Theorem imsval 27540

Description: Value of the induced metric of a normed complex vector space. (Contributed by NM, 11-Sep-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
imsval.3	|- M = ( -v ` U )
imsval.6	|- N = ( normCV ` U )
imsval.8	|- D = ( IndMet ` U )

Assertion

Ref	Expression
imsval	|- ( U e. NrmCVec -> D = ( N o. M ) )

Proof of Theorem imsval

Dummy variable u is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . 4 |- ( u = U -> ( normCV ` u ) = ( normCV ` U ) )
2		fveq2 6191	. . . 4 |- ( u = U -> ( -v ` u ) = ( -v ` U ) )
3	1, 2	coeq12d 5286	. . 3 |- ( u = U -> ( ( normCV ` u ) o. ( -v ` u ) ) = ( ( normCV ` U ) o. ( -v ` U ) ) )
4		df-ims 27456	. . 3 |- IndMet = ( u e. NrmCVec |-> ( ( normCV ` u ) o. ( -v ` u ) ) )
5		fvex 6201	. . . 4 |- ( normCV ` U ) e. _V
6		fvex 6201	. . . 4 |- ( -v ` U ) e. _V
7	5, 6	coex 7118	. . 3 |- ( ( normCV ` U ) o. ( -v ` U ) ) e. _V
8	3, 4, 7	fvmpt 6282	. 2 |- ( U e. NrmCVec -> ( IndMet ` U ) = ( ( normCV ` U ) o. ( -v ` U ) ) )
9		imsval.8	. 2 |- D = ( IndMet ` U )
10		imsval.6	. . 3 |- N = ( normCV ` U )
11		imsval.3	. . 3 |- M = ( -v ` U )
12	10, 11	coeq12i 5285	. 2 |- ( N o. M ) = ( ( normCV ` U ) o. ( -v ` U ) )
13	8, 9, 12	3eqtr4g 2681	1 |- ( U e. NrmCVec -> D = ( N o. M ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   o. ccom 5118   ` cfv 5888   NrmCVeccnv 27439   -vcnsb 27444   norm_CVcnmcv 27445   IndMetcims 27446

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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

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-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fv 5896  df-ims 27456

This theorem is referenced by:  imsdval  27541  imsdf  27544  cnims  27548  hhims  28029  hhssims  28132