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Mirrors > Home > MPE Home > Th. List > bncms | Structured version Visualization version GIF version |
Description: A Banach space is a complete metric space. (Contributed by Mario Carneiro, 15-Oct-2015.) |
Ref | Expression |
---|---|
bncms | ⊢ (𝑊 ∈ Ban → 𝑊 ∈ CMetSp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2821 | . . 3 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
2 | 1 | isbn 23935 | . 2 ⊢ (𝑊 ∈ Ban ↔ (𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp ∧ (Scalar‘𝑊) ∈ CMetSp)) |
3 | 2 | simp2bi 1142 | 1 ⊢ (𝑊 ∈ Ban → 𝑊 ∈ CMetSp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 ‘cfv 6349 Scalarcsca 16562 NrmVeccnvc 23185 CMetSpccms 23929 Bancbn 23930 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-iota 6308 df-fv 6357 df-bn 23933 |
This theorem is referenced by: bncmet 23944 lssbn 23949 hlcms 23963 bncssbn 23971 sitgclbn 31596 |
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