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Theorem nmval 22771
Description: The value of the norm function. 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 6917 . 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 22770 . 2 𝑁 = (𝑥𝑋 ↦ (𝑥𝐷 0 ))
7 ovex 6942 . 2 (𝐴𝐷 0 ) ∈ V
81, 6, 7fvmpt 6533 1 (𝐴𝑋 → (𝑁𝐴) = (𝐴𝐷 0 ))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1656  wcel 2164  cfv 6127  (class class class)co 6910  Basecbs 16229  distcds 16321  0gc0g 16460  normcnm 22758
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-fv 6135  df-ov 6913  df-nm 22764
This theorem is referenced by:  nmval2  22773  ngpds2  22787  isngp4  22793  nmge0  22798  nmeq0  22799  nminv  22802  nmmtri  22803  nmrtri  22805
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