Theorem nmval 24711

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 = (0g‘𝑊)
nmfval.d	⊢ 𝐷 = (dist‘𝑊)

Assertion

Ref	Expression
nmval	⊢ (𝐴 ∈ 𝑋 → (𝑁‘𝐴) = (𝐴𝐷 0 ))

Proof of Theorem nmval

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 7415	. 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 = (0g‘𝑊)
5		nmfval.d	. . 3 ⊢ 𝐷 = (dist‘𝑊)
6	2, 3, 4, 5	nmfval 24710	. 2 ⊢ 𝑁 = (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 ))
7		ovex 7441	. 2 ⊢ (𝐴𝐷 0 ) ∈ V
8	1, 6, 7	fvmpt 6987	1 ⊢ (𝐴 ∈ 𝑋 → (𝑁‘𝐴) = (𝐴𝐷 0 ))

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149  ‘cfv 6534  (class class class)co 7408  Basecbs 17265  distcds 17315  0_gc0g 17488  normcnm 24698

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6490  df-fun 6536  df-fn 6537  df-f 6538  df-fv 6542  df-ov 7411  df-nm 24704

This theorem is referenced by:  nmval2  24714  ngpds2  24728  isngp4  24734  nmge0  24739  nmeq0  24740  nminv  24743  nmmtri  24744  nmrtri  24746