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Theorem imsdval2 22181
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
StepHypRef Expression
1 imsdval2.1 . . 3  |-  X  =  ( BaseSet `  U )
2 eqid 2438 . . 3  |-  ( -v
`  U )  =  ( -v `  U
)
3 imsdval2.6 . . 3  |-  N  =  ( normCV `  U )
4 imsdval2.8 . . 3  |-  D  =  ( IndMet `  U )
51, 2, 3, 4imsdval 22180 . 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
)
81, 6, 7, 2nvmval 22125 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A ( -v `  U ) B )  =  ( A G ( -u 1 S B ) ) )
98fveq2d 5734 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A
( -v `  U
) B ) )  =  ( N `  ( A G ( -u
1 S B ) ) ) )
105, 9eqtrd 2470 1  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  =  ( N `  ( A G ( -u 1 S B ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 937    = wceq 1653    e. wcel 1726   ` cfv 5456  (class class class)co 6083   1c1 8993   -ucneg 9294   NrmCVeccnv 22065   +vcpv 22066   BaseSetcba 22067   .s
OLDcns 22068   -vcnsb 22070   normCVcnmcv 22071   IndMetcims 22072
This theorem is referenced by:  imsmetlem  22184  nmcvcn  22193  smcnlem  22195
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-po 4505  df-so 4506  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-pnf 9124  df-mnf 9125  df-ltxr 9127  df-sub 9295  df-neg 9296  df-grpo 21781  df-gid 21782  df-ginv 21783  df-gdiv 21784  df-ablo 21872  df-vc 22027  df-nv 22073  df-va 22076  df-ba 22077  df-sm 22078  df-0v 22079  df-vs 22080  df-nmcv 22081  df-ims 22082
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