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Theorem latabs2 14156
Description: Lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (chabs2 22056 analog.) (Contributed by NM, 8-Nov-2011.)
Hypotheses
Ref Expression
latabs1.b  |-  B  =  ( Base `  K
)
latabs1.j  |-  .\/  =  ( join `  K )
latabs1.m  |-  ./\  =  ( meet `  K )
Assertion
Ref Expression
latabs2  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  ( X  .\/  Y ) )  =  X )

Proof of Theorem latabs2
StepHypRef Expression
1 latabs1.b . . 3  |-  B  =  ( Base `  K
)
2 eqid 2258 . . 3  |-  ( le
`  K )  =  ( le `  K
)
3 latabs1.j . . 3  |-  .\/  =  ( join `  K )
41, 2, 3latlej1 14128 . 2  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  X ( le `  K ) ( X 
.\/  Y ) )
51, 3latjcl 14118 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .\/  Y
)  e.  B )
6 latabs1.m . . . 4  |-  ./\  =  ( meet `  K )
71, 2, 6latleeqm1 14147 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  ( X  .\/  Y )  e.  B )  -> 
( X ( le
`  K ) ( X  .\/  Y )  <-> 
( X  ./\  ( X  .\/  Y ) )  =  X ) )
85, 7syld3an3 1232 . 2  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X ( le
`  K ) ( X  .\/  Y )  <-> 
( X  ./\  ( X  .\/  Y ) )  =  X ) )
94, 8mpbid 203 1  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  ( X  .\/  Y ) )  =  X )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ w3a 939    = wceq 1619    e. wcel 1621   class class class wbr 3997   ` cfv 4673  (class class class)co 5792   Basecbs 13110   lecple 13177   joincjn 14040   meetcmee 14041   Latclat 14113
This theorem is referenced by:  latdisdlem  14254  cmtbr3N  28611  cdlemc6  29552  cdlemkid1  30278  cdlemkid2  30280
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-iota 6225  df-undef 6264  df-riota 6272  df-poset 14042  df-lub 14070  df-glb 14071  df-join 14072  df-meet 14073  df-lat 14114
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