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Theorem omlfh3N 34349
Description: Foulis-Holland Theorem, part 3. Dual of omlfh1N 34348. (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 2467 . . . . . . 7  |-  ( oc
`  K )  =  ( oc `  K
)
3 omlfh1.c . . . . . . 7  |-  C  =  ( cm `  K
)
41, 2, 3cmt4N 34342 . . . . . 6  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X C Y  <-> 
( ( oc `  K ) `  X
) C ( ( oc `  K ) `
 Y ) ) )
543adant3r3 1207 . . . . 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 34342 . . . . . 6  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Z  e.  B )  ->  ( X C Z  <-> 
( ( oc `  K ) `  X
) C ( ( oc `  K ) `
 Z ) ) )
763adant3r2 1206 . . . . 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 710 . . . 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 457 . . . . 5  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  K  e.  OML )
10 omlop 34331 . . . . . . . 8  |-  ( K  e.  OML  ->  K  e.  OP )
1110adantr 465 . . . . . . 7  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  K  e.  OP )
12 simpr1 1002 . . . . . . 7  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  X  e.  B )
131, 2opoccl 34284 . . . . . . 7  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( ( oc `  K ) `  X
)  e.  B )
1411, 12, 13syl2anc 661 . . . . . 6  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( oc `  K
) `  X )  e.  B )
15 simpr2 1003 . . . . . . 7  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Y  e.  B )
161, 2opoccl 34284 . . . . . . 7  |-  ( ( K  e.  OP  /\  Y  e.  B )  ->  ( ( oc `  K ) `  Y
)  e.  B )
1711, 15, 16syl2anc 661 . . . . . 6  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( oc `  K
) `  Y )  e.  B )
18 simpr3 1004 . . . . . . 7  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Z  e.  B )
191, 2opoccl 34284 . . . . . . 7  |-  ( ( K  e.  OP  /\  Z  e.  B )  ->  ( ( oc `  K ) `  Z
)  e.  B )
2011, 18, 19syl2anc 661 . . . . . 6  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( oc `  K
) `  Z )  e.  B )
2114, 17, 203jca 1176 . . . . 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 34348 . . . . . . 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 5875 . . . . . 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 1195 . . . . 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 60 . . . 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 215 . . 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 1193 . 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 34330 . . . . . 6  |-  ( K  e.  OML  ->  K  e.  OL )
3130adantr 465 . . . . 5  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  K  e.  OL )
32 omllat 34332 . . . . . . 7  |-  ( K  e.  OML  ->  K  e.  Lat )
3332adantr 465 . . . . . 6  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  K  e.  Lat )
341, 22latjcl 15550 . . . . . 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 1228 . . . . 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 34308 . . . . 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 1228 . . . 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 34314 . . . . . 6  |-  ( ( K  e.  OL  /\  Y  e.  B  /\  Z  e.  B )  ->  ( ( oc `  K ) `  (
( ( oc `  K ) `  Y
)  .\/  ( ( oc `  K ) `  Z ) ) )  =  ( Y  ./\  Z ) )
3931, 15, 18, 38syl3anc 1228 . . . . 5  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( oc `  K
) `  ( (
( oc `  K
) `  Y )  .\/  ( ( oc `  K ) `  Z
) ) )  =  ( Y  ./\  Z
) )
4039oveq2d 6310 . . . 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 2509 . . 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 1016 . 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 15551 . . . . . 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 1228 . . . . 5  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( ( oc `  K ) `  X
)  ./\  ( ( oc `  K ) `  Y ) )  e.  B )
451, 23latmcl 15551 . . . . . 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 1228 . . . . 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 34311 . . . . 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 1228 . . . 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 34310 . . . . . 6  |-  ( ( K  e.  OL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( oc `  K ) `  (
( ( oc `  K ) `  X
)  ./\  ( ( oc `  K ) `  Y ) ) )  =  ( X  .\/  Y ) )
5031, 12, 15, 49syl3anc 1228 . . . . 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 34310 . . . . . 6  |-  ( ( K  e.  OL  /\  X  e.  B  /\  Z  e.  B )  ->  ( ( oc `  K ) `  (
( ( oc `  K ) `  X
)  ./\  ( ( oc `  K ) `  Z ) ) )  =  ( X  .\/  Z ) )
5231, 12, 18, 51syl3anc 1228 . . . . 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 6312 . . . 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 2509 . . 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 1016 . 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 2518 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   class class class wbr 4452   ` cfv 5593  (class class class)co 6294   Basecbs 14502   occoc 14575   joincjn 15443   meetcmee 15444   Latclat 15544   OPcops 34262   cmccmtN 34263   OLcol 34264   OMLcoml 34265
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4251  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-riota 6255  df-ov 6297  df-oprab 6298  df-poset 15445  df-lub 15473  df-glb 15474  df-join 15475  df-meet 15476  df-p0 15538  df-lat 15545  df-oposet 34266  df-cmtN 34267  df-ol 34268  df-oml 34269
This theorem is referenced by: (None)
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