Theorem nmfval 24636

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 6862	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
3		nmfval.x	. . . . . 6 ⊢ 𝑋 = (Base‘𝑊)
4	2, 3	eqtr4di 2814	. . . . 5 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝑋)
5		fveq2 6862	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (dist‘𝑤) = (dist‘𝑊))
6		nmfval.d	. . . . . . 7 ⊢ 𝐷 = (dist‘𝑊)
7	5, 6	eqtr4di 2814	. . . . . 6 ⊢ (𝑤 = 𝑊 → (dist‘𝑤) = 𝐷)
8		eqidd 2762	. . . . . 6 ⊢ (𝑤 = 𝑊 → 𝑥 = 𝑥)
9		fveq2 6862	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (0g‘𝑤) = (0g‘𝑊))
10		nmfval.z	. . . . . . 7 ⊢ 0 = (0g‘𝑊)
11	9, 10	eqtr4di 2814	. . . . . 6 ⊢ (𝑤 = 𝑊 → (0g‘𝑤) = 0 )
12	7, 8, 11	oveq123d 7412	. . . . 5 ⊢ (𝑤 = 𝑊 → (𝑥(dist‘𝑤)(0g‘𝑤)) = (𝑥𝐷 0 ))
13	4, 12	mpteq12dv 5184	. . . 4 ⊢ (𝑤 = 𝑊 → (𝑥 ∈ (Base‘𝑤) ↦ (𝑥(dist‘𝑤)(0g‘𝑤))) = (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 )))
14		df-nm 24630	. . . 4 ⊢ norm = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘𝑤) ↦ (𝑥(dist‘𝑤)(0g‘𝑤))))
15		eqid 2761	. . . . . 6 ⊢ (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 )) = (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 ))
16		df-ov 7394	. . . . . . . 8 ⊢ (𝑥𝐷 0 ) = (𝐷‘⟨𝑥, 0 ⟩)
17		fvrn0 6890	. . . . . . . 8 ⊢ (𝐷‘⟨𝑥, 0 ⟩) ∈ (ran 𝐷 ∪ {∅})
18	16, 17	eqeltri 2857	. . . . . . 7 ⊢ (𝑥𝐷 0 ) ∈ (ran 𝐷 ∪ {∅})
19	18	a1i 11	. . . . . 6 ⊢ (𝑥 ∈ 𝑋 → (𝑥𝐷 0 ) ∈ (ran 𝐷 ∪ {∅}))
20	15, 19	fmpti 7088	. . . . 5 ⊢ (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 )):𝑋⟶(ran 𝐷 ∪ {∅})
21	3	fvexi 6876	. . . . 5 ⊢ 𝑋 ∈ V
22	6	fvexi 6876	. . . . . . 7 ⊢ 𝐷 ∈ V
23	22	rnex 7886	. . . . . 6 ⊢ ran 𝐷 ∈ V
24		p0ex 5338	. . . . . 6 ⊢ {∅} ∈ V
25	23, 24	unex 7722	. . . . 5 ⊢ (ran 𝐷 ∪ {∅}) ∈ V
26		fex2 7912	. . . . 5 ⊢ (((𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 )):𝑋⟶(ran 𝐷 ∪ {∅}) ∧ 𝑋 ∈ V ∧ (ran 𝐷 ∪ {∅}) ∈ V) → (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 )) ∈ V)
27	20, 21, 25, 26	mp3an 1481	. . . 4 ⊢ (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 )) ∈ V
28	13, 14, 27	fvmpt 6970	. . 3 ⊢ (𝑊 ∈ V → (norm‘𝑊) = (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 )))
29		fvprc 6854	. . . . 5 ⊢ (¬ 𝑊 ∈ V → (norm‘𝑊) = ∅)
30		mpt0 6658	. . . . 5 ⊢ (𝑥 ∈ ∅ ↦ (𝑥𝐷 0 )) = ∅
31	29, 30	eqtr4di 2814	. . . 4 ⊢ (¬ 𝑊 ∈ V → (norm‘𝑊) = (𝑥 ∈ ∅ ↦ (𝑥𝐷 0 )))
32		fvprc 6854	. . . . . 6 ⊢ (¬ 𝑊 ∈ V → (Base‘𝑊) = ∅)
33	3, 32	eqtrid 2808	. . . . 5 ⊢ (¬ 𝑊 ∈ V → 𝑋 = ∅)
34	33	mpteq1d 5187	. . . 4 ⊢ (¬ 𝑊 ∈ V → (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 )) = (𝑥 ∈ ∅ ↦ (𝑥𝐷 0 )))
35	31, 34	eqtr4d 2799	. . 3 ⊢ (¬ 𝑊 ∈ V → (norm‘𝑊) = (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 )))
36	28, 35	pm2.61i 183	. 2 ⊢ (norm‘𝑊) = (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 ))
37	1, 36	eqtri 2784	1 ⊢ 𝑁 = (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 ))

Syntax hints:  ¬ wn 3   = wceq 1559   ∈ wcel 2141  Vcvv 3453   ∪ cun 3900  ∅c0 4283  {csn 4579  ⟨cop 4585   ↦ cmpt 5178  ran crn 5644  ⟶wf 6512  ‘cfv 6516  (class class class)co 7391  Basecbs 17236  distcds 17286  0_gc0g 17459  normcnm 24624

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7713

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-fv 6524  df-ov 7394  df-nm 24630

This theorem is referenced by:  nmval  24637  nmfval0  24638  nmpropd  24642  subgnm  24681  tngnm  24699  cnfldnm  24826  nmcn  24893  ressnm  33103