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Theorem omlfh3N 29449
Description: Foulis-Holland Theorem, part 3. Dual of omlfh1N 29448. (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
omlfh1.b  |-  B  =  ( Base `  K
)
omlfh1.j  |-  .\/  =  ( join `  K )
omlfh1.m  |-  ./\  =  ( meet `  K )
omlfh1.c  |-  C  =  ( cm `  K
)
Assertion
Ref Expression
omlfh3N  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X C Y  /\  X C Z ) )  -> 
( X  .\/  ( Y  ./\  Z ) )  =  ( ( X 
.\/  Y )  ./\  ( X  .\/  Z ) ) )

Proof of Theorem omlfh3N
StepHypRef Expression
1 omlfh1.b . . . . . . 7  |-  B  =  ( Base `  K
)
2 eqid 2283 . . . . . . 7  |-  ( oc
`  K )  =  ( oc `  K
)
3 omlfh1.c . . . . . . 7  |-  C  =  ( cm `  K
)
41, 2, 3cmt4N 29442 . . . . . 6  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X C Y  <-> 
( ( oc `  K ) `  X
) C ( ( oc `  K ) `
 Y ) ) )
543adant3r3 1162 . . . . 5  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X C Y  <->  ( ( oc `  K ) `  X ) C ( ( oc `  K
) `  Y )
) )
61, 2, 3cmt4N 29442 . . . . . 6  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Z  e.  B )  ->  ( X C Z  <-> 
( ( oc `  K ) `  X
) C ( ( oc `  K ) `
 Z ) ) )
763adant3r2 1161 . . . . 5  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X C Z  <->  ( ( oc `  K ) `  X ) C ( ( oc `  K
) `  Z )
) )
85, 7anbi12d 691 . . . 4  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X C Y  /\  X C Z )  <->  ( ( ( oc `  K ) `
 X ) C ( ( oc `  K ) `  Y
)  /\  ( ( oc `  K ) `  X ) C ( ( oc `  K
) `  Z )
) ) )
9 simpl 443 . . . . 5  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  K  e.  OML )
10 omlop 29431 . . . . . . . 8  |-  ( K  e.  OML  ->  K  e.  OP )
1110adantr 451 . . . . . . 7  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  K  e.  OP )
12 simpr1 961 . . . . . . 7  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  X  e.  B )
131, 2opoccl 29384 . . . . . . 7  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( ( oc `  K ) `  X
)  e.  B )
1411, 12, 13syl2anc 642 . . . . . 6  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( oc `  K
) `  X )  e.  B )
15 simpr2 962 . . . . . . 7  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Y  e.  B )
161, 2opoccl 29384 . . . . . . 7  |-  ( ( K  e.  OP  /\  Y  e.  B )  ->  ( ( oc `  K ) `  Y
)  e.  B )
1711, 15, 16syl2anc 642 . . . . . 6  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( oc `  K
) `  Y )  e.  B )
18 simpr3 963 . . . . . . 7  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Z  e.  B )
191, 2opoccl 29384 . . . . . . 7  |-  ( ( K  e.  OP  /\  Z  e.  B )  ->  ( ( oc `  K ) `  Z
)  e.  B )
2011, 18, 19syl2anc 642 . . . . . 6  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( oc `  K
) `  Z )  e.  B )
2114, 17, 203jca 1132 . . . . 5  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( ( oc `  K ) `  X
)  e.  B  /\  ( ( oc `  K ) `  Y
)  e.  B  /\  ( ( oc `  K ) `  Z
)  e.  B ) )
22 omlfh1.j . . . . . . . 8  |-  .\/  =  ( join `  K )
23 omlfh1.m . . . . . . . 8  |-  ./\  =  ( meet `  K )
241, 22, 23, 3omlfh1N 29448 . . . . . . 7  |-  ( ( K  e.  OML  /\  ( ( ( oc
`  K ) `  X )  e.  B  /\  ( ( oc `  K ) `  Y
)  e.  B  /\  ( ( oc `  K ) `  Z
)  e.  B )  /\  ( ( ( oc `  K ) `
 X ) C ( ( oc `  K ) `  Y
)  /\  ( ( oc `  K ) `  X ) C ( ( oc `  K
) `  Z )
) )  ->  (
( ( oc `  K ) `  X
)  ./\  ( (
( oc `  K
) `  Y )  .\/  ( ( oc `  K ) `  Z
) ) )  =  ( ( ( ( oc `  K ) `
 X )  ./\  ( ( oc `  K ) `  Y
) )  .\/  (
( ( oc `  K ) `  X
)  ./\  ( ( oc `  K ) `  Z ) ) ) )
2524fveq2d 5529 . . . . . 6  |-  ( ( K  e.  OML  /\  ( ( ( oc
`  K ) `  X )  e.  B  /\  ( ( oc `  K ) `  Y
)  e.  B  /\  ( ( oc `  K ) `  Z
)  e.  B )  /\  ( ( ( oc `  K ) `
 X ) C ( ( oc `  K ) `  Y
)  /\  ( ( oc `  K ) `  X ) C ( ( oc `  K
) `  Z )
) )  ->  (
( oc `  K
) `  ( (
( oc `  K
) `  X )  ./\  ( ( ( oc
`  K ) `  Y )  .\/  (
( oc `  K
) `  Z )
) ) )  =  ( ( oc `  K ) `  (
( ( ( oc
`  K ) `  X )  ./\  (
( oc `  K
) `  Y )
)  .\/  ( (
( oc `  K
) `  X )  ./\  ( ( oc `  K ) `  Z
) ) ) ) )
26253exp 1150 . . . . 5  |-  ( K  e.  OML  ->  (
( ( ( oc
`  K ) `  X )  e.  B  /\  ( ( oc `  K ) `  Y
)  e.  B  /\  ( ( oc `  K ) `  Z
)  e.  B )  ->  ( ( ( ( oc `  K
) `  X ) C ( ( oc
`  K ) `  Y )  /\  (
( oc `  K
) `  X ) C ( ( oc
`  K ) `  Z ) )  -> 
( ( oc `  K ) `  (
( ( oc `  K ) `  X
)  ./\  ( (
( oc `  K
) `  Y )  .\/  ( ( oc `  K ) `  Z
) ) ) )  =  ( ( oc
`  K ) `  ( ( ( ( oc `  K ) `
 X )  ./\  ( ( oc `  K ) `  Y
) )  .\/  (
( ( oc `  K ) `  X
)  ./\  ( ( oc `  K ) `  Z ) ) ) ) ) ) )
279, 21, 26sylc 56 . . . 4  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( ( ( oc
`  K ) `  X ) C ( ( oc `  K
) `  Y )  /\  ( ( oc `  K ) `  X
) C ( ( oc `  K ) `
 Z ) )  ->  ( ( oc
`  K ) `  ( ( ( oc
`  K ) `  X )  ./\  (
( ( oc `  K ) `  Y
)  .\/  ( ( oc `  K ) `  Z ) ) ) )  =  ( ( oc `  K ) `
 ( ( ( ( oc `  K
) `  X )  ./\  ( ( oc `  K ) `  Y
) )  .\/  (
( ( oc `  K ) `  X
)  ./\  ( ( oc `  K ) `  Z ) ) ) ) ) )
288, 27sylbid 206 . . 3  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X C Y  /\  X C Z )  ->  ( ( oc `  K ) `  ( ( ( oc
`  K ) `  X )  ./\  (
( ( oc `  K ) `  Y
)  .\/  ( ( oc `  K ) `  Z ) ) ) )  =  ( ( oc `  K ) `
 ( ( ( ( oc `  K
) `  X )  ./\  ( ( oc `  K ) `  Y
) )  .\/  (
( ( oc `  K ) `  X
)  ./\  ( ( oc `  K ) `  Z ) ) ) ) ) )
29283impia 1148 . 2  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X C Y  /\  X C Z ) )  -> 
( ( oc `  K ) `  (
( ( oc `  K ) `  X
)  ./\  ( (
( oc `  K
) `  Y )  .\/  ( ( oc `  K ) `  Z
) ) ) )  =  ( ( oc
`  K ) `  ( ( ( ( oc `  K ) `
 X )  ./\  ( ( oc `  K ) `  Y
) )  .\/  (
( ( oc `  K ) `  X
)  ./\  ( ( oc `  K ) `  Z ) ) ) ) )
30 omlol 29430 . . . . . 6  |-  ( K  e.  OML  ->  K  e.  OL )
3130adantr 451 . . . . 5  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  K  e.  OL )
32 omllat 29432 . . . . . . 7  |-  ( K  e.  OML  ->  K  e.  Lat )
3332adantr 451 . . . . . 6  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  K  e.  Lat )
341, 22latjcl 14156 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( ( oc `  K ) `  Y
)  e.  B  /\  ( ( oc `  K ) `  Z
)  e.  B )  ->  ( ( ( oc `  K ) `
 Y )  .\/  ( ( oc `  K ) `  Z
) )  e.  B
)
3533, 17, 20, 34syl3anc 1182 . . . . 5  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( ( oc `  K ) `  Y
)  .\/  ( ( oc `  K ) `  Z ) )  e.  B )
361, 22, 23, 2oldmm2 29408 . . . . 5  |-  ( ( K  e.  OL  /\  X  e.  B  /\  ( ( ( oc
`  K ) `  Y )  .\/  (
( oc `  K
) `  Z )
)  e.  B )  ->  ( ( oc
`  K ) `  ( ( ( oc
`  K ) `  X )  ./\  (
( ( oc `  K ) `  Y
)  .\/  ( ( oc `  K ) `  Z ) ) ) )  =  ( X 
.\/  ( ( oc
`  K ) `  ( ( ( oc
`  K ) `  Y )  .\/  (
( oc `  K
) `  Z )
) ) ) )
3731, 12, 35, 36syl3anc 1182 . . . 4  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( oc `  K
) `  ( (
( oc `  K
) `  X )  ./\  ( ( ( oc
`  K ) `  Y )  .\/  (
( oc `  K
) `  Z )
) ) )  =  ( X  .\/  (
( oc `  K
) `  ( (
( oc `  K
) `  Y )  .\/  ( ( oc `  K ) `  Z
) ) ) ) )
381, 22, 23, 2oldmj4 29414 . . . . . 6  |-  ( ( K  e.  OL  /\  Y  e.  B  /\  Z  e.  B )  ->  ( ( oc `  K ) `  (
( ( oc `  K ) `  Y
)  .\/  ( ( oc `  K ) `  Z ) ) )  =  ( Y  ./\  Z ) )
3931, 15, 18, 38syl3anc 1182 . . . . 5  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( oc `  K
) `  ( (
( oc `  K
) `  Y )  .\/  ( ( oc `  K ) `  Z
) ) )  =  ( Y  ./\  Z
) )
4039oveq2d 5874 . . . 4  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  .\/  ( ( oc
`  K ) `  ( ( ( oc
`  K ) `  Y )  .\/  (
( oc `  K
) `  Z )
) ) )  =  ( X  .\/  ( Y  ./\  Z ) ) )
4137, 40eqtr2d 2316 . . 3  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  .\/  ( Y  ./\  Z ) )  =  ( ( oc `  K
) `  ( (
( oc `  K
) `  X )  ./\  ( ( ( oc
`  K ) `  Y )  .\/  (
( oc `  K
) `  Z )
) ) ) )
42413adant3 975 . 2  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X C Y  /\  X C Z ) )  -> 
( X  .\/  ( Y  ./\  Z ) )  =  ( ( oc
`  K ) `  ( ( ( oc
`  K ) `  X )  ./\  (
( ( oc `  K ) `  Y
)  .\/  ( ( oc `  K ) `  Z ) ) ) ) )
431, 23latmcl 14157 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( ( oc `  K ) `  X
)  e.  B  /\  ( ( oc `  K ) `  Y
)  e.  B )  ->  ( ( ( oc `  K ) `
 X )  ./\  ( ( oc `  K ) `  Y
) )  e.  B
)
4433, 14, 17, 43syl3anc 1182 . . . . 5  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( ( oc `  K ) `  X
)  ./\  ( ( oc `  K ) `  Y ) )  e.  B )
451, 23latmcl 14157 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( ( oc `  K ) `  X
)  e.  B  /\  ( ( oc `  K ) `  Z
)  e.  B )  ->  ( ( ( oc `  K ) `
 X )  ./\  ( ( oc `  K ) `  Z
) )  e.  B
)
4633, 14, 20, 45syl3anc 1182 . . . . 5  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( ( oc `  K ) `  X
)  ./\  ( ( oc `  K ) `  Z ) )  e.  B )
471, 22, 23, 2oldmj1 29411 . . . . 5  |-  ( ( K  e.  OL  /\  ( ( ( oc
`  K ) `  X )  ./\  (
( oc `  K
) `  Y )
)  e.  B  /\  ( ( ( oc
`  K ) `  X )  ./\  (
( oc `  K
) `  Z )
)  e.  B )  ->  ( ( oc
`  K ) `  ( ( ( ( oc `  K ) `
 X )  ./\  ( ( oc `  K ) `  Y
) )  .\/  (
( ( oc `  K ) `  X
)  ./\  ( ( oc `  K ) `  Z ) ) ) )  =  ( ( ( oc `  K
) `  ( (
( oc `  K
) `  X )  ./\  ( ( oc `  K ) `  Y
) ) )  ./\  ( ( oc `  K ) `  (
( ( oc `  K ) `  X
)  ./\  ( ( oc `  K ) `  Z ) ) ) ) )
4831, 44, 46, 47syl3anc 1182 . . . 4  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( oc `  K
) `  ( (
( ( oc `  K ) `  X
)  ./\  ( ( oc `  K ) `  Y ) )  .\/  ( ( ( oc
`  K ) `  X )  ./\  (
( oc `  K
) `  Z )
) ) )  =  ( ( ( oc
`  K ) `  ( ( ( oc
`  K ) `  X )  ./\  (
( oc `  K
) `  Y )
) )  ./\  (
( oc `  K
) `  ( (
( oc `  K
) `  X )  ./\  ( ( oc `  K ) `  Z
) ) ) ) )
491, 22, 23, 2oldmm4 29410 . . . . . 6  |-  ( ( K  e.  OL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( oc `  K ) `  (
( ( oc `  K ) `  X
)  ./\  ( ( oc `  K ) `  Y ) ) )  =  ( X  .\/  Y ) )
5031, 12, 15, 49syl3anc 1182 . . . . 5  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( oc `  K
) `  ( (
( oc `  K
) `  X )  ./\  ( ( oc `  K ) `  Y
) ) )  =  ( X  .\/  Y
) )
511, 22, 23, 2oldmm4 29410 . . . . . 6  |-  ( ( K  e.  OL  /\  X  e.  B  /\  Z  e.  B )  ->  ( ( oc `  K ) `  (
( ( oc `  K ) `  X
)  ./\  ( ( oc `  K ) `  Z ) ) )  =  ( X  .\/  Z ) )
5231, 12, 18, 51syl3anc 1182 . . . . 5  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( oc `  K
) `  ( (
( oc `  K
) `  X )  ./\  ( ( oc `  K ) `  Z
) ) )  =  ( X  .\/  Z
) )
5350, 52oveq12d 5876 . . . 4  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( ( oc `  K ) `  (
( ( oc `  K ) `  X
)  ./\  ( ( oc `  K ) `  Y ) ) ) 
./\  ( ( oc
`  K ) `  ( ( ( oc
`  K ) `  X )  ./\  (
( oc `  K
) `  Z )
) ) )  =  ( ( X  .\/  Y )  ./\  ( X  .\/  Z ) ) )
5448, 53eqtr2d 2316 . . 3  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  .\/  Y
)  ./\  ( X  .\/  Z ) )  =  ( ( oc `  K ) `  (
( ( ( oc
`  K ) `  X )  ./\  (
( oc `  K
) `  Y )
)  .\/  ( (
( oc `  K
) `  X )  ./\  ( ( oc `  K ) `  Z
) ) ) ) )
55543adant3 975 . 2  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X C Y  /\  X C Z ) )  -> 
( ( X  .\/  Y )  ./\  ( X  .\/  Z ) )  =  ( ( oc `  K ) `  (
( ( ( oc
`  K ) `  X )  ./\  (
( oc `  K
) `  Y )
)  .\/  ( (
( oc `  K
) `  X )  ./\  ( ( oc `  K ) `  Z
) ) ) ) )
5629, 42, 553eqtr4d 2325 1  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X C Y  /\  X C Z ) )  -> 
( X  .\/  ( Y  ./\  Z ) )  =  ( ( X 
.\/  Y )  ./\  ( X  .\/  Z ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   occoc 13216   joincjn 14078   meetcmee 14079   Latclat 14151   OPcops 29362   cmccmtN 29363   OLcol 29364   OMLcoml 29365
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-p0 14145  df-lat 14152  df-oposet 29366  df-cmtN 29367  df-ol 29368  df-oml 29369
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