Theorem nmfval0 24485

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 24486 proved from this theorem and grpidcl 18904) or more generally monoids (see mndidcl 18683), or pointed sets). (Contributed by Mario Carneiro, 2-Oct-2015.) Extract this result from the proof of nmfval2 24486. (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 24483	. 2 ⊢ 𝑁 = (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 ))
6		nmfval0.e	. . . . 5 ⊢ 𝐸 = (𝐷 ↾ (𝑋 × 𝑋))
7	6	oveqi 7403	. . . 4 ⊢ (𝑥𝐸 0 ) = (𝑥(𝐷 ↾ (𝑋 × 𝑋)) 0 )
8		ovres 7558	. . . . 5 ⊢ ((𝑥 ∈ 𝑋 ∧ 0 ∈ 𝑋) → (𝑥(𝐷 ↾ (𝑋 × 𝑋)) 0 ) = (𝑥𝐷 0 ))
9	8	ancoms 458	. . . 4 ⊢ (( 0 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑥(𝐷 ↾ (𝑋 × 𝑋)) 0 ) = (𝑥𝐷 0 ))
10	7, 9	eqtr2id 2778	. . 3 ⊢ (( 0 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑥𝐷 0 ) = (𝑥𝐸 0 ))
11	10	mpteq2dva 5203	. 2 ⊢ ( 0 ∈ 𝑋 → (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 )) = (𝑥 ∈ 𝑋 ↦ (𝑥𝐸 0 )))
12	5, 11	eqtrid 2777	1 ⊢ ( 0 ∈ 𝑋 → 𝑁 = (𝑥 ∈ 𝑋 ↦ (𝑥𝐸 0 )))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ↦ cmpt 5191   × cxp 5639   ↾ cres 5643  ‘cfv 6514  (class class class)co 7390  Basecbs 17186  distcds 17236  0_gc0g 17409  normcnm 24471

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-fv 6522  df-ov 7393  df-nm 24477

This theorem is referenced by:  nmfval2  24486