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Theorem bncmet 25226
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 𝑋 = (Baseβ€˜π‘€)
iscms.2 𝐷 = ((distβ€˜π‘€) β†Ύ (𝑋 Γ— 𝑋))
Assertion
Ref Expression
bncmet (𝑀 ∈ Ban β†’ 𝐷 ∈ (CMetβ€˜π‘‹))

Proof of Theorem bncmet
StepHypRef Expression
1 bncms 25223 . 2 (𝑀 ∈ Ban β†’ 𝑀 ∈ CMetSp)
2 iscms.1 . . 3 𝑋 = (Baseβ€˜π‘€)
3 iscms.2 . . 3 𝐷 = ((distβ€˜π‘€) β†Ύ (𝑋 Γ— 𝑋))
42, 3cmscmet 25225 . 2 (𝑀 ∈ CMetSp β†’ 𝐷 ∈ (CMetβ€˜π‘‹))
51, 4syl 17 1 (𝑀 ∈ Ban β†’ 𝐷 ∈ (CMetβ€˜π‘‹))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1533   ∈ wcel 2098   Γ— cxp 5667   β†Ύ cres 5671  β€˜cfv 6536  Basecbs 17151  distcds 17213  CMetccmet 25133  CMetSpccms 25211  Bancbn 25212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-nul 5299
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-xp 5675  df-res 5681  df-iota 6488  df-fv 6544  df-cms 25214  df-bn 25215
This theorem is referenced by: (None)
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