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Theorem latmlem1 14512
Description: Add meet to both sides of a lattice ordering. (ssrin 3568 analog.) (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 14507 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Z  e.  B )  ->  ( X  ./\  Z
)  .<_  X )
543adant3r2 1164 . . . 4  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  ./\  Z )  .<_  X )
6 simpl 445 . . . . 5  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  K  e.  Lat )
71, 3latmcl 14482 . . . . . 6  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Z  e.  B )  ->  ( X  ./\  Z
)  e.  B )
873adant3r2 1164 . . . . 5  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  ./\  Z )  e.  B )
9 simpr1 964 . . . . 5  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  X  e.  B )
10 simpr2 965 . . . . 5  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Y  e.  B )
111, 2lattr 14487 . . . . 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 1187 . . . 4  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( ( X  ./\  Z )  .<_  X  /\  X  .<_  Y )  -> 
( X  ./\  Z
)  .<_  Y ) )
135, 12mpand 658 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  .<_  Y  ->  ( X  ./\  Z )  .<_  Y ) )
141, 2, 3latmle2 14508 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Z  e.  B )  ->  ( X  ./\  Z
)  .<_  Z )
15143adant3r2 1164 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  ./\  Z )  .<_  Z )
1613, 15jctird 530 . 2  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  .<_  Y  ->  (
( X  ./\  Z
)  .<_  Y  /\  ( X  ./\  Z )  .<_  Z ) ) )
17 simpr3 966 . . . 4  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Z  e.  B )
188, 10, 173jca 1135 . . 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 14509 . . 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 458 . 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 207 1  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  .<_  Y  ->  ( X  ./\  Z )  .<_  ( Y  ./\  Z ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   class class class wbr 4214   ` cfv 5456  (class class class)co 6083   Basecbs 13471   lecple 13538   meetcmee 14404   Latclat 14476
This theorem is referenced by:  latmlem2  14513  latmlem12  14514  dalem25  30557  dalawlem2  30731  dalawlem11  30740  dalawlem12  30741  cdleme22d  31202  cdleme30a  31237  cdleme32c  31302  cdleme32e  31304  trlcolem  31585  cdlemk5u  31720  cdlemk39  31775  cdlemm10N  31978  cdlemn2  32055  dihord1  32078
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-undef 6545  df-riota 6551  df-poset 14405  df-glb 14434  df-meet 14436  df-lat 14477
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