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Theorem latabs2 14293
Description: Lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (chabs2 22210 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 2358 . . 3  |-  ( le
`  K )  =  ( le `  K
)
3 latabs1.j . . 3  |-  .\/  =  ( join `  K )
41, 2, 3latlej1 14265 . 2  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  X ( le `  K ) ( X 
.\/  Y ) )
51, 3latjcl 14255 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .\/  Y
)  e.  B )
6 latabs1.m . . . 4  |-  ./\  =  ( meet `  K )
71, 2, 6latleeqm1 14284 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  ( X  .\/  Y )  e.  B )  -> 
( X ( le
`  K ) ( X  .\/  Y )  <-> 
( X  ./\  ( X  .\/  Y ) )  =  X ) )
85, 7syld3an3 1227 . 2  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X ( le
`  K ) ( X  .\/  Y )  <-> 
( X  ./\  ( X  .\/  Y ) )  =  X ) )
94, 8mpbid 201 1  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  ( X  .\/  Y ) )  =  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ w3a 934    = wceq 1642    e. wcel 1710   class class class wbr 4104   ` cfv 5337  (class class class)co 5945   Basecbs 13245   lecple 13312   joincjn 14177   meetcmee 14178   Latclat 14250
This theorem is referenced by:  latdisdlem  14391  cmtbr3N  29513  cdlemc6  30454  cdlemkid1  31180  cdlemkid2  31182
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-undef 6385  df-riota 6391  df-poset 14179  df-lub 14207  df-glb 14208  df-join 14209  df-meet 14210  df-lat 14251
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