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Theorem nvnd 22180
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 22115 . . . 4  |-  ( U  e.  NrmCVec  ->  Z  e.  X
)
43adantr 452 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  Z  e.  X )
5 eqid 2436 . . . 4  |-  ( -v
`  U )  =  ( -v `  U
)
6 nvnd.6 . . . 4  |-  N  =  ( normCV `  U )
7 nvnd.8 . . . 4  |-  D  =  ( IndMet `  U )
81, 5, 6, 7imsdval 22178 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  Z  e.  X )  ->  ( A D Z )  =  ( N `  ( A ( -v `  U ) Z ) ) )
94, 8mpd3an3 1280 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A D Z )  =  ( N `  ( A ( -v `  U ) Z ) ) )
10 eqid 2436 . . . . . 6  |-  ( +v
`  U )  =  ( +v `  U
)
11 eqid 2436 . . . . . 6  |-  ( .s
OLD `  U )  =  ( .s OLD `  U )
121, 10, 11, 5nvmval 22123 . . . . 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 1280 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A ( -v `  U ) Z )  =  ( A ( +v `  U ) ( -u 1 ( .s OLD `  U
) Z ) ) )
14 neg1cn 10067 . . . . . . 7  |-  -u 1  e.  CC
1511, 2nvsz 22119 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  -u 1  e.  CC )  ->  ( -u 1 ( .s OLD `  U ) Z )  =  Z )
1614, 15mpan2 653 . . . . . 6  |-  ( U  e.  NrmCVec  ->  ( -u 1
( .s OLD `  U
) Z )  =  Z )
1716oveq2d 6097 . . . . 5  |-  ( U  e.  NrmCVec  ->  ( A ( +v `  U ) ( -u 1 ( .s OLD `  U
) Z ) )  =  ( A ( +v `  U ) Z ) )
1817adantr 452 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) Z ) )  =  ( A ( +v `  U
) Z ) )
191, 10, 2nv0rid 22116 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A ( +v `  U ) Z )  =  A )
2013, 18, 193eqtrd 2472 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A ( -v `  U ) Z )  =  A )
2120fveq2d 5732 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( N `  ( A
( -v `  U
) Z ) )  =  ( N `  A ) )
229, 21eqtr2d 2469 1  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( N `  A )  =  ( A D Z ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   ` cfv 5454  (class class class)co 6081   CCcc 8988   1c1 8991   -ucneg 9292   NrmCVeccnv 22063   +vcpv 22064   BaseSetcba 22065   .s
OLDcns 22066   0veccn0v 22067   -vcnsb 22068   normCVcnmcv 22069   IndMetcims 22070
This theorem is referenced by:  nvlmle  22188  ubthlem1  22372
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-po 4503  df-so 4504  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-ltxr 9125  df-sub 9293  df-neg 9294  df-grpo 21779  df-gid 21780  df-ginv 21781  df-gdiv 21782  df-ablo 21870  df-vc 22025  df-nv 22071  df-va 22074  df-ba 22075  df-sm 22076  df-0v 22077  df-vs 22078  df-nmcv 22079  df-ims 22080
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