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Theorem latmlem1 15263
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 15258 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Z  e.  B )  ->  ( X  ./\  Z
)  .<_  X )
543adant3r2 1197 . . . 4  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  ./\  Z )  .<_  X )
6 simpl 457 . . . . 5  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  K  e.  Lat )
71, 3latmcl 15234 . . . . . 6  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Z  e.  B )  ->  ( X  ./\  Z
)  e.  B )
873adant3r2 1197 . . . . 5  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  ./\  Z )  e.  B )
9 simpr1 994 . . . . 5  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  X  e.  B )
10 simpr2 995 . . . . 5  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Y  e.  B )
111, 2lattr 15238 . . . . 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 1220 . . . 4  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( ( X  ./\  Z )  .<_  X  /\  X  .<_  Y )  -> 
( X  ./\  Z
)  .<_  Y ) )
135, 12mpand 675 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  .<_  Y  ->  ( X  ./\  Z )  .<_  Y ) )
141, 2, 3latmle2 15259 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Z  e.  B )  ->  ( X  ./\  Z
)  .<_  Z )
15143adant3r2 1197 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  ./\  Z )  .<_  Z )
1613, 15jctird 544 . 2  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  .<_  Y  ->  (
( X  ./\  Z
)  .<_  Y  /\  ( X  ./\  Z )  .<_  Z ) ) )
17 simpr3 996 . . . 4  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Z  e.  B )
188, 10, 173jca 1168 . . 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 15260 . . 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 470 . 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 214 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 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   class class class wbr 4304   ` cfv 5430  (class class class)co 6103   Basecbs 14186   lecple 14257   meetcmee 15127   Latclat 15227
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-reu 2734  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-poset 15128  df-lub 15156  df-glb 15157  df-join 15158  df-meet 15159  df-lat 15228
This theorem is referenced by:  latmlem2  15264  latmlem12  15265  dalem25  33354  dalawlem2  33528  dalawlem11  33537  dalawlem12  33538  cdleme22d  33999  cdleme30a  34034  cdleme32c  34099  cdleme32e  34101  trlcolem  34382  cdlemk5u  34517  cdlemk39  34572  cdlemm10N  34775  cdlemn2  34852  dihord1  34875
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