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Theorem nmval 23192
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 7152 . 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 23191 . 2 𝑁 = (𝑥𝑋 ↦ (𝑥𝐷 0 ))
7 ovex 7178 . 2 (𝐴𝐷 0 ) ∈ V
81, 6, 7fvmpt 6756 1 (𝐴𝑋 → (𝑁𝐴) = (𝐴𝐷 0 ))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115  cfv 6343  (class class class)co 7145  Basecbs 16479  distcds 16570  0gc0g 16709  normcnm 23179
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7451
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4276  df-if 4450  df-pw 4523  df-sn 4550  df-pr 4552  df-op 4556  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-fv 6351  df-ov 7148  df-nm 23185
This theorem is referenced by:  nmval2  23194  ngpds2  23208  isngp4  23214  nmge0  23219  nmeq0  23220  nminv  23223  nmmtri  23224  nmrtri  23226
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