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Theorem nmfval 22306
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 6150 . . . . . 6 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
3 nmfval.x . . . . . 6 𝑋 = (Base‘𝑊)
42, 3syl6eqr 2673 . . . . 5 (𝑤 = 𝑊 → (Base‘𝑤) = 𝑋)
5 fveq2 6150 . . . . . . 7 (𝑤 = 𝑊 → (dist‘𝑤) = (dist‘𝑊))
6 nmfval.d . . . . . . 7 𝐷 = (dist‘𝑊)
75, 6syl6eqr 2673 . . . . . 6 (𝑤 = 𝑊 → (dist‘𝑤) = 𝐷)
8 eqidd 2622 . . . . . 6 (𝑤 = 𝑊𝑥 = 𝑥)
9 fveq2 6150 . . . . . . 7 (𝑤 = 𝑊 → (0g𝑤) = (0g𝑊))
10 nmfval.z . . . . . . 7 0 = (0g𝑊)
119, 10syl6eqr 2673 . . . . . 6 (𝑤 = 𝑊 → (0g𝑤) = 0 )
127, 8, 11oveq123d 6628 . . . . 5 (𝑤 = 𝑊 → (𝑥(dist‘𝑤)(0g𝑤)) = (𝑥𝐷 0 ))
134, 12mpteq12dv 4695 . . . 4 (𝑤 = 𝑊 → (𝑥 ∈ (Base‘𝑤) ↦ (𝑥(dist‘𝑤)(0g𝑤))) = (𝑥𝑋 ↦ (𝑥𝐷 0 )))
14 df-nm 22300 . . . 4 norm = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘𝑤) ↦ (𝑥(dist‘𝑤)(0g𝑤))))
15 eqid 2621 . . . . . 6 (𝑥𝑋 ↦ (𝑥𝐷 0 )) = (𝑥𝑋 ↦ (𝑥𝐷 0 ))
16 df-ov 6610 . . . . . . . 8 (𝑥𝐷 0 ) = (𝐷‘⟨𝑥, 0 ⟩)
17 fvrn0 6175 . . . . . . . 8 (𝐷‘⟨𝑥, 0 ⟩) ∈ (ran 𝐷 ∪ {∅})
1816, 17eqeltri 2694 . . . . . . 7 (𝑥𝐷 0 ) ∈ (ran 𝐷 ∪ {∅})
1918a1i 11 . . . . . 6 (𝑥𝑋 → (𝑥𝐷 0 ) ∈ (ran 𝐷 ∪ {∅}))
2015, 19fmpti 6341 . . . . 5 (𝑥𝑋 ↦ (𝑥𝐷 0 )):𝑋⟶(ran 𝐷 ∪ {∅})
21 fvex 6160 . . . . . 6 (Base‘𝑊) ∈ V
223, 21eqeltri 2694 . . . . 5 𝑋 ∈ V
23 fvex 6160 . . . . . . . 8 (dist‘𝑊) ∈ V
246, 23eqeltri 2694 . . . . . . 7 𝐷 ∈ V
2524rnex 7050 . . . . . 6 ran 𝐷 ∈ V
26 p0ex 4815 . . . . . 6 {∅} ∈ V
2725, 26unex 6912 . . . . 5 (ran 𝐷 ∪ {∅}) ∈ V
28 fex2 7071 . . . . 5 (((𝑥𝑋 ↦ (𝑥𝐷 0 )):𝑋⟶(ran 𝐷 ∪ {∅}) ∧ 𝑋 ∈ V ∧ (ran 𝐷 ∪ {∅}) ∈ V) → (𝑥𝑋 ↦ (𝑥𝐷 0 )) ∈ V)
2920, 22, 27, 28mp3an 1421 . . . 4 (𝑥𝑋 ↦ (𝑥𝐷 0 )) ∈ V
3013, 14, 29fvmpt 6241 . . 3 (𝑊 ∈ V → (norm‘𝑊) = (𝑥𝑋 ↦ (𝑥𝐷 0 )))
31 fvprc 6144 . . . . 5 𝑊 ∈ V → (norm‘𝑊) = ∅)
32 mpt0 5980 . . . . 5 (𝑥 ∈ ∅ ↦ (𝑥𝐷 0 )) = ∅
3331, 32syl6eqr 2673 . . . 4 𝑊 ∈ V → (norm‘𝑊) = (𝑥 ∈ ∅ ↦ (𝑥𝐷 0 )))
34 fvprc 6144 . . . . . 6 𝑊 ∈ V → (Base‘𝑊) = ∅)
353, 34syl5eq 2667 . . . . 5 𝑊 ∈ V → 𝑋 = ∅)
3635mpteq1d 4700 . . . 4 𝑊 ∈ V → (𝑥𝑋 ↦ (𝑥𝐷 0 )) = (𝑥 ∈ ∅ ↦ (𝑥𝐷 0 )))
3733, 36eqtr4d 2658 . . 3 𝑊 ∈ V → (norm‘𝑊) = (𝑥𝑋 ↦ (𝑥𝐷 0 )))
3830, 37pm2.61i 176 . 2 (norm‘𝑊) = (𝑥𝑋 ↦ (𝑥𝐷 0 ))
391, 38eqtri 2643 1 𝑁 = (𝑥𝑋 ↦ (𝑥𝐷 0 ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1480  wcel 1987  Vcvv 3186  cun 3554  c0 3893  {csn 4150  cop 4156  cmpt 4675  ran crn 5077  wf 5845  cfv 5849  (class class class)co 6607  Basecbs 15784  distcds 15874  0gc0g 16024  normcnm 22294
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3419  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-br 4616  df-opab 4676  df-mpt 4677  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-fv 5857  df-ov 6610  df-nm 22300
This theorem is referenced by:  nmval  22307  nmfval2  22308  nmpropd  22311  subgnm  22350  tngnm  22368  cnfldnm  22495  nmcn  22560  ressnm  29448
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