Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  iscbn Structured version   Unicode version

Theorem iscbn 25603
 Description: A complex Banach space is a normed complex vector space with a complete induced metric. (Contributed by NM, 5-Dec-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
iscbn.x
iscbn.8
Assertion
Ref Expression
iscbn

Proof of Theorem iscbn
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 fveq2 5872 . . . 4
2 iscbn.8 . . . 4
31, 2syl6eqr 2526 . . 3
4 fveq2 5872 . . . . 5
5 iscbn.x . . . . 5
64, 5syl6eqr 2526 . . . 4
76fveq2d 5876 . . 3
83, 7eleq12d 2549 . 2
9 df-cbn 25602 . 2
108, 9elrab2 3268 1
 Colors of variables: wff setvar class Syntax hints:   wb 184   wa 369   wceq 1379   wcel 1767  cfv 5594  cms 21561  cnv 25300  cba 25302  cims 25307  ccbn 25601 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-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445 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-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-iota 5557  df-fv 5602  df-cbn 25602 This theorem is referenced by:  cbncms  25604  bnnv  25605  bnsscmcl  25607  cnbn  25608  hhhl  25944  hhssbn  26019
 Copyright terms: Public domain W3C validator