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Theorem nmf2 23947
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 2736 . . . 4 (0g𝑊) = (0g𝑊)
4 nmf2.d . . . 4 𝐷 = (dist‘𝑊)
5 nmf2.e . . . 4 𝐸 = (𝐷 ↾ (𝑋 × 𝑋))
61, 2, 3, 4, 5nmfval2 23945 . . 3 (𝑊 ∈ Grp → 𝑁 = (𝑥𝑋 ↦ (𝑥𝐸(0g𝑊))))
76adantr 481 . 2 ((𝑊 ∈ Grp ∧ 𝐸 ∈ (Met‘𝑋)) → 𝑁 = (𝑥𝑋 ↦ (𝑥𝐸(0g𝑊))))
82, 3grpidcl 18777 . . . 4 (𝑊 ∈ Grp → (0g𝑊) ∈ 𝑋)
9 metcl 23683 . . . . 5 ((𝐸 ∈ (Met‘𝑋) ∧ 𝑥𝑋 ∧ (0g𝑊) ∈ 𝑋) → (𝑥𝐸(0g𝑊)) ∈ ℝ)
1093comr 1125 . . . 4 (((0g𝑊) ∈ 𝑋𝐸 ∈ (Met‘𝑋) ∧ 𝑥𝑋) → (𝑥𝐸(0g𝑊)) ∈ ℝ)
118, 10syl3an1 1163 . . 3 ((𝑊 ∈ Grp ∧ 𝐸 ∈ (Met‘𝑋) ∧ 𝑥𝑋) → (𝑥𝐸(0g𝑊)) ∈ ℝ)
12113expa 1118 . 2 (((𝑊 ∈ Grp ∧ 𝐸 ∈ (Met‘𝑋)) ∧ 𝑥𝑋) → (𝑥𝐸(0g𝑊)) ∈ ℝ)
137, 12fmpt3d 7063 1 ((𝑊 ∈ Grp ∧ 𝐸 ∈ (Met‘𝑋)) → 𝑁:𝑋⟶ℝ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  cmpt 5188   × cxp 5631  cres 5635  wf 6492  cfv 6496  (class class class)co 7356  cr 11049  Basecbs 17082  distcds 17141  0gc0g 17320  Grpcgrp 18747  Metcmet 20780  normcnm 23930
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7671  ax-cnex 11106  ax-resscn 11107
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fv 6504  df-riota 7312  df-ov 7359  df-oprab 7360  df-mpo 7361  df-map 8766  df-0g 17322  df-mgm 18496  df-sgrp 18545  df-mnd 18556  df-grp 18750  df-met 20788  df-nm 23936
This theorem is referenced by:  isngp2  23951  isngp3  23952  nmf  23969
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