Theorem nmfval0 24716

Description: The value of the norm function on a structure containing a zero as the distance restricted to the elements of the base set to zero. Examples of structures containing a "zero" are groups (see nmfval2 24717 proved from this theorem and grpidcl 19032) or more generally monoids (see mndidcl 18807), or pointed sets). (Contributed by Mario Carneiro, 2-Oct-2015.) Extract this result from the proof of nmfval2 24717. (Revised by BJ, 27-Aug-2024.)

Hypotheses

Ref	Expression
nmfval0.n	⊢ 𝑁 = (norm‘𝑊)
nmfval0.x	⊢ 𝑋 = (Base‘𝑊)
nmfval0.z	⊢ 0 = (0g‘𝑊)
nmfval0.d	⊢ 𝐷 = (dist‘𝑊)
nmfval0.e	⊢ 𝐸 = (𝐷 ↾ (𝑋 × 𝑋))

Assertion

Ref	Expression
nmfval0	⊢ ( 0 ∈ 𝑋 → 𝑁 = (𝑥 ∈ 𝑋 ↦ (𝑥𝐸 0 )))

Distinct variable groups:   𝑥,𝐷   𝑥,𝑊   𝑥,𝑋   𝑥, 0

Allowed substitution hints:   𝐸(𝑥)   𝑁(𝑥)

Proof of Theorem nmfval0

Step	Hyp	Ref	Expression
1		nmfval0.n	. . 3 ⊢ 𝑁 = (norm‘𝑊)
2		nmfval0.x	. . 3 ⊢ 𝑋 = (Base‘𝑊)
3		nmfval0.z	. . 3 ⊢ 0 = (0g‘𝑊)
4		nmfval0.d	. . 3 ⊢ 𝐷 = (dist‘𝑊)
5	1, 2, 3, 4	nmfval 24714	. 2 ⊢ 𝑁 = (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 ))
6		nmfval0.e	. . . . 5 ⊢ 𝐸 = (𝐷 ↾ (𝑋 × 𝑋))
7	6	oveqi 7424	. . . 4 ⊢ (𝑥𝐸 0 ) = (𝑥(𝐷 ↾ (𝑋 × 𝑋)) 0 )
8		ovres 7577	. . . . 5 ⊢ ((𝑥 ∈ 𝑋 ∧ 0 ∈ 𝑋) → (𝑥(𝐷 ↾ (𝑋 × 𝑋)) 0 ) = (𝑥𝐷 0 ))
9	8	ancoms 463	. . . 4 ⊢ (( 0 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑥(𝐷 ↾ (𝑋 × 𝑋)) 0 ) = (𝑥𝐷 0 ))
10	7, 9	eqtr2id 2817	. . 3 ⊢ (( 0 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑥𝐷 0 ) = (𝑥𝐸 0 ))
11	10	mpteq2dva 5208	. 2 ⊢ ( 0 ∈ 𝑋 → (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 )) = (𝑥 ∈ 𝑋 ↦ (𝑥𝐸 0 )))
12	5, 11	eqtrid 2816	1 ⊢ ( 0 ∈ 𝑋 → 𝑁 = (𝑥 ∈ 𝑋 ↦ (𝑥𝐸 0 )))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149   ↦ cmpt 5196   × cxp 5660   ↾ cres 5664  ‘cfv 6537  (class class class)co 7411  Basecbs 17269  distcds 17319  0_gc0g 17492  normcnm 24702

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 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

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 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-ov 7414  df-nm 24708

This theorem is referenced by:  nmfval2  24717