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Theorem nmfval 24541
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 = (0g𝑊)
nmfval.d 𝐷 = (dist‘𝑊)
Assertion
Ref Expression
nmfval 𝑁 = (𝑥𝑋 ↦ (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.
StepHypRef Expression
1 nmfval.n . 2 𝑁 = (norm‘𝑊)
2 fveq2 6896 . . . . . 6 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
3 nmfval.x . . . . . 6 𝑋 = (Base‘𝑊)
42, 3eqtr4di 2783 . . . . 5 (𝑤 = 𝑊 → (Base‘𝑤) = 𝑋)
5 fveq2 6896 . . . . . . 7 (𝑤 = 𝑊 → (dist‘𝑤) = (dist‘𝑊))
6 nmfval.d . . . . . . 7 𝐷 = (dist‘𝑊)
75, 6eqtr4di 2783 . . . . . 6 (𝑤 = 𝑊 → (dist‘𝑤) = 𝐷)
8 eqidd 2726 . . . . . 6 (𝑤 = 𝑊𝑥 = 𝑥)
9 fveq2 6896 . . . . . . 7 (𝑤 = 𝑊 → (0g𝑤) = (0g𝑊))
10 nmfval.z . . . . . . 7 0 = (0g𝑊)
119, 10eqtr4di 2783 . . . . . 6 (𝑤 = 𝑊 → (0g𝑤) = 0 )
127, 8, 11oveq123d 7440 . . . . 5 (𝑤 = 𝑊 → (𝑥(dist‘𝑤)(0g𝑤)) = (𝑥𝐷 0 ))
134, 12mpteq12dv 5240 . . . 4 (𝑤 = 𝑊 → (𝑥 ∈ (Base‘𝑤) ↦ (𝑥(dist‘𝑤)(0g𝑤))) = (𝑥𝑋 ↦ (𝑥𝐷 0 )))
14 df-nm 24535 . . . 4 norm = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘𝑤) ↦ (𝑥(dist‘𝑤)(0g𝑤))))
15 eqid 2725 . . . . . 6 (𝑥𝑋 ↦ (𝑥𝐷 0 )) = (𝑥𝑋 ↦ (𝑥𝐷 0 ))
16 df-ov 7422 . . . . . . . 8 (𝑥𝐷 0 ) = (𝐷‘⟨𝑥, 0 ⟩)
17 fvrn0 6926 . . . . . . . 8 (𝐷‘⟨𝑥, 0 ⟩) ∈ (ran 𝐷 ∪ {∅})
1816, 17eqeltri 2821 . . . . . . 7 (𝑥𝐷 0 ) ∈ (ran 𝐷 ∪ {∅})
1918a1i 11 . . . . . 6 (𝑥𝑋 → (𝑥𝐷 0 ) ∈ (ran 𝐷 ∪ {∅}))
2015, 19fmpti 7121 . . . . 5 (𝑥𝑋 ↦ (𝑥𝐷 0 )):𝑋⟶(ran 𝐷 ∪ {∅})
213fvexi 6910 . . . . 5 𝑋 ∈ V
226fvexi 6910 . . . . . . 7 𝐷 ∈ V
2322rnex 7918 . . . . . 6 ran 𝐷 ∈ V
24 p0ex 5384 . . . . . 6 {∅} ∈ V
2523, 24unex 7749 . . . . 5 (ran 𝐷 ∪ {∅}) ∈ V
26 fex2 7942 . . . . 5 (((𝑥𝑋 ↦ (𝑥𝐷 0 )):𝑋⟶(ran 𝐷 ∪ {∅}) ∧ 𝑋 ∈ V ∧ (ran 𝐷 ∪ {∅}) ∈ V) → (𝑥𝑋 ↦ (𝑥𝐷 0 )) ∈ V)
2720, 21, 25, 26mp3an 1457 . . . 4 (𝑥𝑋 ↦ (𝑥𝐷 0 )) ∈ V
2813, 14, 27fvmpt 7004 . . 3 (𝑊 ∈ V → (norm‘𝑊) = (𝑥𝑋 ↦ (𝑥𝐷 0 )))
29 fvprc 6888 . . . . 5 𝑊 ∈ V → (norm‘𝑊) = ∅)
30 mpt0 6698 . . . . 5 (𝑥 ∈ ∅ ↦ (𝑥𝐷 0 )) = ∅
3129, 30eqtr4di 2783 . . . 4 𝑊 ∈ V → (norm‘𝑊) = (𝑥 ∈ ∅ ↦ (𝑥𝐷 0 )))
32 fvprc 6888 . . . . . 6 𝑊 ∈ V → (Base‘𝑊) = ∅)
333, 32eqtrid 2777 . . . . 5 𝑊 ∈ V → 𝑋 = ∅)
3433mpteq1d 5244 . . . 4 𝑊 ∈ V → (𝑥𝑋 ↦ (𝑥𝐷 0 )) = (𝑥 ∈ ∅ ↦ (𝑥𝐷 0 )))
3531, 34eqtr4d 2768 . . 3 𝑊 ∈ V → (norm‘𝑊) = (𝑥𝑋 ↦ (𝑥𝐷 0 )))
3628, 35pm2.61i 182 . 2 (norm‘𝑊) = (𝑥𝑋 ↦ (𝑥𝐷 0 ))
371, 36eqtri 2753 1 𝑁 = (𝑥𝑋 ↦ (𝑥𝐷 0 ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1533  wcel 2098  Vcvv 3461  cun 3942  c0 4322  {csn 4630  cop 4636  cmpt 5232  ran crn 5679  wf 6545  cfv 6549  (class class class)co 7419  Basecbs 17183  distcds 17245  0gc0g 17424  normcnm 24529
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-fv 6557  df-ov 7422  df-nm 24535
This theorem is referenced by:  nmval  24542  nmfval0  24543  nmpropd  24547  subgnm  24586  tngnm  24612  cnfldnm  24739  nmcn  24804  ressnm  32774
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