Theorem imsdval2 22171

Description: Value of the distance function of the induced metric of a normed complex vector space. Equation 1 of [Kreyszig] p. 59. (Contributed by NM, 28-Nov-2006.) (New usage is discouraged.)

Hypotheses

Ref	Expression
imsdval2.1	|- X = ( BaseSet ` U )
imsdval2.2	|- G = ( +v ` U )
imsdval2.4	|- S = ( .s OLD ` U )
imsdval2.6	|- N = ( normCV ` U )
imsdval2.8	|- D = ( IndMet ` U )

Assertion

Ref	Expression
imsdval2	|- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( A D B ) = ( N ` ( A G ( -u 1 S B ) ) ) )

Proof of Theorem imsdval2

Step	Hyp	Ref	Expression
1		imsdval2.1	. . 3 |- X = ( BaseSet ` U )
2		eqid 2435	. . 3 |- ( -v ` U ) = ( -v ` U )
3		imsdval2.6	. . 3 |- N = ( normCV ` U )
4		imsdval2.8	. . 3 |- D = ( IndMet ` U )
5	1, 2, 3, 4	imsdval 22170	. 2 |- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( A D B ) = ( N ` ( A ( -v ` U ) B ) ) )
6		imsdval2.2	. . . 4 |- G = ( +v ` U )
7		imsdval2.4	. . . 4 |- S = ( .s OLD ` U )
8	1, 6, 7, 2	nvmval 22115	. . 3 |- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( A ( -v ` U ) B ) = ( A G ( -u 1 S B ) ) )
9	8	fveq2d 5724	. 2 |- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( N ` ( A ( -v ` U ) B ) ) = ( N ` ( A G ( -u 1 S B ) ) ) )
10	5, 9	eqtrd 2467	1 |- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( A D B ) = ( N ` ( A G ( -u 1 S B ) ) ) )

Syntax hints:   -> wi 4   /\ w3a 936   = wceq 1652   e. wcel 1725   ` cfv 5446  (class class class)co 6073   1c1 8983   -ucneg 9284   NrmCVeccnv 22055   +vcpv 22056   BaseSetcba 22057   .s OLDcns 22058   -vcnsb 22060   norm_CVcnmcv 22061   IndMetcims 22062

This theorem is referenced by:  imsmetlem  22174  nmcvcn  22183  smcnlem  22185

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-ltxr 9117  df-sub 9285  df-neg 9286  df-grpo 21771  df-gid 21772  df-ginv 21773  df-gdiv 21774  df-ablo 21862  df-vc 22017  df-nv 22063  df-va 22066  df-ba 22067  df-sm 22068  df-0v 22069  df-vs 22070  df-nmcv 22071  df-ims 22072