Theorem nmfval 22203

Description: The value of the norm function. (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 6103	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
3		nmfval.x	. . . . . 6 ⊢ 𝑋 = (Base‘𝑊)
4	2, 3	syl6eqr 2662	. . . . 5 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝑋)
5		fveq2 6103	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (dist‘𝑤) = (dist‘𝑊))
6		nmfval.d	. . . . . . 7 ⊢ 𝐷 = (dist‘𝑊)
7	5, 6	syl6eqr 2662	. . . . . 6 ⊢ (𝑤 = 𝑊 → (dist‘𝑤) = 𝐷)
8		eqidd 2611	. . . . . 6 ⊢ (𝑤 = 𝑊 → 𝑥 = 𝑥)
9		fveq2 6103	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (0g‘𝑤) = (0g‘𝑊))
10		nmfval.z	. . . . . . 7 ⊢ 0 = (0g‘𝑊)
11	9, 10	syl6eqr 2662	. . . . . 6 ⊢ (𝑤 = 𝑊 → (0g‘𝑤) = 0 )
12	7, 8, 11	oveq123d 6570	. . . . 5 ⊢ (𝑤 = 𝑊 → (𝑥(dist‘𝑤)(0g‘𝑤)) = (𝑥𝐷 0 ))
13	4, 12	mpteq12dv 4663	. . . 4 ⊢ (𝑤 = 𝑊 → (𝑥 ∈ (Base‘𝑤) ↦ (𝑥(dist‘𝑤)(0g‘𝑤))) = (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 )))
14		df-nm 22197	. . . 4 ⊢ norm = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘𝑤) ↦ (𝑥(dist‘𝑤)(0g‘𝑤))))
15		eqid 2610	. . . . . 6 ⊢ (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 )) = (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 ))
16		df-ov 6552	. . . . . . . 8 ⊢ (𝑥𝐷 0 ) = (𝐷‘⟨𝑥, 0 ⟩)
17		fvrn0 6126	. . . . . . . 8 ⊢ (𝐷‘⟨𝑥, 0 ⟩) ∈ (ran 𝐷 ∪ {∅})
18	16, 17	eqeltri 2684	. . . . . . 7 ⊢ (𝑥𝐷 0 ) ∈ (ran 𝐷 ∪ {∅})
19	18	a1i 11	. . . . . 6 ⊢ (𝑥 ∈ 𝑋 → (𝑥𝐷 0 ) ∈ (ran 𝐷 ∪ {∅}))
20	15, 19	fmpti 6291	. . . . 5 ⊢ (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 )):𝑋⟶(ran 𝐷 ∪ {∅})
21		fvex 6113	. . . . . 6 ⊢ (Base‘𝑊) ∈ V
22	3, 21	eqeltri 2684	. . . . 5 ⊢ 𝑋 ∈ V
23		fvex 6113	. . . . . . . 8 ⊢ (dist‘𝑊) ∈ V
24	6, 23	eqeltri 2684	. . . . . . 7 ⊢ 𝐷 ∈ V
25	24	rnex 6992	. . . . . 6 ⊢ ran 𝐷 ∈ V
26		p0ex 4779	. . . . . 6 ⊢ {∅} ∈ V
27	25, 26	unex 6854	. . . . 5 ⊢ (ran 𝐷 ∪ {∅}) ∈ V
28		fex2 7014	. . . . 5 ⊢ (((𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 )):𝑋⟶(ran 𝐷 ∪ {∅}) ∧ 𝑋 ∈ V ∧ (ran 𝐷 ∪ {∅}) ∈ V) → (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 )) ∈ V)
29	20, 22, 27, 28	mp3an 1416	. . . 4 ⊢ (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 )) ∈ V
30	13, 14, 29	fvmpt 6191	. . 3 ⊢ (𝑊 ∈ V → (norm‘𝑊) = (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 )))
31		fvprc 6097	. . . . 5 ⊢ (¬ 𝑊 ∈ V → (norm‘𝑊) = ∅)
32		mpt0 5934	. . . . 5 ⊢ (𝑥 ∈ ∅ ↦ (𝑥𝐷 0 )) = ∅
33	31, 32	syl6eqr 2662	. . . 4 ⊢ (¬ 𝑊 ∈ V → (norm‘𝑊) = (𝑥 ∈ ∅ ↦ (𝑥𝐷 0 )))
34		fvprc 6097	. . . . . 6 ⊢ (¬ 𝑊 ∈ V → (Base‘𝑊) = ∅)
35	3, 34	syl5eq 2656	. . . . 5 ⊢ (¬ 𝑊 ∈ V → 𝑋 = ∅)
36	35	mpteq1d 4666	. . . 4 ⊢ (¬ 𝑊 ∈ V → (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 )) = (𝑥 ∈ ∅ ↦ (𝑥𝐷 0 )))
37	33, 36	eqtr4d 2647	. . 3 ⊢ (¬ 𝑊 ∈ V → (norm‘𝑊) = (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 )))
38	30, 37	pm2.61i 175	. 2 ⊢ (norm‘𝑊) = (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 ))
39	1, 38	eqtri 2632	1 ⊢ 𝑁 = (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 ))

Syntax hints:  ¬ wn 3   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∪ cun 3538  ∅c0 3874  {csn 4125  ⟨cop 4131   ↦ cmpt 4643  ran crn 5039  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  distcds 15777  0_gc0g 15923  normcnm 22191

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-nm 22197

This theorem is referenced by:  nmval  22204  nmfval2  22205  nmpropd  22208  subgnm  22247  tngnm  22265  cnfldnm  22392  nmcn  22455  ressnm  28982