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Theorem nmf2 22207
 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 2610 . . . . . 6 (0g𝑊) = (0g𝑊)
31, 2grpidcl 17273 . . . . 5 (𝑊 ∈ Grp → (0g𝑊) ∈ 𝑋)
4 metcl 21947 . . . . . 6 ((𝐸 ∈ (Met‘𝑋) ∧ 𝑥𝑋 ∧ (0g𝑊) ∈ 𝑋) → (𝑥𝐸(0g𝑊)) ∈ ℝ)
543comr 1265 . . . . 5 (((0g𝑊) ∈ 𝑋𝐸 ∈ (Met‘𝑋) ∧ 𝑥𝑋) → (𝑥𝐸(0g𝑊)) ∈ ℝ)
63, 5syl3an1 1351 . . . 4 ((𝑊 ∈ Grp ∧ 𝐸 ∈ (Met‘𝑋) ∧ 𝑥𝑋) → (𝑥𝐸(0g𝑊)) ∈ ℝ)
763expa 1257 . . 3 (((𝑊 ∈ Grp ∧ 𝐸 ∈ (Met‘𝑋)) ∧ 𝑥𝑋) → (𝑥𝐸(0g𝑊)) ∈ ℝ)
8 eqid 2610 . . 3 (𝑥𝑋 ↦ (𝑥𝐸(0g𝑊))) = (𝑥𝑋 ↦ (𝑥𝐸(0g𝑊)))
97, 8fmptd 6292 . 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 22205 . . . 4 (𝑊 ∈ Grp → 𝑁 = (𝑥𝑋 ↦ (𝑥𝐸(0g𝑊))))
1413adantr 480 . . 3 ((𝑊 ∈ Grp ∧ 𝐸 ∈ (Met‘𝑋)) → 𝑁 = (𝑥𝑋 ↦ (𝑥𝐸(0g𝑊))))
1514feq1d 5943 . 2 ((𝑊 ∈ Grp ∧ 𝐸 ∈ (Met‘𝑋)) → (𝑁:𝑋⟶ℝ ↔ (𝑥𝑋 ↦ (𝑥𝐸(0g𝑊))):𝑋⟶ℝ))
169, 15mpbird 246 1 ((𝑊 ∈ Grp ∧ 𝐸 ∈ (Met‘𝑋)) → 𝑁:𝑋⟶ℝ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ↦ cmpt 4643   × cxp 5036   ↾ cres 5040  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  ℝcr 9814  Basecbs 15695  distcds 15777  0gc0g 15923  Grpcgrp 17245  Metcme 19553  normcnm 22191 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-met 19561  df-nm 22197 This theorem is referenced by:  isngp2  22211  isngp3  22212  nmf  22229
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