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Theorem rrhval 28138
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 3118 . 2  |-  ( R  e.  V  ->  R  e.  _V )
2 rrhval.1 . . . . . . 7  |-  J  =  ( topGen `  ran  (,) )
32eqcomi 2470 . . . . . 6  |-  ( topGen ` 
ran  (,) )  =  J
43a1i 11 . . . . 5  |-  ( r  =  R  ->  ( topGen `
 ran  (,) )  =  J )
5 fveq2 5872 . . . . . 6  |-  ( r  =  R  ->  ( TopOpen
`  r )  =  ( TopOpen `  R )
)
6 rrhval.2 . . . . . 6  |-  K  =  ( TopOpen `  R )
75, 6syl6eqr 2516 . . . . 5  |-  ( r  =  R  ->  ( TopOpen
`  r )  =  K )
84, 7oveq12d 6314 . . . 4  |-  ( r  =  R  ->  (
( topGen `  ran  (,) )CnExt ( TopOpen `  r )
)  =  ( JCnExt
K ) )
9 fveq2 5872 . . . 4  |-  ( r  =  R  ->  (QQHom `  r )  =  (QQHom `  R ) )
108, 9fveq12d 5878 . . 3  |-  ( r  =  R  ->  (
( ( topGen `  ran  (,) )CnExt ( TopOpen `  r
) ) `  (QQHom `  r ) )  =  ( ( JCnExt K
) `  (QQHom `  R
) ) )
11 df-rrh 28137 . . 3  |- RRHom  =  ( r  e.  _V  |->  ( ( ( topGen `  ran  (,) )CnExt ( TopOpen `  r
) ) `  (QQHom `  r ) ) )
12 fvex 5882 . . 3  |-  ( ( JCnExt K ) `  (QQHom `  R ) )  e.  _V
1310, 11, 12fvmpt 5956 . 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 1395    e. wcel 1819   _Vcvv 3109   ran crn 5009   ` cfv 5594  (class class class)co 6296   (,)cioo 11554   TopOpenctopn 14839   topGenctg 14855  CnExtccnext 20685  QQHomcqqh 28114  RRHomcrrh 28135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6299  df-rrh 28137
This theorem is referenced by:  rrhcn  28139  rrhre  28160
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