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Theorem latmlem1 16339
Description: Add meet to both sides of a lattice ordering. (Contributed by NM, 10-Nov-2011.)
Hypotheses
Ref Expression
latmle.b  |-  B  =  ( Base `  K
)
latmle.l  |-  .<_  =  ( le `  K )
latmle.m  |-  ./\  =  ( meet `  K )
Assertion
Ref Expression
latmlem1  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  .<_  Y  ->  ( X  ./\  Z )  .<_  ( Y  ./\  Z ) ) )

Proof of Theorem latmlem1
StepHypRef Expression
1 latmle.b . . . . . 6  |-  B  =  ( Base `  K
)
2 latmle.l . . . . . 6  |-  .<_  =  ( le `  K )
3 latmle.m . . . . . 6  |-  ./\  =  ( meet `  K )
41, 2, 3latmle1 16334 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Z  e.  B )  ->  ( X  ./\  Z
)  .<_  X )
543adant3r2 1219 . . . 4  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  ./\  Z )  .<_  X )
6 simpl 459 . . . . 5  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  K  e.  Lat )
71, 3latmcl 16310 . . . . . 6  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Z  e.  B )  ->  ( X  ./\  Z
)  e.  B )
873adant3r2 1219 . . . . 5  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  ./\  Z )  e.  B )
9 simpr1 1015 . . . . 5  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  X  e.  B )
10 simpr2 1016 . . . . 5  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Y  e.  B )
111, 2lattr 16314 . . . . 5  |-  ( ( K  e.  Lat  /\  ( ( X  ./\  Z )  e.  B  /\  X  e.  B  /\  Y  e.  B )
)  ->  ( (
( X  ./\  Z
)  .<_  X  /\  X  .<_  Y )  ->  ( X  ./\  Z )  .<_  Y ) )
126, 8, 9, 10, 11syl13anc 1271 . . . 4  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( ( X  ./\  Z )  .<_  X  /\  X  .<_  Y )  -> 
( X  ./\  Z
)  .<_  Y ) )
135, 12mpand 682 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  .<_  Y  ->  ( X  ./\  Z )  .<_  Y ) )
141, 2, 3latmle2 16335 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Z  e.  B )  ->  ( X  ./\  Z
)  .<_  Z )
15143adant3r2 1219 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  ./\  Z )  .<_  Z )
1613, 15jctird 547 . 2  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  .<_  Y  ->  (
( X  ./\  Z
)  .<_  Y  /\  ( X  ./\  Z )  .<_  Z ) ) )
17 simpr3 1017 . . . 4  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Z  e.  B )
188, 10, 173jca 1189 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  ./\  Z
)  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)
191, 2, 3latlem12 16336 . . 3  |-  ( ( K  e.  Lat  /\  ( ( X  ./\  Z )  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( (
( X  ./\  Z
)  .<_  Y  /\  ( X  ./\  Z )  .<_  Z )  <->  ( X  ./\ 
Z )  .<_  ( Y 
./\  Z ) ) )
2018, 19syldan 473 . 2  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( ( X  ./\  Z )  .<_  Y  /\  ( X  ./\  Z ) 
.<_  Z )  <->  ( X  ./\ 
Z )  .<_  ( Y 
./\  Z ) ) )
2116, 20sylibd 218 1  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  .<_  Y  ->  ( X  ./\  Z )  .<_  ( Y  ./\  Z ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 986    = wceq 1446    e. wcel 1889   class class class wbr 4405   ` cfv 5585  (class class class)co 6295   Basecbs 15133   lecple 15209   meetcmee 16202   Latclat 16303
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-reu 2746  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-id 4752  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-oprab 6299  df-poset 16203  df-lub 16232  df-glb 16233  df-join 16234  df-meet 16235  df-lat 16304
This theorem is referenced by:  latmlem2  16340  latmlem12  16341  dalem25  33275  dalawlem2  33449  dalawlem11  33458  dalawlem12  33459  cdleme22d  33922  cdleme30a  33957  cdleme32c  34022  cdleme32e  34024  trlcolem  34305  cdlemk5u  34440  cdlemk39  34495  cdlemm10N  34698  cdlemn2  34775  dihord1  34798
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