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Theorem latm32 29347
Description: A rearrangement of lattice meet. (in12 3496 analog.) (Contributed by NM, 13-Nov-2012.)
Hypotheses
Ref Expression
olmass.b  |-  B  =  ( Base `  K
)
olmass.m  |-  ./\  =  ( meet `  K )
Assertion
Ref Expression
latm32  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  ./\  Y
)  ./\  Z )  =  ( ( X 
./\  Z )  ./\  Y ) )

Proof of Theorem latm32
StepHypRef Expression
1 ollat 29329 . . . . 5  |-  ( K  e.  OL  ->  K  e.  Lat )
2 olmass.b . . . . . 6  |-  B  =  ( Base `  K
)
3 olmass.m . . . . . 6  |-  ./\  =  ( meet `  K )
42, 3latmcom 14432 . . . . 5  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Y  ./\  Z
)  =  ( Z 
./\  Y ) )
51, 4syl3an1 1217 . . . 4  |-  ( ( K  e.  OL  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Y  ./\  Z
)  =  ( Z 
./\  Y ) )
653adant3r1 1162 . . 3  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( Y  ./\  Z )  =  ( Z  ./\  Y
) )
76oveq2d 6037 . 2  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  ./\  ( Y  ./\  Z ) )  =  ( X  ./\  ( Z  ./\ 
Y ) ) )
82, 3latmassOLD 29345 . 2  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( 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 simpr1 963 . . 3  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  X  e.  B )
11 simpr3 965 . . 3  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Z  e.  B )
12 simpr2 964 . . 3  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Y  e.  B )
132, 3latmassOLD 29345 . . 3  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  Z  e.  B  /\  Y  e.  B
) )  ->  (
( X  ./\  Z
)  ./\  Y )  =  ( X  ./\  ( Z  ./\  Y ) ) )
149, 10, 11, 12, 13syl13anc 1186 . 2  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  ./\  Z
)  ./\  Y )  =  ( X  ./\  ( Z  ./\  Y ) ) )
157, 8, 143eqtr4d 2430 1  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  ./\  Y
)  ./\  Z )  =  ( ( X 
./\  Z )  ./\  Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   ` cfv 5395  (class class class)co 6021   Basecbs 13397   meetcmee 14330   Latclat 14402   OLcol 29290
This theorem is referenced by:  cdleme20d  30427  dia2dimlem3  31182
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-undef 6480  df-riota 6486  df-poset 14331  df-lub 14359  df-glb 14360  df-join 14361  df-meet 14362  df-lat 14403  df-oposet 29292  df-ol 29294
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