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Theorem nmf2 24563
Description: The norm on a metric group 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.n . . . 4 𝑁 = (norm‘𝑊)
2 nmf2.x . . . 4 𝑋 = (Base‘𝑊)
3 eqid 2725 . . . 4 (0g𝑊) = (0g𝑊)
4 nmf2.d . . . 4 𝐷 = (dist‘𝑊)
5 nmf2.e . . . 4 𝐸 = (𝐷 ↾ (𝑋 × 𝑋))
61, 2, 3, 4, 5nmfval2 24561 . . 3 (𝑊 ∈ Grp → 𝑁 = (𝑥𝑋 ↦ (𝑥𝐸(0g𝑊))))
76adantr 479 . 2 ((𝑊 ∈ Grp ∧ 𝐸 ∈ (Met‘𝑋)) → 𝑁 = (𝑥𝑋 ↦ (𝑥𝐸(0g𝑊))))
82, 3grpidcl 18946 . . . 4 (𝑊 ∈ Grp → (0g𝑊) ∈ 𝑋)
9 metcl 24299 . . . . 5 ((𝐸 ∈ (Met‘𝑋) ∧ 𝑥𝑋 ∧ (0g𝑊) ∈ 𝑋) → (𝑥𝐸(0g𝑊)) ∈ ℝ)
1093comr 1122 . . . 4 (((0g𝑊) ∈ 𝑋𝐸 ∈ (Met‘𝑋) ∧ 𝑥𝑋) → (𝑥𝐸(0g𝑊)) ∈ ℝ)
118, 10syl3an1 1160 . . 3 ((𝑊 ∈ Grp ∧ 𝐸 ∈ (Met‘𝑋) ∧ 𝑥𝑋) → (𝑥𝐸(0g𝑊)) ∈ ℝ)
12113expa 1115 . 2 (((𝑊 ∈ Grp ∧ 𝐸 ∈ (Met‘𝑋)) ∧ 𝑥𝑋) → (𝑥𝐸(0g𝑊)) ∈ ℝ)
137, 12fmpt3d 7125 1 ((𝑊 ∈ Grp ∧ 𝐸 ∈ (Met‘𝑋)) → 𝑁:𝑋⟶ℝ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  cmpt 5232   × cxp 5676  cres 5680  wf 6545  cfv 6549  (class class class)co 7419  cr 11144  Basecbs 17199  distcds 17261  0gc0g 17440  Grpcgrp 18914  Metcmet 21299  normcnm 24546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11201  ax-resscn 11202
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-map 8847  df-0g 17442  df-mgm 18619  df-sgrp 18698  df-mnd 18714  df-grp 18917  df-met 21307  df-nm 24552
This theorem is referenced by:  isngp2  24567  isngp3  24568  nmf  24585
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