Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  latmassOLD Unicode version

Theorem latmassOLD 30041
Description: Ortholattice meet is associative. (This can also be proved for lattices with a longer proof.) (inass 3392 analog.) (Contributed by NM, 7-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
olmass.b  |-  B  =  ( Base `  K
)
olmass.m  |-  ./\  =  ( meet `  K )
Assertion
Ref Expression
latmassOLD  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  ./\  Y
)  ./\  Z )  =  ( X  ./\  ( Y  ./\  Z ) ) )

Proof of Theorem latmassOLD
StepHypRef Expression
1 simpl 443 . . . 4  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  K  e.  OL )
2 ollat 30025 . . . . . 6  |-  ( K  e.  OL  ->  K  e.  Lat )
32adantr 451 . . . . 5  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  K  e.  Lat )
4 olop 30026 . . . . . . 7  |-  ( K  e.  OL  ->  K  e.  OP )
54adantr 451 . . . . . 6  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  K  e.  OP )
6 simpr1 961 . . . . . 6  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  X  e.  B )
7 olmass.b . . . . . . 7  |-  B  =  ( Base `  K
)
8 eqid 2296 . . . . . . 7  |-  ( oc
`  K )  =  ( oc `  K
)
97, 8opoccl 30006 . . . . . 6  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( ( oc `  K ) `  X
)  e.  B )
105, 6, 9syl2anc 642 . . . . 5  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( oc `  K
) `  X )  e.  B )
11 simpr2 962 . . . . . 6  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Y  e.  B )
127, 8opoccl 30006 . . . . . 6  |-  ( ( K  e.  OP  /\  Y  e.  B )  ->  ( ( oc `  K ) `  Y
)  e.  B )
135, 11, 12syl2anc 642 . . . . 5  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( oc `  K
) `  Y )  e.  B )
14 eqid 2296 . . . . . 6  |-  ( join `  K )  =  (
join `  K )
157, 14latjcl 14172 . . . . 5  |-  ( ( K  e.  Lat  /\  ( ( oc `  K ) `  X
)  e.  B  /\  ( ( oc `  K ) `  Y
)  e.  B )  ->  ( ( ( oc `  K ) `
 X ) (
join `  K )
( ( oc `  K ) `  Y
) )  e.  B
)
163, 10, 13, 15syl3anc 1182 . . . 4  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( ( oc `  K ) `  X
) ( join `  K
) ( ( oc
`  K ) `  Y ) )  e.  B )
17 simpr3 963 . . . 4  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Z  e.  B )
18 olmass.m . . . . 5  |-  ./\  =  ( meet `  K )
197, 14, 18, 8oldmj3 30035 . . . 4  |-  ( ( K  e.  OL  /\  ( ( ( oc
`  K ) `  X ) ( join `  K ) ( ( oc `  K ) `
 Y ) )  e.  B  /\  Z  e.  B )  ->  (
( oc `  K
) `  ( (
( ( oc `  K ) `  X
) ( join `  K
) ( ( oc
`  K ) `  Y ) ) (
join `  K )
( ( oc `  K ) `  Z
) ) )  =  ( ( ( oc
`  K ) `  ( ( ( oc
`  K ) `  X ) ( join `  K ) ( ( oc `  K ) `
 Y ) ) )  ./\  Z )
)
201, 16, 17, 19syl3anc 1182 . . 3  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( oc `  K
) `  ( (
( ( oc `  K ) `  X
) ( join `  K
) ( ( oc
`  K ) `  Y ) ) (
join `  K )
( ( oc `  K ) `  Z
) ) )  =  ( ( ( oc
`  K ) `  ( ( ( oc
`  K ) `  X ) ( join `  K ) ( ( oc `  K ) `
 Y ) ) )  ./\  Z )
)
217, 8opoccl 30006 . . . . . 6  |-  ( ( K  e.  OP  /\  Z  e.  B )  ->  ( ( oc `  K ) `  Z
)  e.  B )
225, 17, 21syl2anc 642 . . . . 5  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( oc `  K
) `  Z )  e.  B )
237, 14latjass 14217 . . . . 5  |-  ( ( K  e.  Lat  /\  ( ( ( oc
`  K ) `  X )  e.  B  /\  ( ( oc `  K ) `  Y
)  e.  B  /\  ( ( oc `  K ) `  Z
)  e.  B ) )  ->  ( (
( ( oc `  K ) `  X
) ( join `  K
) ( ( oc
`  K ) `  Y ) ) (
join `  K )
( ( oc `  K ) `  Z
) )  =  ( ( ( oc `  K ) `  X
) ( join `  K
) ( ( ( oc `  K ) `
 Y ) (
join `  K )
( ( oc `  K ) `  Z
) ) ) )
243, 10, 13, 22, 23syl13anc 1184 . . . 4  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( ( ( oc
`  K ) `  X ) ( join `  K ) ( ( oc `  K ) `
 Y ) ) ( join `  K
) ( ( oc
`  K ) `  Z ) )  =  ( ( ( oc
`  K ) `  X ) ( join `  K ) ( ( ( oc `  K
) `  Y )
( join `  K )
( ( oc `  K ) `  Z
) ) ) )
2524fveq2d 5545 . . 3  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( oc `  K
) `  ( (
( ( oc `  K ) `  X
) ( join `  K
) ( ( oc
`  K ) `  Y ) ) (
join `  K )
( ( oc `  K ) `  Z
) ) )  =  ( ( oc `  K ) `  (
( ( oc `  K ) `  X
) ( join `  K
) ( ( ( oc `  K ) `
 Y ) (
join `  K )
( ( oc `  K ) `  Z
) ) ) ) )
267, 14, 18, 8oldmj4 30036 . . . . 5  |-  ( ( K  e.  OL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( oc `  K ) `  (
( ( oc `  K ) `  X
) ( join `  K
) ( ( oc
`  K ) `  Y ) ) )  =  ( X  ./\  Y ) )
27263adant3r3 1162 . . . 4  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( oc `  K
) `  ( (
( oc `  K
) `  X )
( join `  K )
( ( oc `  K ) `  Y
) ) )  =  ( X  ./\  Y
) )
2827oveq1d 5889 . . 3  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( ( oc `  K ) `  (
( ( oc `  K ) `  X
) ( join `  K
) ( ( oc
`  K ) `  Y ) ) ) 
./\  Z )  =  ( ( X  ./\  Y )  ./\  Z )
)
2920, 25, 283eqtr3rd 2337 . 2  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  ./\  Y
)  ./\  Z )  =  ( ( oc
`  K ) `  ( ( ( oc
`  K ) `  X ) ( join `  K ) ( ( ( oc `  K
) `  Y )
( join `  K )
( ( oc `  K ) `  Z
) ) ) ) )
307, 14latjcl 14172 . . . 4  |-  ( ( K  e.  Lat  /\  ( ( oc `  K ) `  Y
)  e.  B  /\  ( ( oc `  K ) `  Z
)  e.  B )  ->  ( ( ( oc `  K ) `
 Y ) (
join `  K )
( ( oc `  K ) `  Z
) )  e.  B
)
313, 13, 22, 30syl3anc 1182 . . 3  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( ( oc `  K ) `  Y
) ( join `  K
) ( ( oc
`  K ) `  Z ) )  e.  B )
327, 14, 18, 8oldmj2 30034 . . 3  |-  ( ( K  e.  OL  /\  X  e.  B  /\  ( ( ( oc
`  K ) `  Y ) ( join `  K ) ( ( oc `  K ) `
 Z ) )  e.  B )  -> 
( ( oc `  K ) `  (
( ( oc `  K ) `  X
) ( join `  K
) ( ( ( oc `  K ) `
 Y ) (
join `  K )
( ( oc `  K ) `  Z
) ) ) )  =  ( X  ./\  ( ( oc `  K ) `  (
( ( oc `  K ) `  Y
) ( join `  K
) ( ( oc
`  K ) `  Z ) ) ) ) )
331, 6, 31, 32syl3anc 1182 . 2  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( oc `  K
) `  ( (
( oc `  K
) `  X )
( join `  K )
( ( ( oc
`  K ) `  Y ) ( join `  K ) ( ( oc `  K ) `
 Z ) ) ) )  =  ( X  ./\  ( ( oc `  K ) `  ( ( ( oc
`  K ) `  Y ) ( join `  K ) ( ( oc `  K ) `
 Z ) ) ) ) )
347, 14, 18, 8oldmj4 30036 . . . 4  |-  ( ( K  e.  OL  /\  Y  e.  B  /\  Z  e.  B )  ->  ( ( oc `  K ) `  (
( ( oc `  K ) `  Y
) ( join `  K
) ( ( oc
`  K ) `  Z ) ) )  =  ( Y  ./\  Z ) )
35343adant3r1 1160 . . 3  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( oc `  K
) `  ( (
( oc `  K
) `  Y )
( join `  K )
( ( oc `  K ) `  Z
) ) )  =  ( Y  ./\  Z
) )
3635oveq2d 5890 . 2  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  ./\  ( ( oc
`  K ) `  ( ( ( oc
`  K ) `  Y ) ( join `  K ) ( ( oc `  K ) `
 Z ) ) ) )  =  ( X  ./\  ( Y  ./\ 
Z ) ) )
3729, 33, 363eqtrd 2332 1  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  ./\  Y
)  ./\  Z )  =  ( X  ./\  ( 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   occoc 13232   joincjn 14094   meetcmee 14095   Latclat 14167   OPcops 29984   OLcol 29986
This theorem is referenced by:  latm12  30042  latm32  30043  latmrot  30044  latm4  30045  cmtcomlemN  30060  cmtbr3N  30066  omlfh1N  30070  dalawlem2  30683  dalawlem7  30688  dalawlem11  30692  dalawlem12  30693  lhp2at0  30843  cdleme20d  31123  cdleme23b  31161  cdlemh2  31627  dia2dimlem2  31877  dihmeetbclemN  32116
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
  Copyright terms: Public domain W3C validator