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Theorem latmmdiN 29969
Description: Lattice meet distributes over itself. (inindi 3550 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
olmass.b  |-  B  =  ( Base `  K
)
olmass.m  |-  ./\  =  ( meet `  K )
Assertion
Ref Expression
latmmdiN  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  ./\  ( Y  ./\  Z ) )  =  ( ( X  ./\  Y
)  ./\  ( X  ./\ 
Z ) ) )

Proof of Theorem latmmdiN
StepHypRef Expression
1 ollat 29948 . . . . 5  |-  ( K  e.  OL  ->  K  e.  Lat )
21adantr 452 . . . 4  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  K  e.  Lat )
3 simpr1 963 . . . 4  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  X  e.  B )
4 olmass.b . . . . 5  |-  B  =  ( Base `  K
)
5 olmass.m . . . . 5  |-  ./\  =  ( meet `  K )
64, 5latmidm 14507 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B )  ->  ( X  ./\  X
)  =  X )
72, 3, 6syl2anc 643 . . 3  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  ./\  X )  =  X )
87oveq1d 6088 . 2  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  ./\  X
)  ./\  ( Y  ./\ 
Z ) )  =  ( X  ./\  ( Y  ./\  Z ) ) )
9 simpl 444 . . 3  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  K  e.  OL )
10 simpr2 964 . . 3  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Y  e.  B )
11 simpr3 965 . . 3  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Z  e.  B )
124, 5latm4 29968 . . 3  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  X  e.  B
)  /\  ( Y  e.  B  /\  Z  e.  B ) )  -> 
( ( X  ./\  X )  ./\  ( Y  ./\ 
Z ) )  =  ( ( X  ./\  Y )  ./\  ( X  ./\ 
Z ) ) )
139, 3, 3, 10, 11, 12syl122anc 1193 . 2  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  ./\  X
)  ./\  ( Y  ./\ 
Z ) )  =  ( ( X  ./\  Y )  ./\  ( X  ./\ 
Z ) ) )
148, 13eqtr3d 2469 1  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  ./\  ( Y  ./\  Z ) )  =  ( ( X  ./\  Y
)  ./\  ( X  ./\ 
Z ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   ` cfv 5446  (class class class)co 6073   Basecbs 13461   meetcmee 14394   Latclat 14466   OLcol 29909
This theorem is referenced by:  omlfh1N  29993
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-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-poset 14395  df-lub 14423  df-glb 14424  df-join 14425  df-meet 14426  df-lat 14467  df-oposet 29911  df-ol 29913
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