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Theorem bncmet 24855
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 24852 . 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 24854 . 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 1541   ∈ wcel 2106   Γ— cxp 5673   β†Ύ cres 5677  β€˜cfv 6540  Basecbs 17140  distcds 17202  CMetccmet 24762  CMetSpccms 24840  Bancbn 24841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-nul 5305
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-rab 3433  df-v 3476  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-xp 5681  df-res 5687  df-iota 6492  df-fv 6548  df-cms 24843  df-bn 24844
This theorem is referenced by: (None)
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