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Theorem dlatl 14621
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 2436 . . 3  |-  ( Base `  K )  =  (
Base `  K )
2 eqid 2436 . . 3  |-  ( join `  K )  =  (
join `  K )
3 eqid 2436 . . 3  |-  ( meet `  K )  =  (
meet `  K )
41, 2, 3isdlat 14619 . 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 447 1  |-  ( K  e. DLat  ->  K  e.  Lat )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   A.wral 2705   ` cfv 5454  (class class class)co 6081   Basecbs 13469   joincjn 14401   meetcmee 14402   Latclat 14474  DLatcdlat 14617
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-nul 4338
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-iota 5418  df-fv 5462  df-ov 6084  df-dlat 14618
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