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Theorem iscbn 22371
 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 5731 . . . 4
2 iscbn.8 . . . 4
31, 2syl6eqr 2488 . . 3
4 fveq2 5731 . . . . 5
5 iscbn.x . . . . 5
64, 5syl6eqr 2488 . . . 4
76fveq2d 5735 . . 3
83, 7eleq12d 2506 . 2
9 df-cbn 22370 . 2
108, 9elrab2 3096 1
 Colors of variables: wff set class Syntax hints:   wb 178   wa 360   wceq 1653   wcel 1726  cfv 5457  cms 19212  cnv 22068  cba 22070  cims 22075  ccbn 22369 This theorem is referenced by:  cbncms  22372  bnnv  22373  bnsscmcl  22375  cnbn  22376  hhhl  22711  hhssbn  22785 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-iota 5421  df-fv 5465  df-cbn 22370
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