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Theorem xrhval 28661
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 3096 . 2  |-  ( R  e.  V  ->  R  e.  _V )
2 fveq2 5881 . . . . . 6  |-  ( r  =  R  ->  (RRHom `  r )  =  (RRHom `  R ) )
32fveq1d 5883 . . . . 5  |-  ( r  =  R  ->  (
(RRHom `  r ) `  x )  =  ( (RRHom `  R ) `  x ) )
4 fveq2 5881 . . . . . . . 8  |-  ( r  =  R  ->  ( lub `  r )  =  ( lub `  R
) )
5 xrhval.u . . . . . . . 8  |-  U  =  ( lub `  R
)
64, 5syl6eqr 2488 . . . . . . 7  |-  ( r  =  R  ->  ( lub `  r )  =  U )
72imaeq1d 5187 . . . . . . . 8  |-  ( r  =  R  ->  (
(RRHom `  r ) " RR )  =  ( (RRHom `  R ) " RR ) )
8 xrhval.b . . . . . . . 8  |-  B  =  ( (RRHom `  R
) " RR )
97, 8syl6eqr 2488 . . . . . . 7  |-  ( r  =  R  ->  (
(RRHom `  r ) " RR )  =  B )
106, 9fveq12d 5887 . . . . . 6  |-  ( r  =  R  ->  (
( lub `  r
) `  ( (RRHom `  r ) " RR ) )  =  ( U `  B ) )
11 fveq2 5881 . . . . . . . 8  |-  ( r  =  R  ->  ( glb `  r )  =  ( glb `  R
) )
12 xrhval.l . . . . . . . 8  |-  L  =  ( glb `  R
)
1311, 12syl6eqr 2488 . . . . . . 7  |-  ( r  =  R  ->  ( glb `  r )  =  L )
1413, 9fveq12d 5887 . . . . . 6  |-  ( r  =  R  ->  (
( glb `  r
) `  ( (RRHom `  r ) " RR ) )  =  ( L `  B ) )
1510, 14ifeq12d 3935 . . . . 5  |-  ( r  =  R  ->  if ( x  = +oo ,  ( ( lub `  r ) `  (
(RRHom `  r ) " RR ) ) ,  ( ( glb `  r
) `  ( (RRHom `  r ) " RR ) ) )  =  if ( x  = +oo ,  ( U `
 B ) ,  ( L `  B
) ) )
163, 15ifeq12d 3935 . . . 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 ) ) ) )
1716mpteq2dv 4513 . . 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
) ) ) ) )
18 df-xrh 28660 . . 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 ) ) ) ) ) )
19 xrex 11299 . . . 4  |-  RR*  e.  _V
2019mptex 6151 . . 3  |-  ( x  e.  RR*  |->  if ( x  e.  RR , 
( (RRHom `  R
) `  x ) ,  if ( x  = +oo ,  ( U `
 B ) ,  ( L `  B
) ) ) )  e.  _V
2117, 18, 20fvmpt 5964 . 2  |-  ( R  e.  _V  ->  (RR*Hom `  R )  =  ( x  e.  RR*  |->  if ( x  e.  RR , 
( (RRHom `  R
) `  x ) ,  if ( x  = +oo ,  ( U `
 B ) ,  ( L `  B
) ) ) ) )
221, 21syl 17 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 1437    e. wcel 1870   _Vcvv 3087   ifcif 3915    |-> cmpt 4484   "cima 4857   ` cfv 5601   RRcr 9537   +oocpnf 9671   RR*cxr 9673   lubclub 16138   glbcglb 16139  RRHomcrrh 28636  RR*Homcxrh 28659
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-xr 9678  df-xrh 28660
This theorem is referenced by: (None)
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