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Theorem latmlej12 15847
Description: Ordering of a meet and join with a common variable. (Contributed by NM, 4-Oct-2012.)
Hypotheses
Ref Expression
latledi.b  |-  B  =  ( Base `  K
)
latledi.l  |-  .<_  =  ( le `  K )
latledi.j  |-  .\/  =  ( join `  K )
latledi.m  |-  ./\  =  ( meet `  K )
Assertion
Ref Expression
latmlej12  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  ./\  Y )  .<_  ( Z  .\/  X ) )

Proof of Theorem latmlej12
StepHypRef Expression
1 latledi.b . . 3  |-  B  =  ( Base `  K
)
2 latledi.l . . 3  |-  .<_  =  ( le `  K )
3 latledi.j . . 3  |-  .\/  =  ( join `  K )
4 latledi.m . . 3  |-  ./\  =  ( meet `  K )
51, 2, 3, 4latmlej11 15846 . 2  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  ./\  Y )  .<_  ( X  .\/  Z ) )
61, 3latjcom 15815 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Z  e.  B )  ->  ( X  .\/  Z
)  =  ( Z 
.\/  X ) )
763adant3r2 1206 . 2  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  .\/  Z )  =  ( Z  .\/  X
) )
85, 7breqtrd 4480 1  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  ./\  Y )  .<_  ( Z  .\/  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   Basecbs 14643   lecple 14718   joincjn 15699   meetcmee 15700   Latclat 15801
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-poset 15701  df-lub 15730  df-glb 15731  df-join 15732  df-meet 15733  df-lat 15802
This theorem is referenced by:  latmlej22  15849  mod1ile  15861  dalawlem3  35698  dalawlem12  35707
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