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Theorem cmtfvalN 32855
Description: Value of commutes relation. (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
cmtfvalN  |-  ( K  e.  A  ->  C  =  { <. x ,  y
>.  |  ( x  e.  B  /\  y  e.  B  /\  x  =  ( ( x 
./\  y )  .\/  ( x  ./\  (  ._|_  `  y ) ) ) ) } )
Distinct variable groups:    x, y, B    x, K, y
Allowed substitution hints:    A( x, y)    C( x, y)    .\/ ( x, y)    ./\ (
x, y)    ._|_ ( x, y)

Proof of Theorem cmtfvalN
Dummy variable  p is distinct from all other variables.
StepHypRef Expression
1 elex 2981 . 2  |-  ( K  e.  A  ->  K  e.  _V )
2 cmtfval.c . . 3  |-  C  =  ( cm `  K
)
3 fveq2 5691 . . . . . . . 8  |-  ( p  =  K  ->  ( Base `  p )  =  ( Base `  K
) )
4 cmtfval.b . . . . . . . 8  |-  B  =  ( Base `  K
)
53, 4syl6eqr 2493 . . . . . . 7  |-  ( p  =  K  ->  ( Base `  p )  =  B )
65eleq2d 2510 . . . . . 6  |-  ( p  =  K  ->  (
x  e.  ( Base `  p )  <->  x  e.  B ) )
75eleq2d 2510 . . . . . 6  |-  ( p  =  K  ->  (
y  e.  ( Base `  p )  <->  y  e.  B ) )
8 fveq2 5691 . . . . . . . . 9  |-  ( p  =  K  ->  ( join `  p )  =  ( join `  K
) )
9 cmtfval.j . . . . . . . . 9  |-  .\/  =  ( join `  K )
108, 9syl6eqr 2493 . . . . . . . 8  |-  ( p  =  K  ->  ( join `  p )  = 
.\/  )
11 fveq2 5691 . . . . . . . . . 10  |-  ( p  =  K  ->  ( meet `  p )  =  ( meet `  K
) )
12 cmtfval.m . . . . . . . . . 10  |-  ./\  =  ( meet `  K )
1311, 12syl6eqr 2493 . . . . . . . . 9  |-  ( p  =  K  ->  ( meet `  p )  = 
./\  )
1413oveqd 6108 . . . . . . . 8  |-  ( p  =  K  ->  (
x ( meet `  p
) y )  =  ( x  ./\  y
) )
15 eqidd 2444 . . . . . . . . 9  |-  ( p  =  K  ->  x  =  x )
16 fveq2 5691 . . . . . . . . . . 11  |-  ( p  =  K  ->  ( oc `  p )  =  ( oc `  K
) )
17 cmtfval.o . . . . . . . . . . 11  |-  ._|_  =  ( oc `  K )
1816, 17syl6eqr 2493 . . . . . . . . . 10  |-  ( p  =  K  ->  ( oc `  p )  = 
._|_  )
1918fveq1d 5693 . . . . . . . . 9  |-  ( p  =  K  ->  (
( oc `  p
) `  y )  =  (  ._|_  `  y
) )
2013, 15, 19oveq123d 6112 . . . . . . . 8  |-  ( p  =  K  ->  (
x ( meet `  p
) ( ( oc
`  p ) `  y ) )  =  ( x  ./\  (  ._|_  `  y ) ) )
2110, 14, 20oveq123d 6112 . . . . . . 7  |-  ( p  =  K  ->  (
( x ( meet `  p ) y ) ( join `  p
) ( x (
meet `  p )
( ( oc `  p ) `  y
) ) )  =  ( ( x  ./\  y )  .\/  (
x  ./\  (  ._|_  `  y ) ) ) )
2221eqeq2d 2454 . . . . . 6  |-  ( p  =  K  ->  (
x  =  ( ( x ( meet `  p
) y ) (
join `  p )
( x ( meet `  p ) ( ( oc `  p ) `
 y ) ) )  <->  x  =  (
( x  ./\  y
)  .\/  ( x  ./\  (  ._|_  `  y ) ) ) ) )
236, 7, 223anbi123d 1289 . . . . 5  |-  ( p  =  K  ->  (
( x  e.  (
Base `  p )  /\  y  e.  ( Base `  p )  /\  x  =  ( (
x ( meet `  p
) y ) (
join `  p )
( x ( meet `  p ) ( ( oc `  p ) `
 y ) ) ) )  <->  ( x  e.  B  /\  y  e.  B  /\  x  =  ( ( x 
./\  y )  .\/  ( x  ./\  (  ._|_  `  y ) ) ) ) ) )
2423opabbidv 4355 . . . 4  |-  ( p  =  K  ->  { <. x ,  y >.  |  ( x  e.  ( Base `  p )  /\  y  e.  ( Base `  p
)  /\  x  =  ( ( x (
meet `  p )
y ) ( join `  p ) ( x ( meet `  p
) ( ( oc
`  p ) `  y ) ) ) ) }  =  { <. x ,  y >.  |  ( x  e.  B  /\  y  e.  B  /\  x  =  ( ( x  ./\  y )  .\/  (
x  ./\  (  ._|_  `  y ) ) ) ) } )
25 df-cmtN 32822 . . . 4  |-  cm  =  ( p  e.  _V  |->  { <. x ,  y
>.  |  ( x  e.  ( Base `  p
)  /\  y  e.  ( Base `  p )  /\  x  =  (
( x ( meet `  p ) y ) ( join `  p
) ( x (
meet `  p )
( ( oc `  p ) `  y
) ) ) ) } )
26 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 ) ) ) ) )
2726opabbii 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 ) ) ) ) }
28 fvex 5701 . . . . . . . 8  |-  ( Base `  K )  e.  _V
294, 28eqeltri 2513 . . . . . . 7  |-  B  e. 
_V
3029, 29xpex 6508 . . . . . 6  |-  ( B  X.  B )  e. 
_V
31 opabssxp 4911 . . . . . 6  |-  { <. x ,  y >.  |  ( ( x  e.  B  /\  y  e.  B
)  /\  x  =  ( ( x  ./\  y )  .\/  (
x  ./\  (  ._|_  `  y ) ) ) ) }  C_  ( B  X.  B )
3230, 31ssexi 4437 . . . . 5  |-  { <. x ,  y >.  |  ( ( x  e.  B  /\  y  e.  B
)  /\  x  =  ( ( x  ./\  y )  .\/  (
x  ./\  (  ._|_  `  y ) ) ) ) }  e.  _V
3327, 32eqeltri 2513 . . . 4  |-  { <. x ,  y >.  |  ( x  e.  B  /\  y  e.  B  /\  x  =  ( (
x  ./\  y )  .\/  ( x  ./\  (  ._|_  `  y ) ) ) ) }  e.  _V
3424, 25, 33fvmpt 5774 . . 3  |-  ( K  e.  _V  ->  ( cm `  K )  =  { <. x ,  y
>.  |  ( x  e.  B  /\  y  e.  B  /\  x  =  ( ( x 
./\  y )  .\/  ( x  ./\  (  ._|_  `  y ) ) ) ) } )
352, 34syl5eq 2487 . 2  |-  ( K  e.  _V  ->  C  =  { <. x ,  y
>.  |  ( x  e.  B  /\  y  e.  B  /\  x  =  ( ( x 
./\  y )  .\/  ( x  ./\  (  ._|_  `  y ) ) ) ) } )
361, 35syl 16 1  |-  ( K  e.  A  ->  C  =  { <. x ,  y
>.  |  ( x  e.  B  /\  y  e.  B  /\  x  =  ( ( x 
./\  y )  .\/  ( x  ./\  (  ._|_  `  y ) ) ) ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   _Vcvv 2972   {copab 4349    X. cxp 4838   ` 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:  cmtvalN  32856
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