Theorem iscbn 27720

Description: A complex Banach space is a normed complex vector space with a complete induced metric. (Contributed by NM, 5-Dec-2006.) (New usage is discouraged.)

Hypotheses

Ref	Expression
iscbn.x	|- X = ( BaseSet ` U )
iscbn.8	|- D = ( IndMet ` U )

Assertion

Ref	Expression
iscbn	|- ( U e. CBan <-> ( U e. NrmCVec /\ D e. ( CMet ` X ) ) )

Proof of Theorem iscbn

Dummy variable u is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . 4 |- ( u = U -> ( IndMet ` u ) = ( IndMet ` U ) )
2		iscbn.8	. . . 4 |- D = ( IndMet ` U )
3	1, 2	syl6eqr 2674	. . 3 |- ( u = U -> ( IndMet ` u ) = D )
4		fveq2 6191	. . . . 5 |- ( u = U -> ( BaseSet ` u ) = ( BaseSet ` U ) )
5		iscbn.x	. . . . 5 |- X = ( BaseSet ` U )
6	4, 5	syl6eqr 2674	. . . 4 |- ( u = U -> ( BaseSet ` u ) = X )
7	6	fveq2d 6195	. . 3 |- ( u = U -> ( CMet ` ( BaseSet ` u ) ) = ( CMet ` X ) )
8	3, 7	eleq12d 2695	. 2 |- ( u = U -> ( ( IndMet ` u ) e. ( CMet ` ( BaseSet ` u ) ) <-> D e. ( CMet ` X ) ) )
9		df-cbn 27719	. 2 |- CBan = { u e. NrmCVec | ( IndMet ` u ) e. ( CMet ` ( BaseSet ` u ) ) }
10	8, 9	elrab2 3366	1 |- ( U e. CBan <-> ( U e. NrmCVec /\ D e. ( CMet ` X ) ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   ` cfv 5888   CMetcms 23052   NrmCVeccnv 27439   BaseSetcba 27441   IndMetcims 27446   CBanccbn 27718

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-cbn 27719

This theorem is referenced by:  cbncms  27721  bnnv  27722  bnsscmcl  27724  cnbn  27725  hhhl  28061  hhssbn  28137