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Theorem bncmet 19300
 Description: The induced metric on Banach space is complete. (Contributed by NM, 8-Sep-2007.) (Revised by Mario Carneiro, 15-Oct-2015.)
Hypotheses
Ref Expression
iscms.1
iscms.2
Assertion
Ref Expression
bncmet Ban

Proof of Theorem bncmet
StepHypRef Expression
1 bncms 19297 . 2 Ban CMetSp
2 iscms.1 . . 3
3 iscms.2 . . 3
42, 3cmscmet 19299 . 2 CMetSp
51, 4syl 16 1 Ban
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1652   wcel 1725   cxp 4876   cres 4880  cfv 5454  cbs 13469  cds 13538  cms 19207  CMetSpccms 19285  Bancbn 19286 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-nul 4338 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-xp 4884  df-res 4890  df-iota 5418  df-fv 5462  df-cms 19288  df-bn 19289
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