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Theorem latmlem1 14187
Description: Add meet to both sides of a lattice ordering. (ssrin 3394 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 14182 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Z  e.  B )  ->  ( X  ./\  Z
)  .<_  X )
543adant3r2 1161 . . . 4  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  ./\  Z )  .<_  X )
6 simpl 443 . . . . 5  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  K  e.  Lat )
71, 3latmcl 14157 . . . . . 6  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Z  e.  B )  ->  ( X  ./\  Z
)  e.  B )
873adant3r2 1161 . . . . 5  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  ./\  Z )  e.  B )
9 simpr1 961 . . . . 5  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  X  e.  B )
10 simpr2 962 . . . . 5  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Y  e.  B )
111, 2lattr 14162 . . . . 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 1184 . . . 4  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( ( X  ./\  Z )  .<_  X  /\  X  .<_  Y )  -> 
( X  ./\  Z
)  .<_  Y ) )
135, 12mpand 656 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  .<_  Y  ->  ( X  ./\  Z )  .<_  Y ) )
141, 2, 3latmle2 14183 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Z  e.  B )  ->  ( X  ./\  Z
)  .<_  Z )
15143adant3r2 1161 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  ./\  Z )  .<_  Z )
1613, 15jctird 528 . 2  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  .<_  Y  ->  (
( X  ./\  Z
)  .<_  Y  /\  ( X  ./\  Z )  .<_  Z ) ) )
17 simpr3 963 . . . 4  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Z  e.  B )
188, 10, 173jca 1132 . . 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 14184 . . 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 456 . 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 205 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 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   meetcmee 14079   Latclat 14151
This theorem is referenced by:  latmlem2  14188  latmlem12  14189  dalem25  29887  dalawlem2  30061  dalawlem11  30070  dalawlem12  30071  cdleme22d  30532  cdleme30a  30567  cdleme32c  30632  cdleme32e  30634  trlcolem  30915  cdlemk5u  31050  cdlemk39  31105  cdlemm10N  31308  cdlemn2  31385  dihord1  31408
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-glb 14109  df-meet 14111  df-lat 14152
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