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Theorem bncmet 24714
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 24711 . 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 24713 . 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 1542   ∈ wcel 2107   Γ— cxp 5632   β†Ύ cres 5636  β€˜cfv 6497  Basecbs 17084  distcds 17143  CMetccmet 24621  CMetSpccms 24699  Bancbn 24700
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-nul 5264
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-rab 3409  df-v 3448  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-xp 5640  df-res 5646  df-iota 6449  df-fv 6505  df-cms 24702  df-bn 24703
This theorem is referenced by: (None)
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