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Theorem mod1ile 14211
Description: The weak direction of the modular law (e.g. pmod1i 30037, atmod1i1 30046) 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 730 . . . . 5  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X  .<_  Z )  ->  K  e.  Lat )
2 simplr1 997 . . . . 5  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X  .<_  Z )  ->  X  e.  B )
3 simplr2 998 . . . . 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 14166 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  X  .<_  ( X  .\/  Y ) )
81, 2, 3, 7syl3anc 1182 . . . 4  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X  .<_  Z )  ->  X  .<_  ( X  .\/  Y
) )
9 simpr 447 . . . 4  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X  .<_  Z )  ->  X  .<_  Z )
104, 6latjcl 14156 . . . . . 6  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .\/  Y
)  e.  B )
111, 2, 3, 10syl3anc 1182 . . . . 5  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X  .<_  Z )  ->  ( X  .\/  Y )  e.  B )
12 simplr3 999 . . . . 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 14184 . . . . 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 1184 . . . 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 887 . . 3  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X  .<_  Z )  ->  X  .<_  ( ( X  .\/  Y )  ./\  Z )
)
174, 5, 6, 13latmlej12 14197 . . . . 5  |-  ( ( K  e.  Lat  /\  ( Y  e.  B  /\  Z  e.  B  /\  X  e.  B
) )  ->  ( Y  ./\  Z )  .<_  ( X  .\/  Y ) )
181, 3, 12, 2, 17syl13anc 1184 . . . 4  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X  .<_  Z )  ->  ( Y  ./\  Z )  .<_  ( X  .\/  Y ) )
194, 5, 13latmle2 14183 . . . . 5  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Y  ./\  Z
)  .<_  Z )
201, 3, 12, 19syl3anc 1182 . . . 4  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X  .<_  Z )  ->  ( Y  ./\  Z )  .<_  Z )
214, 13latmcl 14157 . . . . . 6  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Y  ./\  Z
)  e.  B )
221, 3, 12, 21syl3anc 1182 . . . . 5  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X  .<_  Z )  ->  ( Y  ./\  Z )  e.  B )
234, 5, 13latlem12 14184 . . . . 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 1184 . . . 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 887 . . 3  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X  .<_  Z )  ->  ( Y  ./\  Z )  .<_  ( ( X  .\/  Y )  ./\  Z )
)
264, 13latmcl 14157 . . . . 5  |-  ( ( K  e.  Lat  /\  ( X  .\/  Y )  e.  B  /\  Z  e.  B )  ->  (
( X  .\/  Y
)  ./\  Z )  e.  B )
271, 11, 12, 26syl3anc 1182 . . . 4  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X  .<_  Z )  ->  (
( X  .\/  Y
)  ./\  Z )  e.  B )
284, 5, 6latjle12 14168 . . . 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 1184 . . 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 887 . 2  |-  ( ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X  .<_  Z )  ->  ( X  .\/  ( Y  ./\  Z ) )  .<_  ( ( X  .\/  Y ) 
./\  Z ) )
3130ex 423 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 set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   joincjn 14078   meetcmee 14079   Latclat 14151
This theorem is referenced by:  mod2ile  14212  hlmod1i  30045
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-lat 14152
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