MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  latmlem12 Structured version   Unicode version

Theorem latmlem12 14504
Description: Add join to both sides of a lattice ordering. (ss2in 3560 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
latmlem12  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  -> 
( ( X  .<_  Y  /\  Z  .<_  W )  ->  ( X  ./\  Z )  .<_  ( Y  ./\ 
W ) ) )

Proof of Theorem latmlem12
StepHypRef Expression
1 simp1 957 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  ->  K  e.  Lat )
2 simp2l 983 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  ->  X  e.  B )
3 simp2r 984 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  ->  Y  e.  B )
4 simp3l 985 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  ->  Z  e.  B )
5 latmle.b . . . 4  |-  B  =  ( Base `  K
)
6 latmle.l . . . 4  |-  .<_  =  ( le `  K )
7 latmle.m . . . 4  |-  ./\  =  ( meet `  K )
85, 6, 7latmlem1 14502 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  .<_  Y  ->  ( X  ./\  Z )  .<_  ( Y  ./\  Z ) ) )
91, 2, 3, 4, 8syl13anc 1186 . 2  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  -> 
( X  .<_  Y  -> 
( X  ./\  Z
)  .<_  ( Y  ./\  Z ) ) )
10 simp3r 986 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  ->  W  e.  B )
115, 6, 7latmlem2 14503 . . 3  |-  ( ( K  e.  Lat  /\  ( Z  e.  B  /\  W  e.  B  /\  Y  e.  B
) )  ->  ( Z  .<_  W  ->  ( Y  ./\  Z )  .<_  ( Y  ./\  W ) ) )
121, 4, 10, 3, 11syl13anc 1186 . 2  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  -> 
( Z  .<_  W  -> 
( Y  ./\  Z
)  .<_  ( Y  ./\  W ) ) )
135, 7latmcl 14472 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Z  e.  B )  ->  ( X  ./\  Z
)  e.  B )
141, 2, 4, 13syl3anc 1184 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  -> 
( X  ./\  Z
)  e.  B )
155, 7latmcl 14472 . . . 4  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Y  ./\  Z
)  e.  B )
161, 3, 4, 15syl3anc 1184 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  -> 
( Y  ./\  Z
)  e.  B )
175, 7latmcl 14472 . . . 4  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  W  e.  B )  ->  ( Y  ./\  W
)  e.  B )
181, 3, 10, 17syl3anc 1184 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  -> 
( Y  ./\  W
)  e.  B )
195, 6lattr 14477 . . 3  |-  ( ( K  e.  Lat  /\  ( ( X  ./\  Z )  e.  B  /\  ( Y  ./\  Z )  e.  B  /\  ( Y  ./\  W )  e.  B ) )  -> 
( ( ( X 
./\  Z )  .<_  ( Y  ./\  Z )  /\  ( Y  ./\  Z )  .<_  ( Y  ./\ 
W ) )  -> 
( X  ./\  Z
)  .<_  ( Y  ./\  W ) ) )
201, 14, 16, 18, 19syl13anc 1186 . 2  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  -> 
( ( ( X 
./\  Z )  .<_  ( Y  ./\  Z )  /\  ( Y  ./\  Z )  .<_  ( Y  ./\ 
W ) )  -> 
( X  ./\  Z
)  .<_  ( Y  ./\  W ) ) )
219, 12, 20syl2and 470 1  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  -> 
( ( X  .<_  Y  /\  Z  .<_  W )  ->  ( X  ./\  Z )  .<_  ( Y  ./\ 
W ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   Basecbs 13461   lecple 13528   meetcmee 14394   Latclat 14466
This theorem is referenced by:  dalem10  30407  dalem55  30461  dalawlem3  30607  dalawlem7  30611  dalawlem11  30615  dalawlem12  30616  cdlemk51  31687
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-poset 14395  df-glb 14424  df-meet 14426  df-lat 14467
  Copyright terms: Public domain W3C validator