Theorem nmfval 24601

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 6906	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
3		nmfval.x	. . . . . 6 ⊢ 𝑋 = (Base‘𝑊)
4	2, 3	eqtr4di 2795	. . . . 5 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝑋)
5		fveq2 6906	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (dist‘𝑤) = (dist‘𝑊))
6		nmfval.d	. . . . . . 7 ⊢ 𝐷 = (dist‘𝑊)
7	5, 6	eqtr4di 2795	. . . . . 6 ⊢ (𝑤 = 𝑊 → (dist‘𝑤) = 𝐷)
8		eqidd 2738	. . . . . 6 ⊢ (𝑤 = 𝑊 → 𝑥 = 𝑥)
9		fveq2 6906	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (0g‘𝑤) = (0g‘𝑊))
10		nmfval.z	. . . . . . 7 ⊢ 0 = (0g‘𝑊)
11	9, 10	eqtr4di 2795	. . . . . 6 ⊢ (𝑤 = 𝑊 → (0g‘𝑤) = 0 )
12	7, 8, 11	oveq123d 7452	. . . . 5 ⊢ (𝑤 = 𝑊 → (𝑥(dist‘𝑤)(0g‘𝑤)) = (𝑥𝐷 0 ))
13	4, 12	mpteq12dv 5233	. . . 4 ⊢ (𝑤 = 𝑊 → (𝑥 ∈ (Base‘𝑤) ↦ (𝑥(dist‘𝑤)(0g‘𝑤))) = (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 )))
14		df-nm 24595	. . . 4 ⊢ norm = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘𝑤) ↦ (𝑥(dist‘𝑤)(0g‘𝑤))))
15		eqid 2737	. . . . . 6 ⊢ (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 )) = (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 ))
16		df-ov 7434	. . . . . . . 8 ⊢ (𝑥𝐷 0 ) = (𝐷‘⟨𝑥, 0 ⟩)
17		fvrn0 6936	. . . . . . . 8 ⊢ (𝐷‘⟨𝑥, 0 ⟩) ∈ (ran 𝐷 ∪ {∅})
18	16, 17	eqeltri 2837	. . . . . . 7 ⊢ (𝑥𝐷 0 ) ∈ (ran 𝐷 ∪ {∅})
19	18	a1i 11	. . . . . 6 ⊢ (𝑥 ∈ 𝑋 → (𝑥𝐷 0 ) ∈ (ran 𝐷 ∪ {∅}))
20	15, 19	fmpti 7132	. . . . 5 ⊢ (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 )):𝑋⟶(ran 𝐷 ∪ {∅})
21	3	fvexi 6920	. . . . 5 ⊢ 𝑋 ∈ V
22	6	fvexi 6920	. . . . . . 7 ⊢ 𝐷 ∈ V
23	22	rnex 7932	. . . . . 6 ⊢ ran 𝐷 ∈ V
24		p0ex 5384	. . . . . 6 ⊢ {∅} ∈ V
25	23, 24	unex 7764	. . . . 5 ⊢ (ran 𝐷 ∪ {∅}) ∈ V
26		fex2 7958	. . . . 5 ⊢ (((𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 )):𝑋⟶(ran 𝐷 ∪ {∅}) ∧ 𝑋 ∈ V ∧ (ran 𝐷 ∪ {∅}) ∈ V) → (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 )) ∈ V)
27	20, 21, 25, 26	mp3an 1463	. . . 4 ⊢ (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 )) ∈ V
28	13, 14, 27	fvmpt 7016	. . 3 ⊢ (𝑊 ∈ V → (norm‘𝑊) = (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 )))
29		fvprc 6898	. . . . 5 ⊢ (¬ 𝑊 ∈ V → (norm‘𝑊) = ∅)
30		mpt0 6710	. . . . 5 ⊢ (𝑥 ∈ ∅ ↦ (𝑥𝐷 0 )) = ∅
31	29, 30	eqtr4di 2795	. . . 4 ⊢ (¬ 𝑊 ∈ V → (norm‘𝑊) = (𝑥 ∈ ∅ ↦ (𝑥𝐷 0 )))
32		fvprc 6898	. . . . . 6 ⊢ (¬ 𝑊 ∈ V → (Base‘𝑊) = ∅)
33	3, 32	eqtrid 2789	. . . . 5 ⊢ (¬ 𝑊 ∈ V → 𝑋 = ∅)
34	33	mpteq1d 5237	. . . 4 ⊢ (¬ 𝑊 ∈ V → (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 )) = (𝑥 ∈ ∅ ↦ (𝑥𝐷 0 )))
35	31, 34	eqtr4d 2780	. . 3 ⊢ (¬ 𝑊 ∈ V → (norm‘𝑊) = (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 )))
36	28, 35	pm2.61i 182	. 2 ⊢ (norm‘𝑊) = (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 ))
37	1, 36	eqtri 2765	1 ⊢ 𝑁 = (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 ))

Syntax hints:  ¬ wn 3   = wceq 1540   ∈ wcel 2108  Vcvv 3480   ∪ cun 3949  ∅c0 4333  {csn 4626  ⟨cop 4632   ↦ cmpt 5225  ran crn 5686  ⟶wf 6557  ‘cfv 6561  (class class class)co 7431  Basecbs 17247  distcds 17306  0_gc0g 17484  normcnm 24589

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-nm 24595

This theorem is referenced by:  nmval  24602  nmfval0  24603  nmpropd  24607  subgnm  24646  tngnm  24672  cnfldnm  24799  nmcn  24866  ressnm  32949