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Theorem nmfval 23198
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 = (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 6649 . . . . . 6 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
3 nmfval.x . . . . . 6 𝑋 = (Base‘𝑊)
42, 3eqtr4di 2854 . . . . 5 (𝑤 = 𝑊 → (Base‘𝑤) = 𝑋)
5 fveq2 6649 . . . . . . 7 (𝑤 = 𝑊 → (dist‘𝑤) = (dist‘𝑊))
6 nmfval.d . . . . . . 7 𝐷 = (dist‘𝑊)
75, 6eqtr4di 2854 . . . . . 6 (𝑤 = 𝑊 → (dist‘𝑤) = 𝐷)
8 eqidd 2802 . . . . . 6 (𝑤 = 𝑊𝑥 = 𝑥)
9 fveq2 6649 . . . . . . 7 (𝑤 = 𝑊 → (0g𝑤) = (0g𝑊))
10 nmfval.z . . . . . . 7 0 = (0g𝑊)
119, 10eqtr4di 2854 . . . . . 6 (𝑤 = 𝑊 → (0g𝑤) = 0 )
127, 8, 11oveq123d 7160 . . . . 5 (𝑤 = 𝑊 → (𝑥(dist‘𝑤)(0g𝑤)) = (𝑥𝐷 0 ))
134, 12mpteq12dv 5118 . . . 4 (𝑤 = 𝑊 → (𝑥 ∈ (Base‘𝑤) ↦ (𝑥(dist‘𝑤)(0g𝑤))) = (𝑥𝑋 ↦ (𝑥𝐷 0 )))
14 df-nm 23192 . . . 4 norm = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘𝑤) ↦ (𝑥(dist‘𝑤)(0g𝑤))))
15 eqid 2801 . . . . . 6 (𝑥𝑋 ↦ (𝑥𝐷 0 )) = (𝑥𝑋 ↦ (𝑥𝐷 0 ))
16 df-ov 7142 . . . . . . . 8 (𝑥𝐷 0 ) = (𝐷‘⟨𝑥, 0 ⟩)
17 fvrn0 6677 . . . . . . . 8 (𝐷‘⟨𝑥, 0 ⟩) ∈ (ran 𝐷 ∪ {∅})
1816, 17eqeltri 2889 . . . . . . 7 (𝑥𝐷 0 ) ∈ (ran 𝐷 ∪ {∅})
1918a1i 11 . . . . . 6 (𝑥𝑋 → (𝑥𝐷 0 ) ∈ (ran 𝐷 ∪ {∅}))
2015, 19fmpti 6857 . . . . 5 (𝑥𝑋 ↦ (𝑥𝐷 0 )):𝑋⟶(ran 𝐷 ∪ {∅})
213fvexi 6663 . . . . 5 𝑋 ∈ V
226fvexi 6663 . . . . . . 7 𝐷 ∈ V
2322rnex 7603 . . . . . 6 ran 𝐷 ∈ V
24 p0ex 5253 . . . . . 6 {∅} ∈ V
2523, 24unex 7453 . . . . 5 (ran 𝐷 ∪ {∅}) ∈ V
26 fex2 7624 . . . . 5 (((𝑥𝑋 ↦ (𝑥𝐷 0 )):𝑋⟶(ran 𝐷 ∪ {∅}) ∧ 𝑋 ∈ V ∧ (ran 𝐷 ∪ {∅}) ∈ V) → (𝑥𝑋 ↦ (𝑥𝐷 0 )) ∈ V)
2720, 21, 25, 26mp3an 1458 . . . 4 (𝑥𝑋 ↦ (𝑥𝐷 0 )) ∈ V
2813, 14, 27fvmpt 6749 . . 3 (𝑊 ∈ V → (norm‘𝑊) = (𝑥𝑋 ↦ (𝑥𝐷 0 )))
29 fvprc 6642 . . . . 5 𝑊 ∈ V → (norm‘𝑊) = ∅)
30 mpt0 6466 . . . . 5 (𝑥 ∈ ∅ ↦ (𝑥𝐷 0 )) = ∅
3129, 30eqtr4di 2854 . . . 4 𝑊 ∈ V → (norm‘𝑊) = (𝑥 ∈ ∅ ↦ (𝑥𝐷 0 )))
32 fvprc 6642 . . . . . 6 𝑊 ∈ V → (Base‘𝑊) = ∅)
333, 32syl5eq 2848 . . . . 5 𝑊 ∈ V → 𝑋 = ∅)
3433mpteq1d 5122 . . . 4 𝑊 ∈ V → (𝑥𝑋 ↦ (𝑥𝐷 0 )) = (𝑥 ∈ ∅ ↦ (𝑥𝐷 0 )))
3531, 34eqtr4d 2839 . . 3 𝑊 ∈ V → (norm‘𝑊) = (𝑥𝑋 ↦ (𝑥𝐷 0 )))
3628, 35pm2.61i 185 . 2 (norm‘𝑊) = (𝑥𝑋 ↦ (𝑥𝐷 0 ))
371, 36eqtri 2824 1 𝑁 = (𝑥𝑋 ↦ (𝑥𝐷 0 ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1538  wcel 2112  Vcvv 3444  cun 3882  c0 4246  {csn 4528  cop 4534  cmpt 5113  ran crn 5524  wf 6324  cfv 6328  (class class class)co 7139  Basecbs 16478  distcds 16569  0gc0g 16708  normcnm 23186
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336  df-ov 7142  df-nm 23192
This theorem is referenced by:  nmval  23199  nmfval2  23200  nmpropd  23203  subgnm  23242  tngnm  23260  cnfldnm  23387  nmcn  23452  ressnm  30667
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