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Theorem cmtvalN 29453
Description: Equivalence for commutes relation. Definition of commutes in [Kalmbach] p. 20. (cmbr 22271 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 29452 . . . . 5  |-  ( K  e.  A  ->  C  =  { <. x ,  y
>.  |  ( x  e.  B  /\  y  e.  B  /\  x  =  ( ( x 
./\  y )  .\/  ( x  ./\  (  ._|_  `  y ) ) ) ) } )
7 df-3an 936 . . . . . 6  |-  ( ( x  e.  B  /\  y  e.  B  /\  x  =  ( (
x  ./\  y )  .\/  ( x  ./\  (  ._|_  `  y ) ) ) )  <->  ( (
x  e.  B  /\  y  e.  B )  /\  x  =  (
( x  ./\  y
)  .\/  ( x  ./\  (  ._|_  `  y ) ) ) ) )
87opabbii 4162 . . . . 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 2406 . . . 4  |-  ( K  e.  A  ->  C  =  { <. x ,  y
>.  |  ( (
x  e.  B  /\  y  e.  B )  /\  x  =  (
( x  ./\  y
)  .\/  ( x  ./\  (  ._|_  `  y ) ) ) ) } )
109breqd 4113 . . 3  |-  ( K  e.  A  ->  ( X C Y  <->  X { <. x ,  y >.  |  ( ( x  e.  B  /\  y  e.  B )  /\  x  =  ( ( x 
./\  y )  .\/  ( x  ./\  (  ._|_  `  y ) ) ) ) } Y ) )
11103ad2ant1 976 . 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 4103 . . . 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 19 . . . . . 6  |-  ( x  =  X  ->  x  =  X )
14 oveq1 5949 . . . . . . 7  |-  ( x  =  X  ->  (
x  ./\  y )  =  ( X  ./\  y ) )
15 oveq1 5949 . . . . . . 7  |-  ( x  =  X  ->  (
x  ./\  (  ._|_  `  y ) )  =  ( X  ./\  (  ._|_  `  y ) ) )
1614, 15oveq12d 5960 . . . . . 6  |-  ( x  =  X  ->  (
( x  ./\  y
)  .\/  ( x  ./\  (  ._|_  `  y ) ) )  =  ( ( X  ./\  y
)  .\/  ( X  ./\  (  ._|_  `  y ) ) ) )
1713, 16eqeq12d 2372 . . . . 5  |-  ( x  =  X  ->  (
x  =  ( ( x  ./\  y )  .\/  ( x  ./\  (  ._|_  `  y ) ) )  <->  X  =  (
( X  ./\  y
)  .\/  ( X  ./\  (  ._|_  `  y ) ) ) ) )
18 oveq2 5950 . . . . . . 7  |-  ( y  =  Y  ->  ( X  ./\  y )  =  ( X  ./\  Y
) )
19 fveq2 5605 . . . . . . . 8  |-  ( y  =  Y  ->  (  ._|_  `  y )  =  (  ._|_  `  Y ) )
2019oveq2d 5958 . . . . . . 7  |-  ( y  =  Y  ->  ( X  ./\  (  ._|_  `  y
) )  =  ( X  ./\  (  ._|_  `  Y ) ) )
2118, 20oveq12d 5960 . . . . . 6  |-  ( y  =  Y  ->  (
( X  ./\  y
)  .\/  ( X  ./\  (  ._|_  `  y ) ) )  =  ( ( X  ./\  Y
)  .\/  ( X  ./\  (  ._|_  `  Y ) ) ) )
2221eqeq2d 2369 . . . . 5  |-  ( y  =  Y  ->  ( X  =  ( ( X  ./\  y )  .\/  ( X  ./\  (  ._|_  `  y ) ) )  <-> 
X  =  ( ( X  ./\  Y )  .\/  ( X  ./\  (  ._|_  `  Y ) ) ) ) )
2317, 22opelopab2 4364 . . . 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 248 . . 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 973 . 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 244 1  |-  ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  B )  ->  ( X C Y  <-> 
X  =  ( ( X  ./\  Y )  .\/  ( X  ./\  (  ._|_  `  Y ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1642    e. wcel 1710   <.cop 3719   class class class wbr 4102   {copab 4155   ` cfv 5334  (class class class)co 5942   Basecbs 13239   occoc 13307   joincjn 14171   meetcmee 14172   cmccmtN 29415
This theorem is referenced by:  cmtcomlemN  29490  cmt2N  29492  cmtbr2N  29495  cmtbr3N  29496
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-iota 5298  df-fun 5336  df-fv 5342  df-ov 5945  df-cmtN 29419
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