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Theorem nmf2 22598
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 2760 . . . . . 6 (0g𝑊) = (0g𝑊)
31, 2grpidcl 17651 . . . . 5 (𝑊 ∈ Grp → (0g𝑊) ∈ 𝑋)
4 metcl 22338 . . . . . 6 ((𝐸 ∈ (Met‘𝑋) ∧ 𝑥𝑋 ∧ (0g𝑊) ∈ 𝑋) → (𝑥𝐸(0g𝑊)) ∈ ℝ)
543comr 1120 . . . . 5 (((0g𝑊) ∈ 𝑋𝐸 ∈ (Met‘𝑋) ∧ 𝑥𝑋) → (𝑥𝐸(0g𝑊)) ∈ ℝ)
63, 5syl3an1 1167 . . . 4 ((𝑊 ∈ Grp ∧ 𝐸 ∈ (Met‘𝑋) ∧ 𝑥𝑋) → (𝑥𝐸(0g𝑊)) ∈ ℝ)
763expa 1112 . . 3 (((𝑊 ∈ Grp ∧ 𝐸 ∈ (Met‘𝑋)) ∧ 𝑥𝑋) → (𝑥𝐸(0g𝑊)) ∈ ℝ)
8 eqid 2760 . . 3 (𝑥𝑋 ↦ (𝑥𝐸(0g𝑊))) = (𝑥𝑋 ↦ (𝑥𝐸(0g𝑊)))
97, 8fmptd 6548 . 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 22596 . . . 4 (𝑊 ∈ Grp → 𝑁 = (𝑥𝑋 ↦ (𝑥𝐸(0g𝑊))))
1413adantr 472 . . 3 ((𝑊 ∈ Grp ∧ 𝐸 ∈ (Met‘𝑋)) → 𝑁 = (𝑥𝑋 ↦ (𝑥𝐸(0g𝑊))))
1514feq1d 6191 . 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 383   = wceq 1632  wcel 2139  cmpt 4881   × cxp 5264  cres 5268  wf 6045  cfv 6049  (class class class)co 6813  cr 10127  Basecbs 16059  distcds 16152  0gc0g 16302  Grpcgrp 17623  Metcme 19934  normcnm 22582
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 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185
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 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-map 8025  df-0g 16304  df-mgm 17443  df-sgrp 17485  df-mnd 17496  df-grp 17626  df-met 19942  df-nm 22588
This theorem is referenced by:  isngp2  22602  isngp3  22603  nmf  22620
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