Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  xrhval Structured version   Unicode version

Theorem xrhval 26449
Description: The value of the embedding from the extended real numbers into a complete lattice. (Contributed by Thierry Arnoux, 19-Feb-2018.)
Hypotheses
Ref Expression
xrhval.b  |-  B  =  ( (RRHom `  R
) " RR )
xrhval.l  |-  L  =  ( glb `  R
)
xrhval.u  |-  U  =  ( lub `  R
)
Assertion
Ref Expression
xrhval  |-  ( R  e.  V  ->  (RR*Hom `  R )  =  ( x  e.  RR*  |->  if ( x  e.  RR , 
( (RRHom `  R
) `  x ) ,  if ( x  = +oo ,  ( U `
 B ) ,  ( L `  B
) ) ) ) )
Distinct variable group:    x, R
Allowed substitution hints:    B( x)    U( x)    L( x)    V( x)

Proof of Theorem xrhval
Dummy variable  r is distinct from all other variables.
StepHypRef Expression
1 elex 2986 . 2  |-  ( R  e.  V  ->  R  e.  _V )
2 eqidd 2444 . . . 4  |-  ( r  =  R  ->  RR*  =  RR* )
3 fveq2 5696 . . . . . 6  |-  ( r  =  R  ->  (RRHom `  r )  =  (RRHom `  R ) )
4 eqidd 2444 . . . . . 6  |-  ( r  =  R  ->  x  =  x )
53, 4fveq12d 5702 . . . . 5  |-  ( r  =  R  ->  (
(RRHom `  r ) `  x )  =  ( (RRHom `  R ) `  x ) )
6 fveq2 5696 . . . . . . . 8  |-  ( r  =  R  ->  ( lub `  r )  =  ( lub `  R
) )
7 xrhval.u . . . . . . . 8  |-  U  =  ( lub `  R
)
86, 7syl6eqr 2493 . . . . . . 7  |-  ( r  =  R  ->  ( lub `  r )  =  U )
93imaeq1d 5173 . . . . . . . 8  |-  ( r  =  R  ->  (
(RRHom `  r ) " RR )  =  ( (RRHom `  R ) " RR ) )
10 xrhval.b . . . . . . . 8  |-  B  =  ( (RRHom `  R
) " RR )
119, 10syl6eqr 2493 . . . . . . 7  |-  ( r  =  R  ->  (
(RRHom `  r ) " RR )  =  B )
128, 11fveq12d 5702 . . . . . 6  |-  ( r  =  R  ->  (
( lub `  r
) `  ( (RRHom `  r ) " RR ) )  =  ( U `  B ) )
13 fveq2 5696 . . . . . . . 8  |-  ( r  =  R  ->  ( glb `  r )  =  ( glb `  R
) )
14 xrhval.l . . . . . . . 8  |-  L  =  ( glb `  R
)
1513, 14syl6eqr 2493 . . . . . . 7  |-  ( r  =  R  ->  ( glb `  r )  =  L )
1615, 11fveq12d 5702 . . . . . 6  |-  ( r  =  R  ->  (
( glb `  r
) `  ( (RRHom `  r ) " RR ) )  =  ( L `  B ) )
1712, 16ifeq12d 3814 . . . . 5  |-  ( r  =  R  ->  if ( x  = +oo ,  ( ( lub `  r ) `  (
(RRHom `  r ) " RR ) ) ,  ( ( glb `  r
) `  ( (RRHom `  r ) " RR ) ) )  =  if ( x  = +oo ,  ( U `
 B ) ,  ( L `  B
) ) )
185, 17ifeq12d 3814 . . . 4  |-  ( r  =  R  ->  if ( x  e.  RR ,  ( (RRHom `  r ) `  x
) ,  if ( x  = +oo , 
( ( lub `  r
) `  ( (RRHom `  r ) " RR ) ) ,  ( ( glb `  r
) `  ( (RRHom `  r ) " RR ) ) ) )  =  if ( x  e.  RR ,  ( (RRHom `  R ) `  x ) ,  if ( x  = +oo ,  ( U `  B ) ,  ( L `  B ) ) ) )
192, 18mpteq12dv 4375 . . 3  |-  ( r  =  R  ->  (
x  e.  RR*  |->  if ( x  e.  RR , 
( (RRHom `  r
) `  x ) ,  if ( x  = +oo ,  ( ( lub `  r ) `
 ( (RRHom `  r ) " RR ) ) ,  ( ( glb `  r
) `  ( (RRHom `  r ) " RR ) ) ) ) )  =  ( x  e.  RR*  |->  if ( x  e.  RR , 
( (RRHom `  R
) `  x ) ,  if ( x  = +oo ,  ( U `
 B ) ,  ( L `  B
) ) ) ) )
20 df-xrh 26448 . . 3  |- RR*Hom  =  ( r  e.  _V  |->  ( x  e.  RR*  |->  if ( x  e.  RR , 
( (RRHom `  r
) `  x ) ,  if ( x  = +oo ,  ( ( lub `  r ) `
 ( (RRHom `  r ) " RR ) ) ,  ( ( glb `  r
) `  ( (RRHom `  r ) " RR ) ) ) ) ) )
21 xrex 10993 . . . 4  |-  RR*  e.  _V
2221mptex 5953 . . 3  |-  ( x  e.  RR*  |->  if ( x  e.  RR , 
( (RRHom `  R
) `  x ) ,  if ( x  = +oo ,  ( U `
 B ) ,  ( L `  B
) ) ) )  e.  _V
2319, 20, 22fvmpt 5779 . 2  |-  ( R  e.  _V  ->  (RR*Hom `  R )  =  ( x  e.  RR*  |->  if ( x  e.  RR , 
( (RRHom `  R
) `  x ) ,  if ( x  = +oo ,  ( U `
 B ) ,  ( L `  B
) ) ) ) )
241, 23syl 16 1  |-  ( R  e.  V  ->  (RR*Hom `  R )  =  ( x  e.  RR*  |->  if ( x  e.  RR , 
( (RRHom `  R
) `  x ) ,  if ( x  = +oo ,  ( U `
 B ) ,  ( L `  B
) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   _Vcvv 2977   ifcif 3796    e. cmpt 4355   "cima 4848   ` cfv 5423   RRcr 9286   +oocpnf 9420   RR*cxr 9422   lubclub 15117   glbcglb 15118  RRHomcrrh 26427  RR*Homcxrh 26447
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-rep 4408  ax-sep 4418  ax-nul 4426  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344
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 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-xr 9427  df-xrh 26448
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator