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Theorem nmf2 22378
Description: The norm is a function from the base set into the reals. (Contributed by Mario Carneiro, 2-Oct-2015.)
Hypotheses
Ref Expression
nmf2.n 𝑁 = (norm‘𝑊)
nmf2.x 𝑋 = (Base‘𝑊)
nmf2.d 𝐷 = (dist‘𝑊)
nmf2.e 𝐸 = (𝐷 ↾ (𝑋 × 𝑋))
Assertion
Ref Expression
nmf2 ((𝑊 ∈ Grp ∧ 𝐸 ∈ (Met‘𝑋)) → 𝑁:𝑋⟶ℝ)

Proof of Theorem nmf2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 nmf2.x . . . . . 6 𝑋 = (Base‘𝑊)
2 eqid 2620 . . . . . 6 (0g𝑊) = (0g𝑊)
31, 2grpidcl 17431 . . . . 5 (𝑊 ∈ Grp → (0g𝑊) ∈ 𝑋)
4 metcl 22118 . . . . . 6 ((𝐸 ∈ (Met‘𝑋) ∧ 𝑥𝑋 ∧ (0g𝑊) ∈ 𝑋) → (𝑥𝐸(0g𝑊)) ∈ ℝ)
543comr 1271 . . . . 5 (((0g𝑊) ∈ 𝑋𝐸 ∈ (Met‘𝑋) ∧ 𝑥𝑋) → (𝑥𝐸(0g𝑊)) ∈ ℝ)
63, 5syl3an1 1357 . . . 4 ((𝑊 ∈ Grp ∧ 𝐸 ∈ (Met‘𝑋) ∧ 𝑥𝑋) → (𝑥𝐸(0g𝑊)) ∈ ℝ)
763expa 1263 . . 3 (((𝑊 ∈ Grp ∧ 𝐸 ∈ (Met‘𝑋)) ∧ 𝑥𝑋) → (𝑥𝐸(0g𝑊)) ∈ ℝ)
8 eqid 2620 . . 3 (𝑥𝑋 ↦ (𝑥𝐸(0g𝑊))) = (𝑥𝑋 ↦ (𝑥𝐸(0g𝑊)))
97, 8fmptd 6371 . 2 ((𝑊 ∈ Grp ∧ 𝐸 ∈ (Met‘𝑋)) → (𝑥𝑋 ↦ (𝑥𝐸(0g𝑊))):𝑋⟶ℝ)
10 nmf2.n . . . . 5 𝑁 = (norm‘𝑊)
11 nmf2.d . . . . 5 𝐷 = (dist‘𝑊)
12 nmf2.e . . . . 5 𝐸 = (𝐷 ↾ (𝑋 × 𝑋))
1310, 1, 2, 11, 12nmfval2 22376 . . . 4 (𝑊 ∈ Grp → 𝑁 = (𝑥𝑋 ↦ (𝑥𝐸(0g𝑊))))
1413adantr 481 . . 3 ((𝑊 ∈ Grp ∧ 𝐸 ∈ (Met‘𝑋)) → 𝑁 = (𝑥𝑋 ↦ (𝑥𝐸(0g𝑊))))
1514feq1d 6017 . 2 ((𝑊 ∈ Grp ∧ 𝐸 ∈ (Met‘𝑋)) → (𝑁:𝑋⟶ℝ ↔ (𝑥𝑋 ↦ (𝑥𝐸(0g𝑊))):𝑋⟶ℝ))
169, 15mpbird 247 1 ((𝑊 ∈ Grp ∧ 𝐸 ∈ (Met‘𝑋)) → 𝑁:𝑋⟶ℝ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1481  wcel 1988  cmpt 4720   × cxp 5102  cres 5106  wf 5872  cfv 5876  (class class class)co 6635  cr 9920  Basecbs 15838  distcds 15931  0gc0g 16081  Grpcgrp 17403  Metcme 19713  normcnm 22362
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-cnex 9977  ax-resscn 9978
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-map 7844  df-0g 16083  df-mgm 17223  df-sgrp 17265  df-mnd 17276  df-grp 17406  df-met 19721  df-nm 22368
This theorem is referenced by:  isngp2  22382  isngp3  22383  nmf  22400
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