Theorem imsdval2 21250

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 2284	. . 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 21249	. 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 21194	. . 3 |- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( A ( -v ` U ) B ) = ( A G ( -u 1 S B ) ) )
9	8	fveq2d 5490	. 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 2316	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 934   = wceq 1623   e. wcel 1685   ` cfv 5221  (class class class)co 5820   1c1 8734   -ucneg 9034   NrmCVeccnv 21134   +vcpv 21135   BaseSetcba 21136   .s OLDcns 21137   -vcnsb 21139   norm_CVcnmcv 21140   IndMetcims 21141

This theorem is referenced by:  imsmetlem  21253  nmcvcn  21262  smcnlem  21264

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-po 4313  df-so 4314  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-iota 6253  df-riota 6300  df-er 6656  df-en 6860  df-dom 6861  df-sdom 6862  df-pnf 8865  df-mnf 8866  df-ltxr 8868  df-sub 9035  df-neg 9036  df-grpo 20852  df-gid 20853  df-ginv 20854  df-gdiv 20855  df-ablo 20943  df-vc 21096  df-nv 21142  df-va 21145  df-ba 21146  df-sm 21147  df-0v 21148  df-vs 21149  df-nmcv 21150  df-ims 21151