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Theorem imsdval2 21217
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  =  ( .s OLD `  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 2258 . . 3  |-  ( -v
`  U )  =  ( -v `  U
)
3 imsdval2.6 . . 3  |-  N  =  ( normCV `  U )
4 imsdval2.8 . . 3  |-  D  =  ( IndMet `  U )
51, 2, 3, 4imsdval 21216 . 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  =  ( .s OLD `  U
)
81, 6, 7, 2nvmval 21161 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A ( -v `  U ) B )  =  ( A G ( -u 1 S B ) ) )
98fveq2d 5462 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A
( -v `  U
) B ) )  =  ( N `  ( A G ( -u
1 S B ) ) ) )
105, 9eqtrd 2290 1  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  =  ( N `  ( A G ( -u 1 S B ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ w3a 939    = wceq 1619    e. wcel 1621   ` cfv 4673  (class class class)co 5792   1c1 8706   -ucneg 9006   NrmCVeccnv 21101   +vcpv 21102   BaseSetcba 21103   .s
OLDcns 21104   -vcnsb 21106   normCVcnmcv 21107   IndMetcims 21108
This theorem is referenced by:  imsmetlem  21220  nmcvcn  21229  smcnlem  21231
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 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781
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 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-po 4286  df-so 4287  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-iota 6225  df-riota 6272  df-er 6628  df-en 6832  df-dom 6833  df-sdom 6834  df-pnf 8837  df-mnf 8838  df-ltxr 8840  df-sub 9007  df-neg 9008  df-grpo 20819  df-gid 20820  df-ginv 20821  df-gdiv 20822  df-ablo 20910  df-vc 21063  df-nv 21109  df-va 21112  df-ba 21113  df-sm 21114  df-0v 21115  df-vs 21116  df-nmcv 21117  df-ims 21118
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