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Theorem cmtvalN 32856
Description: Equivalence for commutes relation. Definition of commutes in [Kalmbach] p. 20. (cmbr 24987 analog.) (Contributed by NM, 6-Nov-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
cmtfval.b  |-  B  =  ( Base `  K
)
cmtfval.j  |-  .\/  =  ( join `  K )
cmtfval.m  |-  ./\  =  ( meet `  K )
cmtfval.o  |-  ._|_  =  ( oc `  K )
cmtfval.c  |-  C  =  ( cm `  K
)
Assertion
Ref Expression
cmtvalN  |-  ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  B )  ->  ( X C Y  <-> 
X  =  ( ( X  ./\  Y )  .\/  ( X  ./\  (  ._|_  `  Y ) ) ) ) )

Proof of Theorem cmtvalN
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cmtfval.b . . . . . 6  |-  B  =  ( Base `  K
)
2 cmtfval.j . . . . . 6  |-  .\/  =  ( join `  K )
3 cmtfval.m . . . . . 6  |-  ./\  =  ( meet `  K )
4 cmtfval.o . . . . . 6  |-  ._|_  =  ( oc `  K )
5 cmtfval.c . . . . . 6  |-  C  =  ( cm `  K
)
61, 2, 3, 4, 5cmtfvalN 32855 . . . . 5  |-  ( K  e.  A  ->  C  =  { <. x ,  y
>.  |  ( x  e.  B  /\  y  e.  B  /\  x  =  ( ( x 
./\  y )  .\/  ( x  ./\  (  ._|_  `  y ) ) ) ) } )
7 df-3an 967 . . . . . 6  |-  ( ( x  e.  B  /\  y  e.  B  /\  x  =  ( (
x  ./\  y )  .\/  ( x  ./\  (  ._|_  `  y ) ) ) )  <->  ( (
x  e.  B  /\  y  e.  B )  /\  x  =  (
( x  ./\  y
)  .\/  ( x  ./\  (  ._|_  `  y ) ) ) ) )
87opabbii 4356 . . . . 5  |-  { <. x ,  y >.  |  ( x  e.  B  /\  y  e.  B  /\  x  =  ( (
x  ./\  y )  .\/  ( x  ./\  (  ._|_  `  y ) ) ) ) }  =  { <. x ,  y
>.  |  ( (
x  e.  B  /\  y  e.  B )  /\  x  =  (
( x  ./\  y
)  .\/  ( x  ./\  (  ._|_  `  y ) ) ) ) }
96, 8syl6eq 2491 . . . 4  |-  ( K  e.  A  ->  C  =  { <. x ,  y
>.  |  ( (
x  e.  B  /\  y  e.  B )  /\  x  =  (
( x  ./\  y
)  .\/  ( x  ./\  (  ._|_  `  y ) ) ) ) } )
109breqd 4303 . . 3  |-  ( K  e.  A  ->  ( X C Y  <->  X { <. x ,  y >.  |  ( ( x  e.  B  /\  y  e.  B )  /\  x  =  ( ( x 
./\  y )  .\/  ( x  ./\  (  ._|_  `  y ) ) ) ) } Y ) )
11103ad2ant1 1009 . 2  |-  ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  B )  ->  ( X C Y  <-> 
X { <. x ,  y >.  |  ( ( x  e.  B  /\  y  e.  B
)  /\  x  =  ( ( x  ./\  y )  .\/  (
x  ./\  (  ._|_  `  y ) ) ) ) } Y ) )
12 df-br 4293 . . . 4  |-  ( X { <. x ,  y
>.  |  ( (
x  e.  B  /\  y  e.  B )  /\  x  =  (
( x  ./\  y
)  .\/  ( x  ./\  (  ._|_  `  y ) ) ) ) } Y  <->  <. X ,  Y >.  e.  { <. x ,  y >.  |  ( ( x  e.  B  /\  y  e.  B
)  /\  x  =  ( ( x  ./\  y )  .\/  (
x  ./\  (  ._|_  `  y ) ) ) ) } )
13 id 22 . . . . . 6  |-  ( x  =  X  ->  x  =  X )
14 oveq1 6098 . . . . . . 7  |-  ( x  =  X  ->  (
x  ./\  y )  =  ( X  ./\  y ) )
15 oveq1 6098 . . . . . . 7  |-  ( x  =  X  ->  (
x  ./\  (  ._|_  `  y ) )  =  ( X  ./\  (  ._|_  `  y ) ) )
1614, 15oveq12d 6109 . . . . . 6  |-  ( x  =  X  ->  (
( x  ./\  y
)  .\/  ( x  ./\  (  ._|_  `  y ) ) )  =  ( ( X  ./\  y
)  .\/  ( X  ./\  (  ._|_  `  y ) ) ) )
1713, 16eqeq12d 2457 . . . . 5  |-  ( x  =  X  ->  (
x  =  ( ( x  ./\  y )  .\/  ( x  ./\  (  ._|_  `  y ) ) )  <->  X  =  (
( X  ./\  y
)  .\/  ( X  ./\  (  ._|_  `  y ) ) ) ) )
18 oveq2 6099 . . . . . . 7  |-  ( y  =  Y  ->  ( X  ./\  y )  =  ( X  ./\  Y
) )
19 fveq2 5691 . . . . . . . 8  |-  ( y  =  Y  ->  (  ._|_  `  y )  =  (  ._|_  `  Y ) )
2019oveq2d 6107 . . . . . . 7  |-  ( y  =  Y  ->  ( X  ./\  (  ._|_  `  y
) )  =  ( X  ./\  (  ._|_  `  Y ) ) )
2118, 20oveq12d 6109 . . . . . 6  |-  ( y  =  Y  ->  (
( X  ./\  y
)  .\/  ( X  ./\  (  ._|_  `  y ) ) )  =  ( ( X  ./\  Y
)  .\/  ( X  ./\  (  ._|_  `  Y ) ) ) )
2221eqeq2d 2454 . . . . 5  |-  ( y  =  Y  ->  ( X  =  ( ( X  ./\  y )  .\/  ( X  ./\  (  ._|_  `  y ) ) )  <-> 
X  =  ( ( X  ./\  Y )  .\/  ( X  ./\  (  ._|_  `  Y ) ) ) ) )
2317, 22opelopab2 4609 . . . 4  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( <. X ,  Y >.  e.  { <. x ,  y >.  |  ( ( x  e.  B  /\  y  e.  B
)  /\  x  =  ( ( x  ./\  y )  .\/  (
x  ./\  (  ._|_  `  y ) ) ) ) }  <->  X  =  ( ( X  ./\  Y )  .\/  ( X 
./\  (  ._|_  `  Y
) ) ) ) )
2412, 23syl5bb 257 . . 3  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( X { <. x ,  y >.  |  ( ( x  e.  B  /\  y  e.  B
)  /\  x  =  ( ( x  ./\  y )  .\/  (
x  ./\  (  ._|_  `  y ) ) ) ) } Y  <->  X  =  ( ( X  ./\  Y )  .\/  ( X 
./\  (  ._|_  `  Y
) ) ) ) )
25243adant1 1006 . 2  |-  ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  B )  ->  ( X { <. x ,  y >.  |  ( ( x  e.  B  /\  y  e.  B
)  /\  x  =  ( ( x  ./\  y )  .\/  (
x  ./\  (  ._|_  `  y ) ) ) ) } Y  <->  X  =  ( ( X  ./\  Y )  .\/  ( X 
./\  (  ._|_  `  Y
) ) ) ) )
2611, 25bitrd 253 1  |-  ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  B )  ->  ( X C Y  <-> 
X  =  ( ( X  ./\  Y )  .\/  ( X  ./\  (  ._|_  `  Y ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   <.cop 3883   class class class wbr 4292   {copab 4349   ` cfv 5418  (class class class)co 6091   Basecbs 14174   occoc 14246   joincjn 15114   meetcmee 15115   cmccmtN 32818
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-iota 5381  df-fun 5420  df-fv 5426  df-ov 6094  df-cmtN 32822
This theorem is referenced by:  cmtcomlemN  32893  cmt2N  32895  cmtbr2N  32898  cmtbr3N  32899
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