Theorem nmfval 24710

Description: The value of the norm function as the distance to zero. (Contributed 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
nmfval	⊢ 𝑁 = (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 ))

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

Allowed substitution hint:   𝑁(𝑥)

Proof of Theorem nmfval

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nmfval.n	. 2 ⊢ 𝑁 = (norm‘𝑊)
2		fveq2 6879	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
3		nmfval.x	. . . . . 6 ⊢ 𝑋 = (Base‘𝑊)
4	2, 3	eqtr4di 2822	. . . . 5 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝑋)
5		fveq2 6879	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (dist‘𝑤) = (dist‘𝑊))
6		nmfval.d	. . . . . . 7 ⊢ 𝐷 = (dist‘𝑊)
7	5, 6	eqtr4di 2822	. . . . . 6 ⊢ (𝑤 = 𝑊 → (dist‘𝑤) = 𝐷)
8		eqidd 2770	. . . . . 6 ⊢ (𝑤 = 𝑊 → 𝑥 = 𝑥)
9		fveq2 6879	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (0g‘𝑤) = (0g‘𝑊))
10		nmfval.z	. . . . . . 7 ⊢ 0 = (0g‘𝑊)
11	9, 10	eqtr4di 2822	. . . . . 6 ⊢ (𝑤 = 𝑊 → (0g‘𝑤) = 0 )
12	7, 8, 11	oveq123d 7429	. . . . 5 ⊢ (𝑤 = 𝑊 → (𝑥(dist‘𝑤)(0g‘𝑤)) = (𝑥𝐷 0 ))
13	4, 12	mpteq12dv 5199	. . . 4 ⊢ (𝑤 = 𝑊 → (𝑥 ∈ (Base‘𝑤) ↦ (𝑥(dist‘𝑤)(0g‘𝑤))) = (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 )))
14		df-nm 24704	. . . 4 ⊢ norm = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘𝑤) ↦ (𝑥(dist‘𝑤)(0g‘𝑤))))
15		eqid 2769	. . . . . 6 ⊢ (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 )) = (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 ))
16		df-ov 7411	. . . . . . . 8 ⊢ (𝑥𝐷 0 ) = (𝐷‘⟨𝑥, 0 ⟩)
17		fvrn0 6907	. . . . . . . 8 ⊢ (𝐷‘⟨𝑥, 0 ⟩) ∈ (ran 𝐷 ∪ {∅})
18	16, 17	eqeltri 2865	. . . . . . 7 ⊢ (𝑥𝐷 0 ) ∈ (ran 𝐷 ∪ {∅})
19	18	a1i 11	. . . . . 6 ⊢ (𝑥 ∈ 𝑋 → (𝑥𝐷 0 ) ∈ (ran 𝐷 ∪ {∅}))
20	15, 19	fmpti 7105	. . . . 5 ⊢ (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 )):𝑋⟶(ran 𝐷 ∪ {∅})
21	3	fvexi 6893	. . . . 5 ⊢ 𝑋 ∈ V
22	6	fvexi 6893	. . . . . . 7 ⊢ 𝐷 ∈ V
23	22	rnex 7903	. . . . . 6 ⊢ ran 𝐷 ∈ V
24		p0ex 5353	. . . . . 6 ⊢ {∅} ∈ V
25	23, 24	unex 7739	. . . . 5 ⊢ (ran 𝐷 ∪ {∅}) ∈ V
26		fex2 7929	. . . . 5 ⊢ (((𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 )):𝑋⟶(ran 𝐷 ∪ {∅}) ∧ 𝑋 ∈ V ∧ (ran 𝐷 ∪ {∅}) ∈ V) → (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 )) ∈ V)
27	20, 21, 25, 26	mp3an 1487	. . . 4 ⊢ (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 )) ∈ V
28	13, 14, 27	fvmpt 6987	. . 3 ⊢ (𝑊 ∈ V → (norm‘𝑊) = (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 )))
29		fvprc 6871	. . . . 5 ⊢ (¬ 𝑊 ∈ V → (norm‘𝑊) = ∅)
30		mpt0 6675	. . . . 5 ⊢ (𝑥 ∈ ∅ ↦ (𝑥𝐷 0 )) = ∅
31	29, 30	eqtr4di 2822	. . . 4 ⊢ (¬ 𝑊 ∈ V → (norm‘𝑊) = (𝑥 ∈ ∅ ↦ (𝑥𝐷 0 )))
32		fvprc 6871	. . . . . 6 ⊢ (¬ 𝑊 ∈ V → (Base‘𝑊) = ∅)
33	3, 32	eqtrid 2816	. . . . 5 ⊢ (¬ 𝑊 ∈ V → 𝑋 = ∅)
34	33	mpteq1d 5202	. . . 4 ⊢ (¬ 𝑊 ∈ V → (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 )) = (𝑥 ∈ ∅ ↦ (𝑥𝐷 0 )))
35	31, 34	eqtr4d 2807	. . 3 ⊢ (¬ 𝑊 ∈ V → (norm‘𝑊) = (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 )))
36	28, 35	pm2.61i 184	. 2 ⊢ (norm‘𝑊) = (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 ))
37	1, 36	eqtri 2792	1 ⊢ 𝑁 = (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 ))

Syntax hints:  ¬ wn 3   = wceq 1567   ∈ wcel 2149  Vcvv 3463   ∪ cun 3911  ∅c0 4294  {csn 4591  ⟨cop 4597   ↦ cmpt 5193  ran crn 5660  ⟶wf 6530  ‘cfv 6534  (class class class)co 7408  Basecbs 17265  distcds 17315  0_gc0g 17488  normcnm 24698

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 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730

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 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6490  df-fun 6536  df-fn 6537  df-f 6538  df-fv 6542  df-ov 7411  df-nm 24704

This theorem is referenced by:  nmval  24711  nmfval0  24712  nmpropd  24716  subgnm  24755  tngnm  24773  cnfldnm  24900  nmcn  24967  ressnm  33221