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Theorem cmtfvalN 30022
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 2809 . 2  |-  ( K  e.  A  ->  K  e.  _V )
2 cmtfval.c . . 3  |-  C  =  ( cm `  K
)
3 fveq2 5541 . . . . . . . 8  |-  ( p  =  K  ->  ( Base `  p )  =  ( Base `  K
) )
4 cmtfval.b . . . . . . . 8  |-  B  =  ( Base `  K
)
53, 4syl6eqr 2346 . . . . . . 7  |-  ( p  =  K  ->  ( Base `  p )  =  B )
65eleq2d 2363 . . . . . 6  |-  ( p  =  K  ->  (
x  e.  ( Base `  p )  <->  x  e.  B ) )
75eleq2d 2363 . . . . . 6  |-  ( p  =  K  ->  (
y  e.  ( Base `  p )  <->  y  e.  B ) )
8 fveq2 5541 . . . . . . . . 9  |-  ( p  =  K  ->  ( join `  p )  =  ( join `  K
) )
9 cmtfval.j . . . . . . . . 9  |-  .\/  =  ( join `  K )
108, 9syl6eqr 2346 . . . . . . . 8  |-  ( p  =  K  ->  ( join `  p )  = 
.\/  )
11 fveq2 5541 . . . . . . . . . 10  |-  ( p  =  K  ->  ( meet `  p )  =  ( meet `  K
) )
12 cmtfval.m . . . . . . . . . 10  |-  ./\  =  ( meet `  K )
1311, 12syl6eqr 2346 . . . . . . . . 9  |-  ( p  =  K  ->  ( meet `  p )  = 
./\  )
1413oveqd 5891 . . . . . . . 8  |-  ( p  =  K  ->  (
x ( meet `  p
) y )  =  ( x  ./\  y
) )
15 eqidd 2297 . . . . . . . . 9  |-  ( p  =  K  ->  x  =  x )
16 fveq2 5541 . . . . . . . . . . 11  |-  ( p  =  K  ->  ( oc `  p )  =  ( oc `  K
) )
17 cmtfval.o . . . . . . . . . . 11  |-  ._|_  =  ( oc `  K )
1816, 17syl6eqr 2346 . . . . . . . . . 10  |-  ( p  =  K  ->  ( oc `  p )  = 
._|_  )
1918fveq1d 5543 . . . . . . . . 9  |-  ( p  =  K  ->  (
( oc `  p
) `  y )  =  (  ._|_  `  y
) )
2013, 15, 19oveq123d 5895 . . . . . . . 8  |-  ( p  =  K  ->  (
x ( meet `  p
) ( ( oc
`  p ) `  y ) )  =  ( x  ./\  (  ._|_  `  y ) ) )
2110, 14, 20oveq123d 5895 . . . . . . 7  |-  ( p  =  K  ->  (
( x ( meet `  p ) y ) ( join `  p
) ( x (
meet `  p )
( ( oc `  p ) `  y
) ) )  =  ( ( x  ./\  y )  .\/  (
x  ./\  (  ._|_  `  y ) ) ) )
2221eqeq2d 2307 . . . . . 6  |-  ( p  =  K  ->  (
x  =  ( ( x ( meet `  p
) y ) (
join `  p )
( x ( meet `  p ) ( ( oc `  p ) `
 y ) ) )  <->  x  =  (
( x  ./\  y
)  .\/  ( x  ./\  (  ._|_  `  y ) ) ) ) )
236, 7, 223anbi123d 1252 . . . . 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 4098 . . . 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 29989 . . . 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 936 . . . . . 6  |-  ( ( x  e.  B  /\  y  e.  B  /\  x  =  ( (
x  ./\  y )  .\/  ( x  ./\  (  ._|_  `  y ) ) ) )  <->  ( (
x  e.  B  /\  y  e.  B )  /\  x  =  (
( x  ./\  y
)  .\/  ( x  ./\  (  ._|_  `  y ) ) ) ) )
2726opabbii 4099 . . . . 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 5555 . . . . . . . 8  |-  ( Base `  K )  e.  _V
294, 28eqeltri 2366 . . . . . . 7  |-  B  e. 
_V
3029, 29xpex 4817 . . . . . 6  |-  ( B  X.  B )  e. 
_V
31 opabssxp 4778 . . . . . 6  |-  { <. x ,  y >.  |  ( ( x  e.  B  /\  y  e.  B
)  /\  x  =  ( ( x  ./\  y )  .\/  (
x  ./\  (  ._|_  `  y ) ) ) ) }  C_  ( B  X.  B )
3230, 31ssexi 4175 . . . . 5  |-  { <. x ,  y >.  |  ( ( x  e.  B  /\  y  e.  B
)  /\  x  =  ( ( x  ./\  y )  .\/  (
x  ./\  (  ._|_  `  y ) ) ) ) }  e.  _V
3327, 32eqeltri 2366 . . . 4  |-  { <. x ,  y >.  |  ( x  e.  B  /\  y  e.  B  /\  x  =  ( (
x  ./\  y )  .\/  ( x  ./\  (  ._|_  `  y ) ) ) ) }  e.  _V
3424, 25, 33fvmpt 5618 . . 3  |-  ( K  e.  _V  ->  ( cm `  K )  =  { <. x ,  y
>.  |  ( x  e.  B  /\  y  e.  B  /\  x  =  ( ( x 
./\  y )  .\/  ( x  ./\  (  ._|_  `  y ) ) ) ) } )
352, 34syl5eq 2340 . 2  |-  ( K  e.  _V  ->  C  =  { <. x ,  y
>.  |  ( x  e.  B  /\  y  e.  B  /\  x  =  ( ( x 
./\  y )  .\/  ( x  ./\  (  ._|_  `  y ) ) ) ) } )
361, 35syl 15 1  |-  ( K  e.  A  ->  C  =  { <. x ,  y
>.  |  ( x  e.  B  /\  y  e.  B  /\  x  =  ( ( x 
./\  y )  .\/  ( x  ./\  (  ._|_  `  y ) ) ) ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   _Vcvv 2801   {copab 4092    X. cxp 4703   ` cfv 5271  (class class class)co 5874   Basecbs 13164   occoc 13232   joincjn 14094   meetcmee 14095   cmccmtN 29985
This theorem is referenced by:  cmtvalN  30023
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-cmtN 29989
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