MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  latjlej12 Unicode version

Theorem latjlej12 14222
Description: Add join to both sides of a lattice ordering. (chlej12i 22109 analog.) (Contributed by NM, 8-Nov-2011.)
Hypotheses
Ref Expression
latlej.b  |-  B  =  ( Base `  K
)
latlej.l  |-  .<_  =  ( le `  K )
latlej.j  |-  .\/  =  ( join `  K )
Assertion
Ref Expression
latjlej12  |-  ( ( 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 latjlej12
StepHypRef Expression
1 simp1 955 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  ->  K  e.  Lat )
2 simp2l 981 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  ->  X  e.  B )
3 simp2r 982 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  ->  Y  e.  B )
4 simp3l 983 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  ->  Z  e.  B )
5 latlej.b . . . 4  |-  B  =  ( Base `  K
)
6 latlej.l . . . 4  |-  .<_  =  ( le `  K )
7 latlej.j . . . 4  |-  .\/  =  ( join `  K )
85, 6, 7latjlej1 14220 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  .<_  Y  ->  ( X  .\/  Z )  .<_  ( Y  .\/  Z ) ) )
91, 2, 3, 4, 8syl13anc 1184 . 2  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  -> 
( X  .<_  Y  -> 
( X  .\/  Z
)  .<_  ( Y  .\/  Z ) ) )
10 simp3r 984 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  ->  W  e.  B )
115, 6, 7latjlej2 14221 . . 3  |-  ( ( K  e.  Lat  /\  ( Z  e.  B  /\  W  e.  B  /\  Y  e.  B
) )  ->  ( Z  .<_  W  ->  ( Y  .\/  Z )  .<_  ( Y  .\/  W ) ) )
121, 4, 10, 3, 11syl13anc 1184 . 2  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  -> 
( Z  .<_  W  -> 
( Y  .\/  Z
)  .<_  ( Y  .\/  W ) ) )
135, 7latjcl 14205 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Z  e.  B )  ->  ( X  .\/  Z
)  e.  B )
141, 2, 4, 13syl3anc 1182 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  -> 
( X  .\/  Z
)  e.  B )
155, 7latjcl 14205 . . . 4  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Y  .\/  Z
)  e.  B )
161, 3, 4, 15syl3anc 1182 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  -> 
( Y  .\/  Z
)  e.  B )
175, 7latjcl 14205 . . . 4  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  W  e.  B )  ->  ( Y  .\/  W
)  e.  B )
181, 3, 10, 17syl3anc 1182 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  -> 
( Y  .\/  W
)  e.  B )
195, 6lattr 14211 . . 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 1184 . 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 469 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 358    /\ w3a 934    = wceq 1633    e. wcel 1701   class class class wbr 4060   ` cfv 5292  (class class class)co 5900   Basecbs 13195   lecple 13262   joincjn 14127   Latclat 14200
This theorem is referenced by:  latledi  14244  dalem-cly  29678  dalem38  29717  dalem44  29723  cdlema1N  29798  pmapjoin  29859  4atexlemc  30076  cdlemg33a  30713
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-undef 6340  df-riota 6346  df-poset 14129  df-lub 14157  df-join 14159  df-lat 14201
  Copyright terms: Public domain W3C validator