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Theorem latm12 30101
Description: A rearrangement of lattice meet. (in12 3554 analog.) (Contributed by NM, 8-Nov-2011.)
Hypotheses
Ref Expression
olmass.b  |-  B  =  ( Base `  K
)
olmass.m  |-  ./\  =  ( meet `  K )
Assertion
Ref Expression
latm12  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  ./\  ( Y  ./\  Z ) )  =  ( Y  ./\  ( X  ./\ 
Z ) ) )

Proof of Theorem latm12
StepHypRef Expression
1 ollat 30084 . . . . 5  |-  ( K  e.  OL  ->  K  e.  Lat )
21adantr 453 . . . 4  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  K  e.  Lat )
3 simpr1 964 . . . 4  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  X  e.  B )
4 simpr2 965 . . . 4  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Y  e.  B )
5 olmass.b . . . . 5  |-  B  =  ( Base `  K
)
6 olmass.m . . . . 5  |-  ./\  =  ( meet `  K )
75, 6latmcom 14509 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  =  ( Y 
./\  X ) )
82, 3, 4, 7syl3anc 1185 . . 3  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  ./\  Y )  =  ( Y  ./\  X
) )
98oveq1d 6099 . 2  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  ./\  Y
)  ./\  Z )  =  ( ( Y 
./\  X )  ./\  Z ) )
105, 6latmassOLD 30100 . 2  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  ./\  Y
)  ./\  Z )  =  ( X  ./\  ( Y  ./\  Z ) ) )
11 simpr3 966 . . . 4  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Z  e.  B )
124, 3, 113jca 1135 . . 3  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( Y  e.  B  /\  X  e.  B  /\  Z  e.  B )
)
135, 6latmassOLD 30100 . . 3  |-  ( ( K  e.  OL  /\  ( Y  e.  B  /\  X  e.  B  /\  Z  e.  B
) )  ->  (
( Y  ./\  X
)  ./\  Z )  =  ( Y  ./\  ( X  ./\  Z ) ) )
1412, 13syldan 458 . 2  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( Y  ./\  X
)  ./\  Z )  =  ( Y  ./\  ( X  ./\  Z ) ) )
159, 10, 143eqtr3d 2478 1  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  ./\  ( Y  ./\  Z ) )  =  ( Y  ./\  ( X  ./\ 
Z ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   ` cfv 5457  (class class class)co 6084   Basecbs 13474   meetcmee 14407   Latclat 14479   OLcol 30045
This theorem is referenced by:  latm4  30104  omlfh1N  30129  dalawlem6  30746
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-undef 6546  df-riota 6552  df-poset 14408  df-lub 14436  df-glb 14437  df-join 14438  df-meet 14439  df-lat 14480  df-oposet 30047  df-ol 30049
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