Theorem isbn 23135

Description: A Banach space is a normed vector space with a complete induced metric. (Contributed by NM, 5-Dec-2006.) (Revised by Mario Carneiro, 15-Oct-2015.)

Hypothesis

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

Assertion

Ref	Expression
isbn	|- ( W e. Ban <-> ( W e. NrmVec /\ W e. CMetSp /\ F e. CMetSp ) )

Proof of Theorem isbn

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elin 3796	. . 3 |- ( W e. (NrmVec i^i CMetSp ) <-> ( W e. NrmVec /\ W e. CMetSp ) )
2	1	anbi1i 731	. 2 |- ( ( W e. (NrmVec i^i CMetSp ) /\ F e. CMetSp ) <-> ( ( W e. NrmVec /\ W e. CMetSp ) /\ F e. CMetSp ) )
3		fveq2 6191	. . . . 5 |- ( w = W -> (Scalar ` w ) = (Scalar ` W ) )
4		isbn.1	. . . . 5 |- F = (Scalar ` W )
5	3, 4	syl6eqr 2674	. . . 4 |- ( w = W -> (Scalar ` w ) = F )
6	5	eleq1d 2686	. . 3 |- ( w = W -> ( (Scalar ` w ) e. CMetSp <-> F e. CMetSp ) )
7		df-bn 23133	. . 3 |- Ban = { w e. (NrmVec i^i CMetSp ) | (Scalar ` w ) e. CMetSp }
8	6, 7	elrab2 3366	. 2 |- ( W e. Ban <-> ( W e. (NrmVec i^i CMetSp ) /\ F e. CMetSp ) )
9		df-3an 1039	. 2 |- ( ( W e. NrmVec /\ W e. CMetSp /\ F e. CMetSp ) <-> ( ( W e. NrmVec /\ W e. CMetSp ) /\ F e. CMetSp ) )
10	2, 8, 9	3bitr4i 292	1 |- ( W e. Ban <-> ( W e. NrmVec /\ W e. CMetSp /\ F e. CMetSp ) )

Syntax hints:   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   i^i cin 3573   ` 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:  bnsca  23136  bnnvc  23137  bncms  23141  lssbn  23148  srabn  23156  ishl2  23166