Theorem bnsca 23136

Description: The scalar field of a Banach space is complete. (Contributed by NM, 8-Sep-2007.) (Revised by Mario Carneiro, 15-Oct-2015.)

Hypothesis

Ref	Expression
isbn.1	|- F = (Scalar ` W )

Assertion

Ref	Expression
bnsca	|- ( W e. Ban -> F e. CMetSp )

Proof of Theorem bnsca

Step	Hyp	Ref	Expression
1		isbn.1	. . 3 |- F = (Scalar ` W )
2	1	isbn 23135	. 2 |- ( W e. Ban <-> ( W e. NrmVec /\ W e. CMetSp /\ F e. CMetSp ) )
3	2	simp3bi 1078	1 |- ( W e. Ban -> F e. CMetSp )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   ` cfv 5888  Scalarcsca 15944  NrmVeccnvc 22386  CMetSpccms 23129  Bancbn 23130

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-bn 23133

This theorem is referenced by:  lssbn  23148  hlprlem  23163