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Theorem nmval 24617
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 7437 . 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 24616 . 2 𝑁 = (𝑥𝑋 ↦ (𝑥𝐷 0 ))
7 ovex 7463 . 2 (𝐴𝐷 0 ) ∈ V
81, 6, 7fvmpt 7015 1 (𝐴𝑋 → (𝑁𝐴) = (𝐴𝐷 0 ))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1536  wcel 2105  cfv 6562  (class class class)co 7430  Basecbs 17244  distcds 17306  0gc0g 17485  normcnm 24604
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-fv 6570  df-ov 7433  df-nm 24610
This theorem is referenced by:  nmval2  24620  ngpds2  24634  isngp4  24640  nmge0  24645  nmeq0  24646  nminv  24649  nmmtri  24650  nmrtri  24652
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