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Theorem nmfval 22615
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 6354 . . . . . 6 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
3 nmfval.x . . . . . 6 𝑋 = (Base‘𝑊)
42, 3syl6eqr 2813 . . . . 5 (𝑤 = 𝑊 → (Base‘𝑤) = 𝑋)
5 fveq2 6354 . . . . . . 7 (𝑤 = 𝑊 → (dist‘𝑤) = (dist‘𝑊))
6 nmfval.d . . . . . . 7 𝐷 = (dist‘𝑊)
75, 6syl6eqr 2813 . . . . . 6 (𝑤 = 𝑊 → (dist‘𝑤) = 𝐷)
8 eqidd 2762 . . . . . 6 (𝑤 = 𝑊𝑥 = 𝑥)
9 fveq2 6354 . . . . . . 7 (𝑤 = 𝑊 → (0g𝑤) = (0g𝑊))
10 nmfval.z . . . . . . 7 0 = (0g𝑊)
119, 10syl6eqr 2813 . . . . . 6 (𝑤 = 𝑊 → (0g𝑤) = 0 )
127, 8, 11oveq123d 6836 . . . . 5 (𝑤 = 𝑊 → (𝑥(dist‘𝑤)(0g𝑤)) = (𝑥𝐷 0 ))
134, 12mpteq12dv 4886 . . . 4 (𝑤 = 𝑊 → (𝑥 ∈ (Base‘𝑤) ↦ (𝑥(dist‘𝑤)(0g𝑤))) = (𝑥𝑋 ↦ (𝑥𝐷 0 )))
14 df-nm 22609 . . . 4 norm = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘𝑤) ↦ (𝑥(dist‘𝑤)(0g𝑤))))
15 eqid 2761 . . . . . 6 (𝑥𝑋 ↦ (𝑥𝐷 0 )) = (𝑥𝑋 ↦ (𝑥𝐷 0 ))
16 df-ov 6818 . . . . . . . 8 (𝑥𝐷 0 ) = (𝐷‘⟨𝑥, 0 ⟩)
17 fvrn0 6379 . . . . . . . 8 (𝐷‘⟨𝑥, 0 ⟩) ∈ (ran 𝐷 ∪ {∅})
1816, 17eqeltri 2836 . . . . . . 7 (𝑥𝐷 0 ) ∈ (ran 𝐷 ∪ {∅})
1918a1i 11 . . . . . 6 (𝑥𝑋 → (𝑥𝐷 0 ) ∈ (ran 𝐷 ∪ {∅}))
2015, 19fmpti 6548 . . . . 5 (𝑥𝑋 ↦ (𝑥𝐷 0 )):𝑋⟶(ran 𝐷 ∪ {∅})
21 fvex 6364 . . . . . 6 (Base‘𝑊) ∈ V
223, 21eqeltri 2836 . . . . 5 𝑋 ∈ V
23 fvex 6364 . . . . . . . 8 (dist‘𝑊) ∈ V
246, 23eqeltri 2836 . . . . . . 7 𝐷 ∈ V
2524rnex 7267 . . . . . 6 ran 𝐷 ∈ V
26 p0ex 5003 . . . . . 6 {∅} ∈ V
2725, 26unex 7123 . . . . 5 (ran 𝐷 ∪ {∅}) ∈ V
28 fex2 7288 . . . . 5 (((𝑥𝑋 ↦ (𝑥𝐷 0 )):𝑋⟶(ran 𝐷 ∪ {∅}) ∧ 𝑋 ∈ V ∧ (ran 𝐷 ∪ {∅}) ∈ V) → (𝑥𝑋 ↦ (𝑥𝐷 0 )) ∈ V)
2920, 22, 27, 28mp3an 1573 . . . 4 (𝑥𝑋 ↦ (𝑥𝐷 0 )) ∈ V
3013, 14, 29fvmpt 6446 . . 3 (𝑊 ∈ V → (norm‘𝑊) = (𝑥𝑋 ↦ (𝑥𝐷 0 )))
31 fvprc 6348 . . . . 5 𝑊 ∈ V → (norm‘𝑊) = ∅)
32 mpt0 6183 . . . . 5 (𝑥 ∈ ∅ ↦ (𝑥𝐷 0 )) = ∅
3331, 32syl6eqr 2813 . . . 4 𝑊 ∈ V → (norm‘𝑊) = (𝑥 ∈ ∅ ↦ (𝑥𝐷 0 )))
34 fvprc 6348 . . . . . 6 𝑊 ∈ V → (Base‘𝑊) = ∅)
353, 34syl5eq 2807 . . . . 5 𝑊 ∈ V → 𝑋 = ∅)
3635mpteq1d 4891 . . . 4 𝑊 ∈ V → (𝑥𝑋 ↦ (𝑥𝐷 0 )) = (𝑥 ∈ ∅ ↦ (𝑥𝐷 0 )))
3733, 36eqtr4d 2798 . . 3 𝑊 ∈ V → (norm‘𝑊) = (𝑥𝑋 ↦ (𝑥𝐷 0 )))
3830, 37pm2.61i 176 . 2 (norm‘𝑊) = (𝑥𝑋 ↦ (𝑥𝐷 0 ))
391, 38eqtri 2783 1 𝑁 = (𝑥𝑋 ↦ (𝑥𝐷 0 ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1632  wcel 2140  Vcvv 3341  cun 3714  c0 4059  {csn 4322  cop 4328  cmpt 4882  ran crn 5268  wf 6046  cfv 6050  (class class class)co 6815  Basecbs 16080  distcds 16173  0gc0g 16323  normcnm 22603
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-sbc 3578  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-br 4806  df-opab 4866  df-mpt 4883  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-fv 6058  df-ov 6818  df-nm 22609
This theorem is referenced by:  nmval  22616  nmfval2  22617  nmpropd  22620  subgnm  22659  tngnm  22677  cnfldnm  22804  nmcn  22869  ressnm  29982
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