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Theorem latmassOLD 29419
Description: Ortholattice meet is associative. (This can also be proved for lattices with a longer proof.) (inass 3379 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 29403 . . . . . 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 29404 . . . . . . 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 2283 . . . . . . 7  |-  ( oc
`  K )  =  ( oc `  K
)
97, 8opoccl 29384 . . . . . 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 29384 . . . . . 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 2283 . . . . . 6  |-  ( join `  K )  =  (
join `  K )
157, 14latjcl 14156 . . . . 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 29413 . . . 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 29384 . . . . . 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 14201 . . . . 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 5529 . . 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 29414 . . . . 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 5873 . . 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 2324 . 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 14156 . . . 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 29412 . . 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 29414 . . . 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 5874 . 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 2319 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 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   Basecbs 13148   occoc 13216   joincjn 14078   meetcmee 14079   Latclat 14151   OPcops 29362   OLcol 29364
This theorem is referenced by:  latm12  29420  latm32  29421  latmrot  29422  latm4  29423  cmtcomlemN  29438  cmtbr3N  29444  omlfh1N  29448  dalawlem2  30061  dalawlem7  30066  dalawlem11  30070  dalawlem12  30071  lhp2at0  30221  cdleme20d  30501  cdleme23b  30539  cdlemh2  31005  dia2dimlem2  31255  dihmeetbclemN  31494
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-lat 14152  df-oposet 29366  df-ol 29368
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