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Theorem nmval 23743
Description: The value of the norm as the distance to zero. Problem 1 of [Kreyszig] p. 63. (Contributed by NM, 4-Dec-2006.) (Revised by Mario Carneiro, 2-Oct-2015.)
Hypotheses
Ref Expression
nmfval.n 𝑁 = (norm‘𝑊)
nmfval.x 𝑋 = (Base‘𝑊)
nmfval.z 0 = (0g𝑊)
nmfval.d 𝐷 = (dist‘𝑊)
Assertion
Ref Expression
nmval (𝐴𝑋 → (𝑁𝐴) = (𝐴𝐷 0 ))

Proof of Theorem nmval
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 oveq1 7278 . 2 (𝑥 = 𝐴 → (𝑥𝐷 0 ) = (𝐴𝐷 0 ))
2 nmfval.n . . 3 𝑁 = (norm‘𝑊)
3 nmfval.x . . 3 𝑋 = (Base‘𝑊)
4 nmfval.z . . 3 0 = (0g𝑊)
5 nmfval.d . . 3 𝐷 = (dist‘𝑊)
62, 3, 4, 5nmfval 23742 . 2 𝑁 = (𝑥𝑋 ↦ (𝑥𝐷 0 ))
7 ovex 7304 . 2 (𝐴𝐷 0 ) ∈ V
81, 6, 7fvmpt 6872 1 (𝐴𝑋 → (𝑁𝐴) = (𝐴𝐷 0 ))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2110  cfv 6432  (class class class)co 7271  Basecbs 16910  distcds 16969  0gc0g 17148  normcnm 23730
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7582
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5163  df-id 5490  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-fv 6440  df-ov 7274  df-nm 23736
This theorem is referenced by:  nmval2  23746  ngpds2  23760  isngp4  23766  nmge0  23771  nmeq0  23772  nminv  23775  nmmtri  23776  nmrtri  23778
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