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Theorem cmtvalN 28531
Description: Equivalence for commutes relation. Definition of commutes in [Kalmbach] p. 20. (cmbr 22106 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
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 28530 . . . . 5  |-  ( K  e.  A  ->  C  =  { <. x ,  y
>.  |  ( x  e.  B  /\  y  e.  B  /\  x  =  ( ( x 
./\  y )  .\/  ( x  ./\  (  ._|_  `  y ) ) ) ) } )
7 df-3an 941 . . . . . 6  |-  ( ( x  e.  B  /\  y  e.  B  /\  x  =  ( (
x  ./\  y )  .\/  ( x  ./\  (  ._|_  `  y ) ) ) )  <->  ( (
x  e.  B  /\  y  e.  B )  /\  x  =  (
( x  ./\  y
)  .\/  ( x  ./\  (  ._|_  `  y ) ) ) ) )
87opabbii 4023 . . . . 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 2304 . . . 4  |-  ( K  e.  A  ->  C  =  { <. x ,  y
>.  |  ( (
x  e.  B  /\  y  e.  B )  /\  x  =  (
( x  ./\  y
)  .\/  ( x  ./\  (  ._|_  `  y ) ) ) ) } )
109breqd 3974 . . 3  |-  ( K  e.  A  ->  ( X C Y  <->  X { <. x ,  y >.  |  ( ( x  e.  B  /\  y  e.  B )  /\  x  =  ( ( x 
./\  y )  .\/  ( x  ./\  (  ._|_  `  y ) ) ) ) } Y ) )
11103ad2ant1 981 . 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 3964 . . . 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 21 . . . . . 6  |-  ( x  =  X  ->  x  =  X )
14 oveq1 5764 . . . . . . 7  |-  ( x  =  X  ->  (
x  ./\  y )  =  ( X  ./\  y ) )
15 oveq1 5764 . . . . . . 7  |-  ( x  =  X  ->  (
x  ./\  (  ._|_  `  y ) )  =  ( X  ./\  (  ._|_  `  y ) ) )
1614, 15oveq12d 5775 . . . . . 6  |-  ( x  =  X  ->  (
( x  ./\  y
)  .\/  ( x  ./\  (  ._|_  `  y ) ) )  =  ( ( X  ./\  y
)  .\/  ( X  ./\  (  ._|_  `  y ) ) ) )
1713, 16eqeq12d 2270 . . . . 5  |-  ( x  =  X  ->  (
x  =  ( ( x  ./\  y )  .\/  ( x  ./\  (  ._|_  `  y ) ) )  <->  X  =  (
( X  ./\  y
)  .\/  ( X  ./\  (  ._|_  `  y ) ) ) ) )
18 oveq2 5765 . . . . . . 7  |-  ( y  =  Y  ->  ( X  ./\  y )  =  ( X  ./\  Y
) )
19 fveq2 5423 . . . . . . . 8  |-  ( y  =  Y  ->  (  ._|_  `  y )  =  (  ._|_  `  Y ) )
2019oveq2d 5773 . . . . . . 7  |-  ( y  =  Y  ->  ( X  ./\  (  ._|_  `  y
) )  =  ( X  ./\  (  ._|_  `  Y ) ) )
2118, 20oveq12d 5775 . . . . . 6  |-  ( y  =  Y  ->  (
( X  ./\  y
)  .\/  ( X  ./\  (  ._|_  `  y ) ) )  =  ( ( X  ./\  Y
)  .\/  ( X  ./\  (  ._|_  `  Y ) ) ) )
2221eqeq2d 2267 . . . . 5  |-  ( y  =  Y  ->  ( X  =  ( ( X  ./\  y )  .\/  ( X  ./\  (  ._|_  `  y ) ) )  <-> 
X  =  ( ( X  ./\  Y )  .\/  ( X  ./\  (  ._|_  `  Y ) ) ) ) )
2317, 22opelopab2 4222 . . . 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 250 . . 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 978 . 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 246 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 6    <-> wb 178    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621   <.cop 3584   class class class wbr 3963   {copab 4016   ` cfv 4638  (class class class)co 5757   Basecbs 13075   occoc 13143   joincjn 14005   meetcmee 14006   cmccmtN 28493
This theorem is referenced by:  cmtcomlemN  28568  cmt2N  28570  cmtbr2N  28573  cmtbr3N  28574
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-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449
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 2520  df-rex 2521  df-rab 2523  df-v 2742  df-sbc 2936  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-br 3964  df-opab 4018  df-mpt 4019  df-id 4246  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fv 4654  df-ov 5760  df-cmtN 28497
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