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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
rrhval.2
Assertion
Ref Expression
rrhval RRHom CnExtQQHom

Proof of Theorem rrhval
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 elex 3118 . 2
2 rrhval.1 . . . . . . 7
32eqcomi 2470 . . . . . 6
43a1i 11 . . . . 5
5 fveq2 5872 . . . . . 6
6 rrhval.2 . . . . . 6
75, 6syl6eqr 2516 . . . . 5
84, 7oveq12d 6314 . . . 4 CnExt CnExt
9 fveq2 5872 . . . 4 QQHom QQHom
108, 9fveq12d 5878 . . 3 CnExtQQHom CnExtQQHom
11 df-rrh 28137 . . 3 RRHom CnExtQQHom
12 fvex 5882 . . 3 CnExtQQHom
1310, 11, 12fvmpt 5956 . 2 RRHom CnExtQQHom
141, 13syl 16 1 RRHom CnExtQQHom
 Colors of variables: wff setvar class Syntax hints:   wi 4   wceq 1395   wcel 1819  cvv 3109   crn 5009  cfv 5594  (class class class)co 6296  cioo 11554  ctopn 14839  ctg 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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