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Theorem latmmdir 30047
Description: Lattice meet distributes over itself. (inindir 3400 analog.) (Contributed by NM, 6-Jun-2012.)
Hypotheses
Ref Expression
olmass.b  |-  B  =  ( Base `  K
)
olmass.m  |-  ./\  =  ( meet `  K )
Assertion
Ref Expression
latmmdir  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  ./\  Y
)  ./\  Z )  =  ( ( X 
./\  Z )  ./\  ( Y  ./\  Z ) ) )

Proof of Theorem latmmdir
StepHypRef Expression
1 ollat 30025 . . . . 5  |-  ( K  e.  OL  ->  K  e.  Lat )
21adantr 451 . . . 4  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  K  e.  Lat )
3 simpr3 963 . . . 4  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Z  e.  B )
4 olmass.b . . . . 5  |-  B  =  ( Base `  K
)
5 olmass.m . . . . 5  |-  ./\  =  ( meet `  K )
64, 5latmidm 14208 . . . 4  |-  ( ( K  e.  Lat  /\  Z  e.  B )  ->  ( Z  ./\  Z
)  =  Z )
72, 3, 6syl2anc 642 . . 3  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( Z  ./\  Z )  =  Z )
87oveq2d 5890 . 2  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  ./\  Y
)  ./\  ( Z  ./\ 
Z ) )  =  ( ( X  ./\  Y )  ./\  Z )
)
9 simpl 443 . . 3  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  K  e.  OL )
10 simpr1 961 . . 3  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  X  e.  B )
11 simpr2 962 . . 3  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Y  e.  B )
124, 5latm4 30045 . . 3  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  Z  e.  B ) )  -> 
( ( X  ./\  Y )  ./\  ( Z  ./\ 
Z ) )  =  ( ( X  ./\  Z )  ./\  ( Y  ./\ 
Z ) ) )
139, 10, 11, 3, 3, 12syl122anc 1191 . 2  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  ./\  Y
)  ./\  ( Z  ./\ 
Z ) )  =  ( ( X  ./\  Z )  ./\  ( Y  ./\ 
Z ) ) )
148, 13eqtr3d 2330 1  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  ./\  Y
)  ./\  Z )  =  ( ( X 
./\  Z )  ./\  ( Y  ./\  Z ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   ` cfv 5271  (class class class)co 5874   Basecbs 13164   meetcmee 14095   Latclat 14167   OLcol 29986
This theorem is referenced by:  dalem24  30508  cdleme0e  31028  cdleme7c  31056  djajN  31949
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-lat 14168  df-oposet 29988  df-ol 29990
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