ( Y 
./\  X ) )
151, 4, 3, 14syl3anc 1219 . . . . 5  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  ( X  ./\  Y )  =  ( Y  ./\  X
) )
1615oveq1d 6205 . . . 4  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  (
( X  ./\  Y
)  .\/  Z )  =  ( ( Y 
./\  X )  .\/  Z ) )
178, 11latmcl 15324 . . . . . 6  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  X  e.  B )  ->  ( Y  ./\  X
)  e.  B )
181, 3, 4, 17syl3anc 1219 . . . . 5  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  ( Y  ./\  X )  e.  B )
198, 10latjcom 15331 . . . . 5  |-  ( ( K  e.  Lat  /\  ( Y  ./\  X )  e.  B  /\  Z  e.  B )  ->  (
( Y  ./\  X
)  .\/  Z )  =  ( Z  .\/  ( Y  ./\  X ) ) )
201, 18, 2, 19syl3anc 1219 . . . 4  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  (
( Y  ./\  X
)  .\/  Z )  =  ( Z  .\/  ( Y  ./\  X ) ) )
2116, 20eqtrd 2492 . . 3  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  (
( X  ./\  Y
)  .\/  Z )  =  ( Z  .\/  ( Y  ./\  X ) ) )
228, 10latjcom 15331 . . . . . 6  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Y  .\/  Z
)  =  ( Z 
.\/  Y ) )
231, 3, 2, 22syl3anc 1219 . . . . 5  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  ( Y  .\/  Z )  =  ( Z  .\/  Y
) )
2423oveq2d 6206 . . . 4  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  ( X  ./\  ( Y  .\/  Z ) )  =  ( X  ./\  ( Z  .\/  Y ) ) )
258, 10latjcl 15323 . . . . . 6  |-  ( ( K  e.  Lat  /\  Z  e.  B  /\  Y  e.  B )  ->  ( Z  .\/  Y
)  e.  B )
261, 2, 3, 25syl3anc 1219 . . . . 5  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  ( Z  .\/  Y )  e.  B )
278, 11latmcom 15347 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  ( Z  .\/  Y )  e.  B )  -> 
( X  ./\  ( Z  .\/  Y ) )  =  ( ( Z 
.\/  Y )  ./\  X ) )
281, 4, 26, 27syl3anc 1219 . . . 4  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  ( X  ./\  ( Z  .\/  Y ) )  =  ( ( Z  .\/  Y
)  ./\  X )
)
2924, 28eqtrd 2492 . . 3  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  ( X  ./\  ( Y  .\/  Z ) )  =  ( ( Z  .\/  Y
)  ./\  X )
)
3013, 21, 293brtr4d 4420 . 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 965    = wceq 1370    e. wcel 1758   class class class wbr 4390   ` cfv 5516  (class class class)co 6190   Basecbs 14276   lecple 14347   joincjn 15216   meetcmee 15217   Latclat 15317
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 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472
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 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-id 4734  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-poset 15218  df-lub 15246  df-glb 15247  df-join 15248  df-meet 15249  df-lat 15318
This theorem is referenced by: (None)
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