Theorem nmval 24746

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 7417	. 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 24745	. 2 ⊢ 𝑁 = (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 ))
7		ovex 7443	. 2 ⊢ (𝐴𝐷 0 ) ∈ V
8	1, 6, 7	fvmpt 6989	1 ⊢ (𝐴 ∈ 𝑋 → (𝑁‘𝐴) = (𝐴𝐷 0 ))

Syntax hints:   → wi 4   = wceq 1570   ∈ wcel 2143  ‘cfv 6536  (class class class)co 7410  Basecbs 17264  distcds 17314  0_gc0g 17487  normcnm 24733

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-ov 7413  df-nm 24739

This theorem is referenced by:  nmval2  24749  ngpds2  24763  isngp4  24769  nmge0  24774  nmeq0  24775  nminv  24778  nmmtri  24779  nmrtri  24781