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Theorem omlfh3N 32533
Description: Foulis-Holland Theorem, part 3. Dual of omlfh1N 32532. (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 2429 . . . . . . 7  |-  ( oc
`  K )  =  ( oc `  K
)
3 omlfh1.c . . . . . . 7  |-  C  =  ( cm `  K
)
41, 2, 3cmt4N 32526 . . . . . 6  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X C Y  <-> 
( ( oc `  K ) `  X
) C ( ( oc `  K ) `
 Y ) ) )
543adant3r3 1216 . . . . 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 32526 . . . . . 6  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Z  e.  B )  ->  ( X C Z  <-> 
( ( oc `  K ) `  X
) C ( ( oc `  K ) `
 Z ) ) )
763adant3r2 1215 . . . . 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 715 . . . 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 458 . . . . 5  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  K  e.  OML )
10 omlop 32515 . . . . . . . 8  |-  ( K  e.  OML  ->  K  e.  OP )
1110adantr 466 . . . . . . 7  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  K  e.  OP )
12 simpr1 1011 . . . . . . 7  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  X  e.  B )
131, 2opoccl 32468 . . . . . . 7  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( ( oc `  K ) `  X
)  e.  B )
1411, 12, 13syl2anc 665 . . . . . 6  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( oc `  K
) `  X )  e.  B )
15 simpr2 1012 . . . . . . 7  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Y  e.  B )
161, 2opoccl 32468 . . . . . . 7  |-  ( ( K  e.  OP  /\  Y  e.  B )  ->  ( ( oc `  K ) `  Y
)  e.  B )
1711, 15, 16syl2anc 665 . . . . . 6  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( oc `  K
) `  Y )  e.  B )
18 simpr3 1013 . . . . . . 7  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Z  e.  B )
191, 2opoccl 32468 . . . . . . 7  |-  ( ( K  e.  OP  /\  Z  e.  B )  ->  ( ( oc `  K ) `  Z
)  e.  B )
2011, 18, 19syl2anc 665 . . . . . 6  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( oc `  K
) `  Z )  e.  B )
2114, 17, 203jca 1185 . . . . 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 32532 . . . . . . 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 5885 . . . . . 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 1204 . . . . 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 62 . . . 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 218 . . 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 1202 . 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 32514 . . . . . 6  |-  ( K  e.  OML  ->  K  e.  OL )
3130adantr 466 . . . . 5  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  K  e.  OL )
32 omllat 32516 . . . . . . 7  |-  ( K  e.  OML  ->  K  e.  Lat )
3332adantr 466 . . . . . 6  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  K  e.  Lat )
341, 22latjcl 16248 . . . . . 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 1264 . . . . 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 32492 . . . . 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 1264 . . . 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 32498 . . . . . 6  |-  ( ( K  e.  OL  /\  Y  e.  B  /\  Z  e.  B )  ->  ( ( oc `  K ) `  (
( ( oc `  K ) `  Y
)  .\/  ( ( oc `  K ) `  Z ) ) )  =  ( Y  ./\  Z ) )
3931, 15, 18, 38syl3anc 1264 . . . . 5  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( oc `  K
) `  ( (
( oc `  K
) `  Y )  .\/  ( ( oc `  K ) `  Z
) ) )  =  ( Y  ./\  Z
) )
4039oveq2d 6321 . . . 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 2471 . . 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 1025 . 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 16249 . . . . . 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 1264 . . . . 5  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( ( oc `  K ) `  X
)  ./\  ( ( oc `  K ) `  Y ) )  e.  B )
451, 23latmcl 16249 . . . . . 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 1264 . . . . 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 32495 . . . . 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 1264 . . . 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 32494 . . . . . 6  |-  ( ( K  e.  OL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( oc `  K ) `  (
( ( oc `  K ) `  X
)  ./\  ( ( oc `  K ) `  Y ) ) )  =  ( X  .\/  Y ) )
5031, 12, 15, 49syl3anc 1264 . . . . 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 32494 . . . . . 6  |-  ( ( K  e.  OL  /\  X  e.  B  /\  Z  e.  B )  ->  ( ( oc `  K ) `  (
( ( oc `  K ) `  X
)  ./\  ( ( oc `  K ) `  Z ) ) )  =  ( X  .\/  Z ) )
5231, 12, 18, 51syl3anc 1264 . . . . 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 6323 . . . 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 2471 . . 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 1025 . 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 2480 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870   class class class wbr 4426   ` cfv 5601  (class class class)co 6305   Basecbs 15084   occoc 15160   joincjn 16140   meetcmee 16141   Latclat 16242   OPcops 32446   cmccmtN 32447   OLcol 32448   OMLcoml 32449
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-preset 16124  df-poset 16142  df-lub 16171  df-glb 16172  df-join 16173  df-meet 16174  df-p0 16236  df-lat 16243  df-oposet 32450  df-cmtN 32451  df-ol 32452  df-oml 32453
This theorem is referenced by: (None)
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