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Theorem dlatl 14298
Description: A distributive lattice is a lattice. (Contributed by Stefan O'Rear, 30-Jan-2015.)
Assertion
Ref Expression
dlatl  |-  ( K  e. DLat  ->  K  e.  Lat )

Proof of Theorem dlatl
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2283 . . 3  |-  ( Base `  K )  =  (
Base `  K )
2 eqid 2283 . . 3  |-  ( join `  K )  =  (
join `  K )
3 eqid 2283 . . 3  |-  ( meet `  K )  =  (
meet `  K )
41, 2, 3isdlat 14296 . 2  |-  ( K  e. DLat 
<->  ( K  e.  Lat  /\ 
A. x  e.  (
Base `  K ) A. y  e.  ( Base `  K ) A. z  e.  ( Base `  K ) ( x ( meet `  K
) ( y (
join `  K )
z ) )  =  ( ( x (
meet `  K )
y ) ( join `  K ) ( x ( meet `  K
) z ) ) ) )
54simplbi 446 1  |-  ( K  e. DLat  ->  K  e.  Lat )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   A.wral 2543   ` cfv 5255  (class class class)co 5858   Basecbs 13148   joincjn 14078   meetcmee 14079   Latclat 14151  DLatcdlat 14294
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-nul 4149
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861  df-dlat 14295
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