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Theorem omlfh3N 30071
Description: Foulis-Holland Theorem, part 3. Dual of omlfh1N 30070. (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 2296 . . . . . . 7  |-  ( oc
`  K )  =  ( oc `  K
)
3 omlfh1.c . . . . . . 7  |-  C  =  ( cm `  K
)
41, 2, 3cmt4N 30064 . . . . . 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 30064 . . . . . 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 30053 . . . . . . . 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 30006 . . . . . . 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 30006 . . . . . . 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 30006 . . . . . . 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 30070 . . . . . . 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 5545 . . . . . 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 30052 . . . . . 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 30054 . . . . . . 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 14172 . . . . . 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 30030 . . . . 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 30036 . . . . . 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 5890 . . . 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 2329 . . 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 14173 . . . . . 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 14173 . . . . . 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 30033 . . . . 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 30032 . . . . . 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 30032 . . . . . 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 5892 . . . 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 2329 . . 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 2338 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 1632    e. wcel 1696   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   occoc 13232   joincjn 14094   meetcmee 14095   Latclat 14167   OPcops 29984   cmccmtN 29985   OLcol 29986   OMLcoml 29987
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-lat 14168  df-oposet 29988  df-cmtN 29989  df-ol 29990  df-oml 29991
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