MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nmfval Structured version   Visualization version   GIF version

Theorem nmfval 22613
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 6332 . . . . . 6 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
3 nmfval.x . . . . . 6 𝑋 = (Base‘𝑊)
42, 3syl6eqr 2823 . . . . 5 (𝑤 = 𝑊 → (Base‘𝑤) = 𝑋)
5 fveq2 6332 . . . . . . 7 (𝑤 = 𝑊 → (dist‘𝑤) = (dist‘𝑊))
6 nmfval.d . . . . . . 7 𝐷 = (dist‘𝑊)
75, 6syl6eqr 2823 . . . . . 6 (𝑤 = 𝑊 → (dist‘𝑤) = 𝐷)
8 eqidd 2772 . . . . . 6 (𝑤 = 𝑊𝑥 = 𝑥)
9 fveq2 6332 . . . . . . 7 (𝑤 = 𝑊 → (0g𝑤) = (0g𝑊))
10 nmfval.z . . . . . . 7 0 = (0g𝑊)
119, 10syl6eqr 2823 . . . . . 6 (𝑤 = 𝑊 → (0g𝑤) = 0 )
127, 8, 11oveq123d 6814 . . . . 5 (𝑤 = 𝑊 → (𝑥(dist‘𝑤)(0g𝑤)) = (𝑥𝐷 0 ))
134, 12mpteq12dv 4867 . . . 4 (𝑤 = 𝑊 → (𝑥 ∈ (Base‘𝑤) ↦ (𝑥(dist‘𝑤)(0g𝑤))) = (𝑥𝑋 ↦ (𝑥𝐷 0 )))
14 df-nm 22607 . . . 4 norm = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘𝑤) ↦ (𝑥(dist‘𝑤)(0g𝑤))))
15 eqid 2771 . . . . . 6 (𝑥𝑋 ↦ (𝑥𝐷 0 )) = (𝑥𝑋 ↦ (𝑥𝐷 0 ))
16 df-ov 6796 . . . . . . . 8 (𝑥𝐷 0 ) = (𝐷‘⟨𝑥, 0 ⟩)
17 fvrn0 6357 . . . . . . . 8 (𝐷‘⟨𝑥, 0 ⟩) ∈ (ran 𝐷 ∪ {∅})
1816, 17eqeltri 2846 . . . . . . 7 (𝑥𝐷 0 ) ∈ (ran 𝐷 ∪ {∅})
1918a1i 11 . . . . . 6 (𝑥𝑋 → (𝑥𝐷 0 ) ∈ (ran 𝐷 ∪ {∅}))
2015, 19fmpti 6525 . . . . 5 (𝑥𝑋 ↦ (𝑥𝐷 0 )):𝑋⟶(ran 𝐷 ∪ {∅})
21 fvex 6342 . . . . . 6 (Base‘𝑊) ∈ V
223, 21eqeltri 2846 . . . . 5 𝑋 ∈ V
23 fvex 6342 . . . . . . . 8 (dist‘𝑊) ∈ V
246, 23eqeltri 2846 . . . . . . 7 𝐷 ∈ V
2524rnex 7247 . . . . . 6 ran 𝐷 ∈ V
26 p0ex 4984 . . . . . 6 {∅} ∈ V
2725, 26unex 7103 . . . . 5 (ran 𝐷 ∪ {∅}) ∈ V
28 fex2 7268 . . . . 5 (((𝑥𝑋 ↦ (𝑥𝐷 0 )):𝑋⟶(ran 𝐷 ∪ {∅}) ∧ 𝑋 ∈ V ∧ (ran 𝐷 ∪ {∅}) ∈ V) → (𝑥𝑋 ↦ (𝑥𝐷 0 )) ∈ V)
2920, 22, 27, 28mp3an 1572 . . . 4 (𝑥𝑋 ↦ (𝑥𝐷 0 )) ∈ V
3013, 14, 29fvmpt 6424 . . 3 (𝑊 ∈ V → (norm‘𝑊) = (𝑥𝑋 ↦ (𝑥𝐷 0 )))
31 fvprc 6326 . . . . 5 𝑊 ∈ V → (norm‘𝑊) = ∅)
32 mpt0 6161 . . . . 5 (𝑥 ∈ ∅ ↦ (𝑥𝐷 0 )) = ∅
3331, 32syl6eqr 2823 . . . 4 𝑊 ∈ V → (norm‘𝑊) = (𝑥 ∈ ∅ ↦ (𝑥𝐷 0 )))
34 fvprc 6326 . . . . . 6 𝑊 ∈ V → (Base‘𝑊) = ∅)
353, 34syl5eq 2817 . . . . 5 𝑊 ∈ V → 𝑋 = ∅)
3635mpteq1d 4872 . . . 4 𝑊 ∈ V → (𝑥𝑋 ↦ (𝑥𝐷 0 )) = (𝑥 ∈ ∅ ↦ (𝑥𝐷 0 )))
3733, 36eqtr4d 2808 . . 3 𝑊 ∈ V → (norm‘𝑊) = (𝑥𝑋 ↦ (𝑥𝐷 0 )))
3830, 37pm2.61i 176 . 2 (norm‘𝑊) = (𝑥𝑋 ↦ (𝑥𝐷 0 ))
391, 38eqtri 2793 1 𝑁 = (𝑥𝑋 ↦ (𝑥𝐷 0 ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1631  wcel 2145  Vcvv 3351  cun 3721  c0 4063  {csn 4316  cop 4322  cmpt 4863  ran crn 5250  wf 6027  cfv 6031  (class class class)co 6793  Basecbs 16064  distcds 16158  0gc0g 16308  normcnm 22601
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-ov 6796  df-nm 22607
This theorem is referenced by:  nmval  22614  nmfval2  22615  nmpropd  22618  subgnm  22657  tngnm  22675  cnfldnm  22802  nmcn  22867  ressnm  29991
  Copyright terms: Public domain W3C validator