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Theorem nvnd 21182
Description: The norm of a normed complex vector space expressed in terms of the distance function of its induced metric. Problem 1 of [Kreyszig] p. 63. (Contributed by NM, 4-Dec-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
nvnd.1  |-  X  =  ( BaseSet `  U )
nvnd.5  |-  Z  =  ( 0vec `  U
)
nvnd.6  |-  N  =  ( normCV `  U )
nvnd.8  |-  D  =  ( IndMet `  U )
Assertion
Ref Expression
nvnd  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( N `  A )  =  ( A D Z ) )

Proof of Theorem nvnd
StepHypRef Expression
1 nvnd.1 . . . . 5  |-  X  =  ( BaseSet `  U )
2 nvnd.5 . . . . 5  |-  Z  =  ( 0vec `  U
)
31, 2nvzcl 21117 . . . 4  |-  ( U  e.  NrmCVec  ->  Z  e.  X
)
43adantr 453 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  Z  e.  X )
5 eqid 2256 . . . 4  |-  ( -v
`  U )  =  ( -v `  U
)
6 nvnd.6 . . . 4  |-  N  =  ( normCV `  U )
7 nvnd.8 . . . 4  |-  D  =  ( IndMet `  U )
81, 5, 6, 7imsdval 21180 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  Z  e.  X )  ->  ( A D Z )  =  ( N `  ( A ( -v `  U ) Z ) ) )
94, 8mpd3an3 1283 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A D Z )  =  ( N `  ( A ( -v `  U ) Z ) ) )
10 eqid 2256 . . . . . 6  |-  ( +v
`  U )  =  ( +v `  U
)
11 eqid 2256 . . . . . 6  |-  ( .s
OLD `  U )  =  ( .s OLD `  U )
121, 10, 11, 5nvmval 21125 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  Z  e.  X )  ->  ( A ( -v `  U ) Z )  =  ( A ( +v `  U ) ( -u 1 ( .s OLD `  U
) Z ) ) )
134, 12mpd3an3 1283 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A ( -v `  U ) Z )  =  ( A ( +v `  U ) ( -u 1 ( .s OLD `  U
) Z ) ) )
14 neg1cn 9746 . . . . . . 7  |-  -u 1  e.  CC
1511, 2nvsz 21121 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  -u 1  e.  CC )  ->  ( -u 1 ( .s OLD `  U ) Z )  =  Z )
1614, 15mpan2 655 . . . . . 6  |-  ( U  e.  NrmCVec  ->  ( -u 1
( .s OLD `  U
) Z )  =  Z )
1716oveq2d 5773 . . . . 5  |-  ( U  e.  NrmCVec  ->  ( A ( +v `  U ) ( -u 1 ( .s OLD `  U
) Z ) )  =  ( A ( +v `  U ) Z ) )
1817adantr 453 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) Z ) )  =  ( A ( +v `  U
) Z ) )
191, 10, 2nv0rid 21118 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A ( +v `  U ) Z )  =  A )
2013, 18, 193eqtrd 2292 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A ( -v `  U ) Z )  =  A )
2120fveq2d 5427 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( N `  ( A
( -v `  U
) Z ) )  =  ( N `  A ) )
229, 21eqtr2d 2289 1  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( N `  A )  =  ( A D Z ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621   ` cfv 4638  (class class class)co 5757   CCcc 8668   1c1 8671   -ucneg 8971   NrmCVeccnv 21065   +vcpv 21066   BaseSetcba 21067   .s
OLDcns 21068   0veccn0v 21069   -vcnsb 21070   normCVcnmcv 21071   IndMetcims 21072
This theorem is referenced by:  nvlmle  21190  ubthlem1  21374
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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