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Theorem xrhval 27660
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 3122 . 2  |-  ( R  e.  V  ->  R  e.  _V )
2 eqidd 2468 . . . 4  |-  ( r  =  R  ->  RR*  =  RR* )
3 fveq2 5866 . . . . . 6  |-  ( r  =  R  ->  (RRHom `  r )  =  (RRHom `  R ) )
4 eqidd 2468 . . . . . 6  |-  ( r  =  R  ->  x  =  x )
53, 4fveq12d 5872 . . . . 5  |-  ( r  =  R  ->  (
(RRHom `  r ) `  x )  =  ( (RRHom `  R ) `  x ) )
6 fveq2 5866 . . . . . . . 8  |-  ( r  =  R  ->  ( lub `  r )  =  ( lub `  R
) )
7 xrhval.u . . . . . . . 8  |-  U  =  ( lub `  R
)
86, 7syl6eqr 2526 . . . . . . 7  |-  ( r  =  R  ->  ( lub `  r )  =  U )
93imaeq1d 5336 . . . . . . . 8  |-  ( r  =  R  ->  (
(RRHom `  r ) " RR )  =  ( (RRHom `  R ) " RR ) )
10 xrhval.b . . . . . . . 8  |-  B  =  ( (RRHom `  R
) " RR )
119, 10syl6eqr 2526 . . . . . . 7  |-  ( r  =  R  ->  (
(RRHom `  r ) " RR )  =  B )
128, 11fveq12d 5872 . . . . . 6  |-  ( r  =  R  ->  (
( lub `  r
) `  ( (RRHom `  r ) " RR ) )  =  ( U `  B ) )
13 fveq2 5866 . . . . . . . 8  |-  ( r  =  R  ->  ( glb `  r )  =  ( glb `  R
) )
14 xrhval.l . . . . . . . 8  |-  L  =  ( glb `  R
)
1513, 14syl6eqr 2526 . . . . . . 7  |-  ( r  =  R  ->  ( glb `  r )  =  L )
1615, 11fveq12d 5872 . . . . . 6  |-  ( r  =  R  ->  (
( glb `  r
) `  ( (RRHom `  r ) " RR ) )  =  ( L `  B ) )
1712, 16ifeq12d 3959 . . . . 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 3959 . . . 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 4525 . . 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 27659 . . 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 11217 . . . 4  |-  RR*  e.  _V
2221mptex 6131 . . 3  |-  ( x  e.  RR*  |->  if ( x  e.  RR , 
( (RRHom `  R
) `  x ) ,  if ( x  = +oo ,  ( U `
 B ) ,  ( L `  B
) ) ) )  e.  _V
2319, 20, 22fvmpt 5950 . 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 1379    e. wcel 1767   _Vcvv 3113   ifcif 3939    |-> cmpt 4505   "cima 5002   ` cfv 5588   RRcr 9491   +oocpnf 9625   RR*cxr 9627   lubclub 15429   glbcglb 15430  RRHomcrrh 27638  RR*Homcxrh 27658
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-xr 9632  df-xrh 27659
This theorem is referenced by: (None)
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