Theorem nmval 23196

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 = (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 7142	. 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 23195	. 2 ⊢ 𝑁 = (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 ))
7		ovex 7168	. 2 ⊢ (𝐴𝐷 0 ) ∈ V
8	1, 6, 7	fvmpt 6745	1 ⊢ (𝐴 ∈ 𝑋 → (𝑁‘𝐴) = (𝐴𝐷 0 ))

Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  ‘cfv 6324  (class class class)co 7135  Basecbs 16475  distcds 16566  0_gc0g 16705  normcnm 23183

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

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 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fv 6332  df-ov 7138  df-nm 23189

This theorem is referenced by:  nmval2  23198  ngpds2  23212  isngp4  23218  nmge0  23223  nmeq0  23224  nminv  23227  nmmtri  23228  nmrtri  23230