Theorem nmfval 23440

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 6695	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
3		nmfval.x	. . . . . 6 ⊢ 𝑋 = (Base‘𝑊)
4	2, 3	eqtr4di 2789	. . . . 5 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝑋)
5		fveq2 6695	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (dist‘𝑤) = (dist‘𝑊))
6		nmfval.d	. . . . . . 7 ⊢ 𝐷 = (dist‘𝑊)
7	5, 6	eqtr4di 2789	. . . . . 6 ⊢ (𝑤 = 𝑊 → (dist‘𝑤) = 𝐷)
8		eqidd 2737	. . . . . 6 ⊢ (𝑤 = 𝑊 → 𝑥 = 𝑥)
9		fveq2 6695	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (0g‘𝑤) = (0g‘𝑊))
10		nmfval.z	. . . . . . 7 ⊢ 0 = (0g‘𝑊)
11	9, 10	eqtr4di 2789	. . . . . 6 ⊢ (𝑤 = 𝑊 → (0g‘𝑤) = 0 )
12	7, 8, 11	oveq123d 7212	. . . . 5 ⊢ (𝑤 = 𝑊 → (𝑥(dist‘𝑤)(0g‘𝑤)) = (𝑥𝐷 0 ))
13	4, 12	mpteq12dv 5125	. . . 4 ⊢ (𝑤 = 𝑊 → (𝑥 ∈ (Base‘𝑤) ↦ (𝑥(dist‘𝑤)(0g‘𝑤))) = (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 )))
14		df-nm 23434	. . . 4 ⊢ norm = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘𝑤) ↦ (𝑥(dist‘𝑤)(0g‘𝑤))))
15		eqid 2736	. . . . . 6 ⊢ (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 )) = (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 ))
16		df-ov 7194	. . . . . . . 8 ⊢ (𝑥𝐷 0 ) = (𝐷‘⟨𝑥, 0 ⟩)
17		fvrn0 6723	. . . . . . . 8 ⊢ (𝐷‘⟨𝑥, 0 ⟩) ∈ (ran 𝐷 ∪ {∅})
18	16, 17	eqeltri 2827	. . . . . . 7 ⊢ (𝑥𝐷 0 ) ∈ (ran 𝐷 ∪ {∅})
19	18	a1i 11	. . . . . 6 ⊢ (𝑥 ∈ 𝑋 → (𝑥𝐷 0 ) ∈ (ran 𝐷 ∪ {∅}))
20	15, 19	fmpti 6907	. . . . 5 ⊢ (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 )):𝑋⟶(ran 𝐷 ∪ {∅})
21	3	fvexi 6709	. . . . 5 ⊢ 𝑋 ∈ V
22	6	fvexi 6709	. . . . . . 7 ⊢ 𝐷 ∈ V
23	22	rnex 7668	. . . . . 6 ⊢ ran 𝐷 ∈ V
24		p0ex 5262	. . . . . 6 ⊢ {∅} ∈ V
25	23, 24	unex 7509	. . . . 5 ⊢ (ran 𝐷 ∪ {∅}) ∈ V
26		fex2 7689	. . . . 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 6796	. . 3 ⊢ (𝑊 ∈ V → (norm‘𝑊) = (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 )))
29		fvprc 6687	. . . . 5 ⊢ (¬ 𝑊 ∈ V → (norm‘𝑊) = ∅)
30		mpt0 6498	. . . . 5 ⊢ (𝑥 ∈ ∅ ↦ (𝑥𝐷 0 )) = ∅
31	29, 30	eqtr4di 2789	. . . 4 ⊢ (¬ 𝑊 ∈ V → (norm‘𝑊) = (𝑥 ∈ ∅ ↦ (𝑥𝐷 0 )))
32		fvprc 6687	. . . . . 6 ⊢ (¬ 𝑊 ∈ V → (Base‘𝑊) = ∅)
33	3, 32	syl5eq 2783	. . . . 5 ⊢ (¬ 𝑊 ∈ V → 𝑋 = ∅)
34	33	mpteq1d 5129	. . . 4 ⊢ (¬ 𝑊 ∈ V → (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 )) = (𝑥 ∈ ∅ ↦ (𝑥𝐷 0 )))
35	31, 34	eqtr4d 2774	. . 3 ⊢ (¬ 𝑊 ∈ V → (norm‘𝑊) = (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 )))
36	28, 35	pm2.61i 185	. 2 ⊢ (norm‘𝑊) = (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 ))
37	1, 36	eqtri 2759	1 ⊢ 𝑁 = (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 ))

Syntax hints:  ¬ wn 3   = wceq 1543   ∈ wcel 2112  Vcvv 3398   ∪ cun 3851  ∅c0 4223  {csn 4527  ⟨cop 4533   ↦ cmpt 5120  ran crn 5537  ⟶wf 6354  ‘cfv 6358  (class class class)co 7191  Basecbs 16666  distcds 16758  0_gc0g 16898  normcnm 23428

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pow 5243  ax-pr 5307  ax-un 7501

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3400  df-sbc 3684  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-pw 4501  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-br 5040  df-opab 5102  df-mpt 5121  df-id 5440  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6316  df-fun 6360  df-fn 6361  df-f 6362  df-fv 6366  df-ov 7194  df-nm 23434

This theorem is referenced by:  nmval  23441  nmfval0  23442  nmpropd  23446  subgnm  23485  tngnm  23503  cnfldnm  23630  nmcn  23695  ressnm  30910