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Theorem rrhval 26425
Description: Value of the canonical homormorphism from the real numbers to a complete space. (Contributed by Thierry Arnoux, 2-Nov-2017.)
Hypotheses
Ref Expression
rrhval.1  |-  J  =  ( topGen `  ran  (,) )
rrhval.2  |-  K  =  ( TopOpen `  R )
Assertion
Ref Expression
rrhval  |-  ( R  e.  V  ->  (RRHom `  R )  =  ( ( JCnExt K ) `
 (QQHom `  R
) ) )

Proof of Theorem rrhval
Dummy variable  r is distinct from all other variables.
StepHypRef Expression
1 elex 2981 . 2  |-  ( R  e.  V  ->  R  e.  _V )
2 rrhval.1 . . . . . . 7  |-  J  =  ( topGen `  ran  (,) )
32eqcomi 2447 . . . . . 6  |-  ( topGen ` 
ran  (,) )  =  J
43a1i 11 . . . . 5  |-  ( r  =  R  ->  ( topGen `
 ran  (,) )  =  J )
5 fveq2 5691 . . . . . 6  |-  ( r  =  R  ->  ( TopOpen
`  r )  =  ( TopOpen `  R )
)
6 rrhval.2 . . . . . 6  |-  K  =  ( TopOpen `  R )
75, 6syl6eqr 2493 . . . . 5  |-  ( r  =  R  ->  ( TopOpen
`  r )  =  K )
84, 7oveq12d 6109 . . . 4  |-  ( r  =  R  ->  (
( topGen `  ran  (,) )CnExt ( TopOpen `  r )
)  =  ( JCnExt
K ) )
9 fveq2 5691 . . . 4  |-  ( r  =  R  ->  (QQHom `  r )  =  (QQHom `  R ) )
108, 9fveq12d 5697 . . 3  |-  ( r  =  R  ->  (
( ( topGen `  ran  (,) )CnExt ( TopOpen `  r
) ) `  (QQHom `  r ) )  =  ( ( JCnExt K
) `  (QQHom `  R
) ) )
11 df-rrh 26424 . . 3  |- RRHom  =  ( r  e.  _V  |->  ( ( ( topGen `  ran  (,) )CnExt ( TopOpen `  r
) ) `  (QQHom `  r ) ) )
12 fvex 5701 . . 3  |-  ( ( JCnExt K ) `  (QQHom `  R ) )  e.  _V
1310, 11, 12fvmpt 5774 . 2  |-  ( R  e.  _V  ->  (RRHom `  R )  =  ( ( JCnExt K ) `
 (QQHom `  R
) ) )
141, 13syl 16 1  |-  ( R  e.  V  ->  (RRHom `  R )  =  ( ( JCnExt K ) `
 (QQHom `  R
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   _Vcvv 2972   ran crn 4841   ` cfv 5418  (class class class)co 6091   (,)cioo 11300   TopOpenctopn 14360   topGenctg 14376  CnExtccnext 19631  QQHomcqqh 26401  RRHomcrrh 26422
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-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pr 4531
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-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-rrh 26424
This theorem is referenced by:  rrhcn  26426  rrhre  26447
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