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Theorem omllaw2N 28585
Description: Variation of orthomodular law. Definition of OML law in [Kalmbach] p. 22. (pjoml2i 22128 analog.) (Contributed by NM, 6-Nov-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
omllaw.b  |-  B  =  ( Base `  K
)
omllaw.l  |-  .<_  =  ( le `  K )
omllaw.j  |-  .\/  =  ( join `  K )
omllaw.m  |-  ./\  =  ( meet `  K )
omllaw.o  |-  ._|_  =  ( oc `  K )
Assertion
Ref Expression
omllaw2N  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  -> 
( X  .\/  (
(  ._|_  `  X )  ./\  Y ) )  =  Y ) )

Proof of Theorem omllaw2N
StepHypRef Expression
1 omllaw.b . . 3  |-  B  =  ( Base `  K
)
2 omllaw.l . . 3  |-  .<_  =  ( le `  K )
3 omllaw.j . . 3  |-  .\/  =  ( join `  K )
4 omllaw.m . . 3  |-  ./\  =  ( meet `  K )
5 omllaw.o . . 3  |-  ._|_  =  ( oc `  K )
61, 2, 3, 4, 5omllaw 28584 . 2  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  ->  Y  =  ( X  .\/  ( Y  ./\  (  ._|_  `  X ) ) ) ) )
7 eqcom 2258 . . 3  |-  ( ( X  .\/  ( ( 
._|_  `  X )  ./\  Y ) )  =  Y  <-> 
Y  =  ( X 
.\/  ( (  ._|_  `  X )  ./\  Y
) ) )
8 omllat 28583 . . . . . . 7  |-  ( K  e.  OML  ->  K  e.  Lat )
983ad2ant1 981 . . . . . 6  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  Lat )
10 omlop 28582 . . . . . . . 8  |-  ( K  e.  OML  ->  K  e.  OP )
111, 5opoccl 28535 . . . . . . . 8  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  (  ._|_  `  X )  e.  B )
1210, 11sylan 459 . . . . . . 7  |-  ( ( K  e.  OML  /\  X  e.  B )  ->  (  ._|_  `  X )  e.  B )
13123adant3 980 . . . . . 6  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  (  ._|_  `  X )  e.  B )
14 simp3 962 . . . . . 6  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  Y  e.  B )
151, 4latmcom 14129 . . . . . 6  |-  ( ( K  e.  Lat  /\  (  ._|_  `  X )  e.  B  /\  Y  e.  B )  ->  (
(  ._|_  `  X )  ./\  Y )  =  ( Y  ./\  (  ._|_  `  X ) ) )
169, 13, 14, 15syl3anc 1187 . . . . 5  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( (  ._|_  `  X
)  ./\  Y )  =  ( Y  ./\  (  ._|_  `  X )
) )
1716oveq2d 5794 . . . 4  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .\/  (
(  ._|_  `  X )  ./\  Y ) )  =  ( X  .\/  ( Y  ./\  (  ._|_  `  X
) ) ) )
1817eqeq2d 2267 . . 3  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( Y  =  ( X  .\/  ( ( 
._|_  `  X )  ./\  Y ) )  <->  Y  =  ( X  .\/  ( Y 
./\  (  ._|_  `  X
) ) ) ) )
197, 18syl5bb 250 . 2  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .\/  ( (  ._|_  `  X
)  ./\  Y )
)  =  Y  <->  Y  =  ( X  .\/  ( Y 
./\  (  ._|_  `  X
) ) ) ) )
206, 19sylibrd 227 1  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  -> 
( X  .\/  (
(  ._|_  `  X )  ./\  Y ) )  =  Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ w3a 939    = wceq 1619    e. wcel 1621   class class class wbr 3983   ` cfv 4659  (class class class)co 5778   Basecbs 13096   lecple 13163   occoc 13164   joincjn 14026   meetcmee 14027   Latclat 14099   OPcops 28513   OMLcoml 28516
This theorem is referenced by:  omllaw5N  28588  cmtcomlemN  28589  cmtbr3N  28595
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4091  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-rex 2522  df-reu 2523  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-op 3609  df-uni 3788  df-iun 3867  df-br 3984  df-opab 4038  df-mpt 4039  df-id 4267  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-1st 6042  df-2nd 6043  df-meet 14059  df-lat 14100  df-oposet 28517  df-ol 28519  df-oml 28520
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