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Theorem mod2ile 14228
Description: The weak direction of the modular law (e.g. pmod2iN 30660) 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 730 . . . . 5  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  K  e.  Lat )
2 simplr3 999 . . . . . 6  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  Z  e.  B )
3 simplr2 998 . . . . . 6  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  Y  e.  B )
4 simplr1 997 . . . . . 6  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  X  e.  B )
52, 3, 43jca 1132 . . . . 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 518 . . . 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 447 . . . 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 14227 . . . 4  |-  ( ( K  e.  Lat  /\  ( Z  e.  B  /\  Y  e.  B  /\  X  e.  B
) )  ->  ( Z  .<_  X  ->  ( Z  .\/  ( Y  ./\  X ) )  .<_  ( ( Z  .\/  Y ) 
./\  X ) ) )
136, 7, 12sylc 56 . . 3  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  ( Z  .\/  ( Y  ./\  X ) )  .<_  ( ( Z  .\/  Y ) 
./\  X ) )
148, 11latmcom 14197 . . . . . 6  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  =  ( Y 
./\  X ) )
151, 4, 3, 14syl3anc 1182 . . . . 5  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  ( X  ./\  Y )  =  ( Y  ./\  X
) )
1615oveq1d 5889 . . . 4  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  (
( X  ./\  Y
)  .\/  Z )  =  ( ( Y 
./\  X )  .\/  Z ) )
178, 11latmcl 14173 . . . . . 6  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  X  e.  B )  ->  ( Y  ./\  X
)  e.  B )
181, 3, 4, 17syl3anc 1182 . . . . 5  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  ( Y  ./\  X )  e.  B )
198, 10latjcom 14181 . . . . 5  |-  ( ( K  e.  Lat  /\  ( Y  ./\  X )  e.  B  /\  Z  e.  B )  ->  (
( Y  ./\  X
)  .\/  Z )  =  ( Z  .\/  ( Y  ./\  X ) ) )
201, 18, 2, 19syl3anc 1182 . . . 4  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  (
( Y  ./\  X
)  .\/  Z )  =  ( Z  .\/  ( Y  ./\  X ) ) )
2116, 20eqtrd 2328 . . 3  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  (
( X  ./\  Y
)  .\/  Z )  =  ( Z  .\/  ( Y  ./\  X ) ) )
228, 10latjcom 14181 . . . . . 6  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Y  .\/  Z
)  =  ( Z 
.\/  Y ) )
231, 3, 2, 22syl3anc 1182 . . . . 5  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  ( Y  .\/  Z )  =  ( Z  .\/  Y
) )
2423oveq2d 5890 . . . 4  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  ( X  ./\  ( Y  .\/  Z ) )  =  ( X  ./\  ( Z  .\/  Y ) ) )
258, 10latjcl 14172 . . . . . 6  |-  ( ( K  e.  Lat  /\  Z  e.  B  /\  Y  e.  B )  ->  ( Z  .\/  Y
)  e.  B )
261, 2, 3, 25syl3anc 1182 . . . . 5  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  ( Z  .\/  Y )  e.  B )
278, 11latmcom 14197 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  ( Z  .\/  Y )  e.  B )  -> 
( X  ./\  ( Z  .\/  Y ) )  =  ( ( Z 
.\/  Y )  ./\  X ) )
281, 4, 26, 27syl3anc 1182 . . . 4  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  ( X  ./\  ( Z  .\/  Y ) )  =  ( ( Z  .\/  Y
)  ./\  X )
)
2924, 28eqtrd 2328 . . 3  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  ( X  ./\  ( Y  .\/  Z ) )  =  ( ( Z  .\/  Y
)  ./\  X )
)
3013, 21, 293brtr4d 4069 . 2  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  (
( X  ./\  Y
)  .\/  Z )  .<_  ( X  ./\  ( Y  .\/  Z ) ) )
3130ex 423 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 358    /\ w3a 934    = wceq 1632    e. wcel 1696   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   joincjn 14094   meetcmee 14095   Latclat 14167
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-lat 14168
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