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Theorem mod2ile 15863
Description: The weak direction of the modular law (e.g. pmod2iN 35716) 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 1040 . . . . . 6  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  Z  e.  B )
3 simplr2 1039 . . . . . 6  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  Y  e.  B )
4 simplr1 1038 . . . . . 6  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  X  e.  B )
52, 3, 43jca 1176 . . . . 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 15862 . . . 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 15832 . . . . . 6  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  =  ( Y 
./\  X ) )
151, 4, 3, 14syl3anc 1228 . . . . 5  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  ( X  ./\  Y )  =  ( Y  ./\  X
) )
1615oveq1d 6311 . . . 4  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  (
( X  ./\  Y
)  .\/  Z )  =  ( ( Y 
./\  X )  .\/  Z ) )
178, 11latmcl 15809 . . . . . 6  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  X  e.  B )  ->  ( Y  ./\  X
)  e.  B )
181, 3, 4, 17syl3anc 1228 . . . . 5  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  ( Y  ./\  X )  e.  B )
198, 10latjcom 15816 . . . . 5  |-  ( ( K  e.  Lat  /\  ( Y  ./\  X )  e.  B  /\  Z  e.  B )  ->  (
( Y  ./\  X
)  .\/  Z )  =  ( Z  .\/  ( Y  ./\  X ) ) )
201, 18, 2, 19syl3anc 1228 . . . 4  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  (
( Y  ./\  X
)  .\/  Z )  =  ( Z  .\/  ( Y  ./\  X ) ) )
2116, 20eqtrd 2498 . . 3  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  (
( X  ./\  Y
)  .\/  Z )  =  ( Z  .\/  ( Y  ./\  X ) ) )
228, 10latjcom 15816 . . . . . 6  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Y  .\/  Z
)  =  ( Z 
.\/  Y ) )
231, 3, 2, 22syl3anc 1228 . . . . 5  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  ( Y  .\/  Z )  =  ( Z  .\/  Y
) )
2423oveq2d 6312 . . . 4  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  ( X  ./\  ( Y  .\/  Z ) )  =  ( X  ./\  ( Z  .\/  Y ) ) )
258, 10latjcl 15808 . . . . . 6  |-  ( ( K  e.  Lat  /\  Z  e.  B  /\  Y  e.  B )  ->  ( Z  .\/  Y
)  e.  B )
261, 2, 3, 25syl3anc 1228 . . . . 5  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  ( Z  .\/  Y )  e.  B )
278, 11latmcom 15832 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  ( Z  .\/  Y )  e.  B )  -> 
( X  ./\  ( Z  .\/  Y ) )  =  ( ( Z 
.\/  Y )  ./\  X ) )
281, 4, 26, 27syl3anc 1228 . . . 4  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  ( X  ./\  ( Z  .\/  Y ) )  =  ( ( Z  .\/  Y
)  ./\  X )
)
2924, 28eqtrd 2498 . . 3  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  Z  .<_  X )  ->  ( X  ./\  ( Y  .\/  Z ) )  =  ( ( Z  .\/  Y
)  ./\  X )
)
3013, 21, 293brtr4d 4486 . 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 973    = wceq 1395    e. wcel 1819   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   Basecbs 14644   lecple 14719   joincjn 15700   meetcmee 15701   Latclat 15802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-poset 15702  df-lub 15731  df-glb 15732  df-join 15733  df-meet 15734  df-lat 15803
This theorem is referenced by: (None)
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