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Theorem nvnd 21257
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 21192 . . . 4  |-  ( U  e.  NrmCVec  ->  Z  e.  X
)
43adantr 451 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  Z  e.  X )
5 eqid 2283 . . . 4  |-  ( -v
`  U )  =  ( -v `  U
)
6 nvnd.6 . . . 4  |-  N  =  ( normCV `  U )
7 nvnd.8 . . . 4  |-  D  =  ( IndMet `  U )
81, 5, 6, 7imsdval 21255 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  Z  e.  X )  ->  ( A D Z )  =  ( N `  ( A ( -v `  U ) Z ) ) )
94, 8mpd3an3 1278 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A D Z )  =  ( N `  ( A ( -v `  U ) Z ) ) )
10 eqid 2283 . . . . . 6  |-  ( +v
`  U )  =  ( +v `  U
)
11 eqid 2283 . . . . . 6  |-  ( .s
OLD `  U )  =  ( .s OLD `  U )
121, 10, 11, 5nvmval 21200 . . . . 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 1278 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A ( -v `  U ) Z )  =  ( A ( +v `  U ) ( -u 1 ( .s OLD `  U
) Z ) ) )
14 neg1cn 9813 . . . . . . 7  |-  -u 1  e.  CC
1511, 2nvsz 21196 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  -u 1  e.  CC )  ->  ( -u 1 ( .s OLD `  U ) Z )  =  Z )
1614, 15mpan2 652 . . . . . 6  |-  ( U  e.  NrmCVec  ->  ( -u 1
( .s OLD `  U
) Z )  =  Z )
1716oveq2d 5874 . . . . 5  |-  ( U  e.  NrmCVec  ->  ( A ( +v `  U ) ( -u 1 ( .s OLD `  U
) Z ) )  =  ( A ( +v `  U ) Z ) )
1817adantr 451 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) Z ) )  =  ( A ( +v `  U
) Z ) )
191, 10, 2nv0rid 21193 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A ( +v `  U ) Z )  =  A )
2013, 18, 193eqtrd 2319 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A ( -v `  U ) Z )  =  A )
2120fveq2d 5529 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( N `  ( A
( -v `  U
) Z ) )  =  ( N `  A ) )
229, 21eqtr2d 2316 1  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( N `  A )  =  ( A D Z ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   CCcc 8735   1c1 8738   -ucneg 9038   NrmCVeccnv 21140   +vcpv 21141   BaseSetcba 21142   .s
OLDcns 21143   0veccn0v 21144   -vcnsb 21145   normCVcnmcv 21146   IndMetcims 21147
This theorem is referenced by:  nvlmle  21265  ubthlem1  21449
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 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813
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 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-ltxr 8872  df-sub 9039  df-neg 9040  df-grpo 20858  df-gid 20859  df-ginv 20860  df-gdiv 20861  df-ablo 20949  df-vc 21102  df-nv 21148  df-va 21151  df-ba 21152  df-sm 21153  df-0v 21154  df-vs 21155  df-nmcv 21156  df-ims 21157
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