Theorem nmval 24547

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 7426	. 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 24546	. 2 ⊢ 𝑁 = (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 ))
7		ovex 7452	. 2 ⊢ (𝐴𝐷 0 ) ∈ V
8	1, 6, 7	fvmpt 7004	1 ⊢ (𝐴 ∈ 𝑋 → (𝑁‘𝐴) = (𝐴𝐷 0 ))

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098  ‘cfv 6549  (class class class)co 7419  Basecbs 17188  distcds 17250  0_gc0g 17429  normcnm 24534

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-fv 6557  df-ov 7422  df-nm 24540

This theorem is referenced by:  nmval2  24550  ngpds2  24564  isngp4  24570  nmge0  24575  nmeq0  24576  nminv  24579  nmmtri  24580  nmrtri  24582