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Theorem latmlem1 16271
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 16266 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Z  e.  B )  ->  ( X  ./\  Z
)  .<_  X )
543adant3r2 1215 . . . 4  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  ./\  Z )  .<_  X )
6 simpl 458 . . . . 5  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  K  e.  Lat )
71, 3latmcl 16242 . . . . . 6  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Z  e.  B )  ->  ( X  ./\  Z
)  e.  B )
873adant3r2 1215 . . . . 5  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  ./\  Z )  e.  B )
9 simpr1 1011 . . . . 5  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  X  e.  B )
10 simpr2 1012 . . . . 5  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Y  e.  B )
111, 2lattr 16246 . . . . 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 1266 . . . 4  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( ( X  ./\  Z )  .<_  X  /\  X  .<_  Y )  -> 
( X  ./\  Z
)  .<_  Y ) )
135, 12mpand 679 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  .<_  Y  ->  ( X  ./\  Z )  .<_  Y ) )
141, 2, 3latmle2 16267 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Z  e.  B )  ->  ( X  ./\  Z
)  .<_  Z )
15143adant3r2 1215 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  ./\  Z )  .<_  Z )
1613, 15jctird 546 . 2  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  .<_  Y  ->  (
( X  ./\  Z
)  .<_  Y  /\  ( X  ./\  Z )  .<_  Z ) ) )
17 simpr3 1013 . . . 4  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Z  e.  B )
188, 10, 173jca 1185 . . 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 16268 . . 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 472 . 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 217 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1867   class class class wbr 4417   ` cfv 5592  (class class class)co 6296   Basecbs 15073   lecple 15149   meetcmee 16134   Latclat 16235
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-poset 16135  df-lub 16164  df-glb 16165  df-join 16166  df-meet 16167  df-lat 16236
This theorem is referenced by:  latmlem2  16272  latmlem12  16273  dalem25  32972  dalawlem2  33146  dalawlem11  33155  dalawlem12  33156  cdleme22d  33619  cdleme30a  33654  cdleme32c  33719  cdleme32e  33721  trlcolem  34002  cdlemk5u  34137  cdlemk39  34192  cdlemm10N  34395  cdlemn2  34472  dihord1  34495
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