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Theorem latmassOLD 30028
Description: Ortholattice meet is associative. (This can also be proved for lattices with a longer proof.) (inass 3552 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 445 . . . 4  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  K  e.  OL )
2 ollat 30012 . . . . . 6  |-  ( K  e.  OL  ->  K  e.  Lat )
32adantr 453 . . . . 5  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  K  e.  Lat )
4 olop 30013 . . . . . . 7  |-  ( K  e.  OL  ->  K  e.  OP )
54adantr 453 . . . . . 6  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  K  e.  OP )
6 simpr1 964 . . . . . 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 2437 . . . . . . 7  |-  ( oc
`  K )  =  ( oc `  K
)
97, 8opoccl 29993 . . . . . 6  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( ( oc `  K ) `  X
)  e.  B )
105, 6, 9syl2anc 644 . . . . 5  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( oc `  K
) `  X )  e.  B )
11 simpr2 965 . . . . . 6  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Y  e.  B )
127, 8opoccl 29993 . . . . . 6  |-  ( ( K  e.  OP  /\  Y  e.  B )  ->  ( ( oc `  K ) `  Y
)  e.  B )
135, 11, 12syl2anc 644 . . . . 5  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( oc `  K
) `  Y )  e.  B )
14 eqid 2437 . . . . . 6  |-  ( join `  K )  =  (
join `  K )
157, 14latjcl 14480 . . . . 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 1185 . . . 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 966 . . . 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 30022 . . . 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 1185 . . 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 29993 . . . . . 6  |-  ( ( K  e.  OP  /\  Z  e.  B )  ->  ( ( oc `  K ) `  Z
)  e.  B )
225, 17, 21syl2anc 644 . . . . 5  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( oc `  K
) `  Z )  e.  B )
237, 14latjass 14525 . . . . 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 1187 . . . 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 5733 . . 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 30023 . . . . 5  |-  ( ( K  e.  OL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( oc `  K ) `  (
( ( oc `  K ) `  X
) ( join `  K
) ( ( oc
`  K ) `  Y ) ) )  =  ( X  ./\  Y ) )
27263adant3r3 1165 . . . 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 6097 . . 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 2478 . 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 14480 . . . 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 1185 . . 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 30021 . . 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 1185 . 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 30023 . . . 4  |-  ( ( K  e.  OL  /\  Y  e.  B  /\  Z  e.  B )  ->  ( ( oc `  K ) `  (
( ( oc `  K ) `  Y
) ( join `  K
) ( ( oc
`  K ) `  Z ) ) )  =  ( Y  ./\  Z ) )
35343adant3r1 1163 . . 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 6098 . 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 2473 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 360    /\ w3a 937    = wceq 1653    e. wcel 1726   ` cfv 5455  (class class class)co 6082   Basecbs 13470   occoc 13538   joincjn 14402   meetcmee 14403   Latclat 14475   OPcops 29971   OLcol 29973
This theorem is referenced by:  latm12  30029  latm32  30030  latmrot  30031  latm4  30032  cmtcomlemN  30047  cmtbr3N  30053  omlfh1N  30057  dalawlem2  30670  dalawlem7  30675  dalawlem11  30679  dalawlem12  30680  lhp2at0  30830  cdleme20d  31110  cdleme23b  31148  cdlemh2  31614  dia2dimlem2  31864  dihmeetbclemN  32103
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
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 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-undef 6544  df-riota 6550  df-poset 14404  df-lub 14432  df-glb 14433  df-join 14434  df-meet 14435  df-lat 14476  df-oposet 29975  df-ol 29977
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