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Theorem imsdval2 24083
Description: Value of the distance function of the induced metric of a normed complex vector space. Equation 1 of [Kreyszig] p. 59. (Contributed by NM, 28-Nov-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
imsdval2.1  |-  X  =  ( BaseSet `  U )
imsdval2.2  |-  G  =  ( +v `  U
)
imsdval2.4  |-  S  =  ( .sOLD `  U )
imsdval2.6  |-  N  =  ( normCV `  U )
imsdval2.8  |-  D  =  ( IndMet `  U )
Assertion
Ref Expression
imsdval2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  =  ( N `  ( A G ( -u 1 S B ) ) ) )

Proof of Theorem imsdval2
StepHypRef Expression
1 imsdval2.1 . . 3  |-  X  =  ( BaseSet `  U )
2 eqid 2443 . . 3  |-  ( -v
`  U )  =  ( -v `  U
)
3 imsdval2.6 . . 3  |-  N  =  ( normCV `  U )
4 imsdval2.8 . . 3  |-  D  =  ( IndMet `  U )
51, 2, 3, 4imsdval 24082 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  =  ( N `  ( A ( -v `  U ) B ) ) )
6 imsdval2.2 . . . 4  |-  G  =  ( +v `  U
)
7 imsdval2.4 . . . 4  |-  S  =  ( .sOLD `  U )
81, 6, 7, 2nvmval 24027 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A ( -v `  U ) B )  =  ( A G ( -u 1 S B ) ) )
98fveq2d 5700 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A
( -v `  U
) B ) )  =  ( N `  ( A G ( -u
1 S B ) ) ) )
105, 9eqtrd 2475 1  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  =  ( N `  ( A G ( -u 1 S B ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 965    = wceq 1369    e. wcel 1756   ` cfv 5423  (class class class)co 6096   1c1 9288   -ucneg 9601   NrmCVeccnv 23967   +vcpv 23968   BaseSetcba 23969   .sOLDcns 23970   -vcnsb 23972   normCVcnmcv 23973   IndMetcims 23974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-po 4646  df-so 4647  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-1st 6582  df-2nd 6583  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-pnf 9425  df-mnf 9426  df-ltxr 9428  df-sub 9602  df-neg 9603  df-grpo 23683  df-gid 23684  df-ginv 23685  df-gdiv 23686  df-ablo 23774  df-vc 23929  df-nv 23975  df-va 23978  df-ba 23979  df-sm 23980  df-0v 23981  df-vs 23982  df-nmcv 23983  df-ims 23984
This theorem is referenced by:  imsmetlem  24086  nmcvcn  24095  smcnlem  24097
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