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Theorem nvnd 21273
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 21208 . . . 4  |-  ( U  e.  NrmCVec  ->  Z  e.  X
)
43adantr 451 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  Z  e.  X )
5 eqid 2296 . . . 4  |-  ( -v
`  U )  =  ( -v `  U
)
6 nvnd.6 . . . 4  |-  N  =  ( normCV `  U )
7 nvnd.8 . . . 4  |-  D  =  ( IndMet `  U )
81, 5, 6, 7imsdval 21271 . . 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 2296 . . . . . 6  |-  ( +v
`  U )  =  ( +v `  U
)
11 eqid 2296 . . . . . 6  |-  ( .s
OLD `  U )  =  ( .s OLD `  U )
121, 10, 11, 5nvmval 21216 . . . . 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 9829 . . . . . . 7  |-  -u 1  e.  CC
1511, 2nvsz 21212 . . . . . . 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 5890 . . . . 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 21209 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A ( +v `  U ) Z )  =  A )
2013, 18, 193eqtrd 2332 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A ( -v `  U ) Z )  =  A )
2120fveq2d 5545 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( N `  ( A
( -v `  U
) Z ) )  =  ( N `  A ) )
229, 21eqtr2d 2329 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 1632    e. wcel 1696   ` cfv 5271  (class class class)co 5874   CCcc 8751   1c1 8754   -ucneg 9054   NrmCVeccnv 21156   +vcpv 21157   BaseSetcba 21158   .s
OLDcns 21159   0veccn0v 21160   -vcnsb 21161   normCVcnmcv 21162   IndMetcims 21163
This theorem is referenced by:  nvlmle  21281  ubthlem1  21465
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-ltxr 8888  df-sub 9055  df-neg 9056  df-grpo 20874  df-gid 20875  df-ginv 20876  df-gdiv 20877  df-ablo 20965  df-vc 21118  df-nv 21164  df-va 21167  df-ba 21168  df-sm 21169  df-0v 21170  df-vs 21171  df-nmcv 21172  df-ims 21173
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