Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cmtvalN Unicode version

Theorem cmtvalN 28668
Description: Equivalence for commutes relation. Definition of commutes in [Kalmbach] p. 20. (cmbr 22155 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 ) ) ) ) )
Dummy variables  x  y are mutually distinct and distinct from all other variables.

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 28667 . . . . 5  |-  ( K  e.  A  ->  C  =  { <. x ,  y
>.  |  ( x  e.  B  /\  y  e.  B  /\  x  =  ( ( x 
./\  y )  .\/  ( x  ./\  (  ._|_  `  y ) ) ) ) } )
7 df-3an 938 . . . . . 6  |-  ( ( x  e.  B  /\  y  e.  B  /\  x  =  ( (
x  ./\  y )  .\/  ( x  ./\  (  ._|_  `  y ) ) ) )  <->  ( (
x  e.  B  /\  y  e.  B )  /\  x  =  (
( x  ./\  y
)  .\/  ( x  ./\  (  ._|_  `  y ) ) ) ) )
87opabbii 4084 . . . . 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 2332 . . . 4  |-  ( K  e.  A  ->  C  =  { <. x ,  y
>.  |  ( (
x  e.  B  /\  y  e.  B )  /\  x  =  (
( x  ./\  y
)  .\/  ( x  ./\  (  ._|_  `  y ) ) ) ) } )
109breqd 4035 . . 3  |-  ( K  e.  A  ->  ( X C Y  <->  X { <. x ,  y >.  |  ( ( x  e.  B  /\  y  e.  B )  /\  x  =  ( ( x 
./\  y )  .\/  ( x  ./\  (  ._|_  `  y ) ) ) ) } Y ) )
11103ad2ant1 978 . 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 4025 . . . 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 5826 . . . . . . 7  |-  ( x  =  X  ->  (
x  ./\  y )  =  ( X  ./\  y ) )
15 oveq1 5826 . . . . . . 7  |-  ( x  =  X  ->  (
x  ./\  (  ._|_  `  y ) )  =  ( X  ./\  (  ._|_  `  y ) ) )
1614, 15oveq12d 5837 . . . . . 6  |-  ( x  =  X  ->  (
( x  ./\  y
)  .\/  ( x  ./\  (  ._|_  `  y ) ) )  =  ( ( X  ./\  y
)  .\/  ( X  ./\  (  ._|_  `  y ) ) ) )
1713, 16eqeq12d 2298 . . . . 5  |-  ( x  =  X  ->  (
x  =  ( ( x  ./\  y )  .\/  ( x  ./\  (  ._|_  `  y ) ) )  <->  X  =  (
( X  ./\  y
)  .\/  ( X  ./\  (  ._|_  `  y ) ) ) ) )
18 oveq2 5827 . . . . . . 7  |-  ( y  =  Y  ->  ( X  ./\  y )  =  ( X  ./\  Y
) )
19 fveq2 5485 . . . . . . . 8  |-  ( y  =  Y  ->  (  ._|_  `  y )  =  (  ._|_  `  Y ) )
2019oveq2d 5835 . . . . . . 7  |-  ( y  =  Y  ->  ( X  ./\  (  ._|_  `  y
) )  =  ( X  ./\  (  ._|_  `  Y ) ) )
2118, 20oveq12d 5837 . . . . . 6  |-  ( y  =  Y  ->  (
( X  ./\  y
)  .\/  ( X  ./\  (  ._|_  `  y ) ) )  =  ( ( X  ./\  Y
)  .\/  ( X  ./\  (  ._|_  `  Y ) ) ) )
2221eqeq2d 2295 . . . . 5  |-  ( y  =  Y  ->  ( X  =  ( ( X  ./\  y )  .\/  ( X  ./\  (  ._|_  `  y ) ) )  <-> 
X  =  ( ( X  ./\  Y )  .\/  ( X  ./\  (  ._|_  `  Y ) ) ) ) )
2317, 22opelopab2 4284 . . . 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 975 . 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 936    = wceq 1624    e. wcel 1685   <.cop 3644   class class class wbr 4024   {copab 4077   ` cfv 5221  (class class class)co 5819   Basecbs 13142   occoc 13210   joincjn 14072   meetcmee 14073   cmccmtN 28630
This theorem is referenced by:  cmtcomlemN  28705  cmt2N  28707  cmtbr2N  28710  cmtbr3N  28711
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fv 5229  df-ov 5822  df-cmtN 28634
  Copyright terms: Public domain W3C validator