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Theorem imsdval2 21181
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 2256 . . 3  |-  ( -v
`  U )  =  ( -v `  U
)
3 imsdval2.6 . . 3  |-  N  =  ( normCV `  U )
4 imsdval2.8 . . 3  |-  D  =  ( IndMet `  U )
51, 2, 3, 4imsdval 21180 . 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 21125 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A ( -v `  U ) B )  =  ( A G ( -u 1 S B ) ) )
98fveq2d 5427 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A
( -v `  U
) B ) )  =  ( N `  ( A G ( -u
1 S B ) ) ) )
105, 9eqtrd 2288 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 6    /\ w3a 939    = wceq 1619    e. wcel 1621   ` cfv 4638  (class class class)co 5757   1c1 8671   -ucneg 8971   NrmCVeccnv 21065   +vcpv 21066   BaseSetcba 21067   .s
OLDcns 21068   -vcnsb 21070   normCVcnmcv 21071   IndMetcims 21072
This theorem is referenced by:  imsmetlem  21184  nmcvcn  21193  smcnlem  21195
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-resscn 8727  ax-1cn 8728  ax-icn 8729  ax-addcl 8730  ax-addrcl 8731  ax-mulcl 8732  ax-mulrcl 8733  ax-mulcom 8734  ax-addass 8735  ax-mulass 8736  ax-distr 8737  ax-i2m1 8738  ax-1ne0 8739  ax-1rid 8740  ax-rnegex 8741  ax-rrecex 8742  ax-cnre 8743  ax-pre-lttri 8744  ax-pre-lttrn 8745  ax-pre-ltadd 8746
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-id 4246  df-po 4251  df-so 4252  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-1st 6021  df-2nd 6022  df-iota 6190  df-riota 6237  df-er 6593  df-en 6797  df-dom 6798  df-sdom 6799  df-pnf 8802  df-mnf 8803  df-ltxr 8805  df-sub 8972  df-neg 8973  df-grpo 20783  df-gid 20784  df-ginv 20785  df-gdiv 20786  df-ablo 20874  df-vc 21027  df-nv 21073  df-va 21076  df-ba 21077  df-sm 21078  df-0v 21079  df-vs 21080  df-nmcv 21081  df-ims 21082
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