Theorem nmfval0 24819

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 24820 proved from this theorem and grpidcl 19092) or more generally monoids (see mndidcl 18855), or pointed sets). (Contributed by Mario Carneiro, 2-Oct-2015.) Extract this result from the proof of nmfval2 24820. (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 24817	. 2 ⊢ 𝑁 = (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 ))
6		nmfval0.e	. . . . 5 ⊢ 𝐸 = (𝐷 ↾ (𝑋 × 𝑋))
7	6	oveqi 7427	. . . 4 ⊢ (𝑥𝐸 0 ) = (𝑥(𝐷 ↾ (𝑋 × 𝑋)) 0 )
8		ovres 7580	. . . . 5 ⊢ ((𝑥 ∈ 𝑋 ∧ 0 ∈ 𝑋) → (𝑥(𝐷 ↾ (𝑋 × 𝑋)) 0 ) = (𝑥𝐷 0 ))
9	8	ancoms 464	. . . 4 ⊢ (( 0 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑥(𝐷 ↾ (𝑋 × 𝑋)) 0 ) = (𝑥𝐷 0 ))
10	7, 9	eqtr2id 2808	. . 3 ⊢ (( 0 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑥𝐷 0 ) = (𝑥𝐸 0 ))
11	10	mpteq2dva 5198	. 2 ⊢ ( 0 ∈ 𝑋 → (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 )) = (𝑥 ∈ 𝑋 ↦ (𝑥𝐸 0 )))
12	5, 11	eqtrid 2807	1 ⊢ ( 0 ∈ 𝑋 → 𝑁 = (𝑥 ∈ 𝑋 ↦ (𝑥𝐸 0 )))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ↦ cmpt 5186   × cxp 5653   ↾ cres 5657  ‘cfv 6533  (class class class)co 7414  Basecbs 17304  distcds 17354  0_gc0g 17527  normcnm 24805

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541  df-ov 7417  df-nm 24811

This theorem is used by:  nmfval2  24820