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Theorem omlfh3N 29994
Description: Foulis-Holland Theorem, part 3. Dual of omlfh1N 29993. (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 2435 . . . . . . 7  |-  ( oc
`  K )  =  ( oc `  K
)
3 omlfh1.c . . . . . . 7  |-  C  =  ( cm `  K
)
41, 2, 3cmt4N 29987 . . . . . 6  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X C Y  <-> 
( ( oc `  K ) `  X
) C ( ( oc `  K ) `
 Y ) ) )
543adant3r3 1164 . . . . 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 29987 . . . . . 6  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Z  e.  B )  ->  ( X C Z  <-> 
( ( oc `  K ) `  X
) C ( ( oc `  K ) `
 Z ) ) )
763adant3r2 1163 . . . . 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 692 . . . 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 444 . . . . 5  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  K  e.  OML )
10 omlop 29976 . . . . . . . 8  |-  ( K  e.  OML  ->  K  e.  OP )
1110adantr 452 . . . . . . 7  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  K  e.  OP )
12 simpr1 963 . . . . . . 7  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  X  e.  B )
131, 2opoccl 29929 . . . . . . 7  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( ( oc `  K ) `  X
)  e.  B )
1411, 12, 13syl2anc 643 . . . . . 6  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( oc `  K
) `  X )  e.  B )
15 simpr2 964 . . . . . . 7  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Y  e.  B )
161, 2opoccl 29929 . . . . . . 7  |-  ( ( K  e.  OP  /\  Y  e.  B )  ->  ( ( oc `  K ) `  Y
)  e.  B )
1711, 15, 16syl2anc 643 . . . . . 6  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( oc `  K
) `  Y )  e.  B )
18 simpr3 965 . . . . . . 7  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Z  e.  B )
191, 2opoccl 29929 . . . . . . 7  |-  ( ( K  e.  OP  /\  Z  e.  B )  ->  ( ( oc `  K ) `  Z
)  e.  B )
2011, 18, 19syl2anc 643 . . . . . 6  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( oc `  K
) `  Z )  e.  B )
2114, 17, 203jca 1134 . . . . 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 29993 . . . . . . 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 5724 . . . . . 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 1152 . . . . 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 58 . . . 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 207 . . 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 1150 . 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 29975 . . . . . 6  |-  ( K  e.  OML  ->  K  e.  OL )
3130adantr 452 . . . . 5  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  K  e.  OL )
32 omllat 29977 . . . . . . 7  |-  ( K  e.  OML  ->  K  e.  Lat )
3332adantr 452 . . . . . 6  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  K  e.  Lat )
341, 22latjcl 14471 . . . . . 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 1184 . . . . 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 29953 . . . . 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 1184 . . . 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 29959 . . . . . 6  |-  ( ( K  e.  OL  /\  Y  e.  B  /\  Z  e.  B )  ->  ( ( oc `  K ) `  (
( ( oc `  K ) `  Y
)  .\/  ( ( oc `  K ) `  Z ) ) )  =  ( Y  ./\  Z ) )
3931, 15, 18, 38syl3anc 1184 . . . . 5  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( oc `  K
) `  ( (
( oc `  K
) `  Y )  .\/  ( ( oc `  K ) `  Z
) ) )  =  ( Y  ./\  Z
) )
4039oveq2d 6089 . . . 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 2468 . . 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 977 . 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 14472 . . . . . 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 1184 . . . . 5  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( ( oc `  K ) `  X
)  ./\  ( ( oc `  K ) `  Y ) )  e.  B )
451, 23latmcl 14472 . . . . . 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 1184 . . . . 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 29956 . . . . 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 1184 . . . 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 29955 . . . . . 6  |-  ( ( K  e.  OL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( oc `  K ) `  (
( ( oc `  K ) `  X
)  ./\  ( ( oc `  K ) `  Y ) ) )  =  ( X  .\/  Y ) )
5031, 12, 15, 49syl3anc 1184 . . . . 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 29955 . . . . . 6  |-  ( ( K  e.  OL  /\  X  e.  B  /\  Z  e.  B )  ->  ( ( oc `  K ) `  (
( ( oc `  K ) `  X
)  ./\  ( ( oc `  K ) `  Z ) ) )  =  ( X  .\/  Z ) )
5231, 12, 18, 51syl3anc 1184 . . . . 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 6091 . . . 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 2468 . . 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 977 . 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 2477 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 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   Basecbs 13461   occoc 13529   joincjn 14393   meetcmee 14394   Latclat 14466   OPcops 29907   cmccmtN 29908   OLcol 29909   OMLcoml 29910
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-poset 14395  df-lub 14423  df-glb 14424  df-join 14425  df-meet 14426  df-p0 14460  df-lat 14467  df-oposet 29911  df-cmtN 29912  df-ol 29913  df-oml 29914
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