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Theorem cmtvalN 34409
Description: Equivalence for commutes relation. Definition of commutes in [Kalmbach] p. 20. (cmbr 26325 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 34408 . . . . 5  |-  ( K  e.  A  ->  C  =  { <. x ,  y
>.  |  ( x  e.  B  /\  y  e.  B  /\  x  =  ( ( x 
./\  y )  .\/  ( x  ./\  (  ._|_  `  y ) ) ) ) } )
7 df-3an 975 . . . . . 6  |-  ( ( x  e.  B  /\  y  e.  B  /\  x  =  ( (
x  ./\  y )  .\/  ( x  ./\  (  ._|_  `  y ) ) ) )  <->  ( (
x  e.  B  /\  y  e.  B )  /\  x  =  (
( x  ./\  y
)  .\/  ( x  ./\  (  ._|_  `  y ) ) ) ) )
87opabbii 4517 . . . . 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 2524 . . . 4  |-  ( K  e.  A  ->  C  =  { <. x ,  y
>.  |  ( (
x  e.  B  /\  y  e.  B )  /\  x  =  (
( x  ./\  y
)  .\/  ( x  ./\  (  ._|_  `  y ) ) ) ) } )
109breqd 4464 . . 3  |-  ( K  e.  A  ->  ( X C Y  <->  X { <. x ,  y >.  |  ( ( x  e.  B  /\  y  e.  B )  /\  x  =  ( ( x 
./\  y )  .\/  ( x  ./\  (  ._|_  `  y ) ) ) ) } Y ) )
11103ad2ant1 1017 . 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 4454 . . . 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 6302 . . . . . . 7  |-  ( x  =  X  ->  (
x  ./\  y )  =  ( X  ./\  y ) )
15 oveq1 6302 . . . . . . 7  |-  ( x  =  X  ->  (
x  ./\  (  ._|_  `  y ) )  =  ( X  ./\  (  ._|_  `  y ) ) )
1614, 15oveq12d 6313 . . . . . 6  |-  ( x  =  X  ->  (
( x  ./\  y
)  .\/  ( x  ./\  (  ._|_  `  y ) ) )  =  ( ( X  ./\  y
)  .\/  ( X  ./\  (  ._|_  `  y ) ) ) )
1713, 16eqeq12d 2489 . . . . 5  |-  ( x  =  X  ->  (
x  =  ( ( x  ./\  y )  .\/  ( x  ./\  (  ._|_  `  y ) ) )  <->  X  =  (
( X  ./\  y
)  .\/  ( X  ./\  (  ._|_  `  y ) ) ) ) )
18 oveq2 6303 . . . . . . 7  |-  ( y  =  Y  ->  ( X  ./\  y )  =  ( X  ./\  Y
) )
19 fveq2 5872 . . . . . . . 8  |-  ( y  =  Y  ->  (  ._|_  `  y )  =  (  ._|_  `  Y ) )
2019oveq2d 6311 . . . . . . 7  |-  ( y  =  Y  ->  ( X  ./\  (  ._|_  `  y
) )  =  ( X  ./\  (  ._|_  `  Y ) ) )
2118, 20oveq12d 6313 . . . . . 6  |-  ( y  =  Y  ->  (
( X  ./\  y
)  .\/  ( X  ./\  (  ._|_  `  y ) ) )  =  ( ( X  ./\  Y
)  .\/  ( X  ./\  (  ._|_  `  Y ) ) ) )
2221eqeq2d 2481 . . . . 5  |-  ( y  =  Y  ->  ( X  =  ( ( X  ./\  y )  .\/  ( X  ./\  (  ._|_  `  y ) ) )  <-> 
X  =  ( ( X  ./\  Y )  .\/  ( X  ./\  (  ._|_  `  Y ) ) ) ) )
2317, 22opelopab2 4774 . . . 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 1014 . 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 973    = wceq 1379    e. wcel 1767   <.cop 4039   class class class wbr 4453   {copab 4510   ` cfv 5594  (class class class)co 6295   Basecbs 14507   occoc 14580   joincjn 15448   meetcmee 15449   cmccmtN 34371
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6298  df-cmtN 34375
This theorem is referenced by:  cmtcomlemN  34446  cmt2N  34448  cmtbr2N  34451  cmtbr3N  34452
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