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Theorem nvnd 21087
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 21022 . . . 4  |-  ( U  e.  NrmCVec  ->  Z  e.  X
)
43adantr 453 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  Z  e.  X )
5 eqid 2253 . . . 4  |-  ( -v
`  U )  =  ( -v `  U
)
6 nvnd.6 . . . 4  |-  N  =  ( normCV `  U )
7 nvnd.8 . . . 4  |-  D  =  ( IndMet `  U )
81, 5, 6, 7imsdval 21085 . . 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 2253 . . . . . 6  |-  ( +v
`  U )  =  ( +v `  U
)
11 eqid 2253 . . . . . 6  |-  ( .s
OLD `  U )  =  ( .s OLD `  U )
121, 10, 11, 5nvmval 21030 . . . . 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 9693 . . . . . . 7  |-  -u 1  e.  CC
1511, 2nvsz 21026 . . . . . . 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 5726 . . . . 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 21023 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A ( +v `  U ) Z )  =  A )
2013, 18, 193eqtrd 2289 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A ( -v `  U ) Z )  =  A )
2120fveq2d 5381 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( N `  ( A
( -v `  U
) Z ) )  =  ( N `  A ) )
229, 21eqtr2d 2286 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 4592  (class class class)co 5710   CCcc 8615   1c1 8618   -ucneg 8918   NrmCVeccnv 20970   +vcpv 20971   BaseSetcba 20972   .s
OLDcns 20973   0veccn0v 20974   -vcnsb 20975   normCVcnmcv 20976   IndMetcims 20977
This theorem is referenced by:  nvlmle  21095  ubthlem1  21279
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 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-po 4207  df-so 4208  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-riota 6190  df-er 6546  df-en 6750  df-dom 6751  df-sdom 6752  df-pnf 8749  df-mnf 8750  df-ltxr 8752  df-sub 8919  df-neg 8920  df-grpo 20688  df-gid 20689  df-ginv 20690  df-gdiv 20691  df-ablo 20779  df-vc 20932  df-nv 20978  df-va 20981  df-ba 20982  df-sm 20983  df-0v 20984  df-vs 20985  df-nmcv 20986  df-ims 20987
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