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Theorem iscbn 24384
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 5773 . . . 4
2 iscbn.8 . . . 4
31, 2syl6eqr 2508 . . 3
4 fveq2 5773 . . . . 5
5 iscbn.x . . . . 5
64, 5syl6eqr 2508 . . . 4
76fveq2d 5777 . . 3
83, 7eleq12d 2530 . 2
9 df-cbn 24383 . 2
108, 9elrab2 3200 1
Colors of variables: wff setvar class
Syntax hints:  <->wb 184  /\wa 369  =wceq 1370  e.wcel 1757  `cfv 5500   cms 20865   cnv 24081   cba 24083   cims 24088   ccbn 24382
This theorem is referenced by:  cbncms  24385  bnnv  24386  bnsscmcl  24388  cnbn  24389  hhhl  24725  hhssbn  24800
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-rex 2798  df-rab 2801  df-v 3054  df-dif 3413  df-un 3415  df-in 3417  df-ss 3424  df-nul 3720  df-if 3874  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4174  df-br 4375  df-iota 5463  df-fv 5508  df-cbn 24383
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