Theorem nmf2 22397

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.

Step	Hyp	Ref	Expression
1		nmf2.x	. . . . . 6 ⊢ 𝑋 = (Base‘𝑊)
2		eqid 2622	. . . . . 6 ⊢ (0g‘𝑊) = (0g‘𝑊)
3	1, 2	grpidcl 17450	. . . . 5 ⊢ (𝑊 ∈ Grp → (0g‘𝑊) ∈ 𝑋)
4		metcl 22137	. . . . . 6 ⊢ ((𝐸 ∈ (Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ (0g‘𝑊) ∈ 𝑋) → (𝑥𝐸(0g‘𝑊)) ∈ ℝ)
5	4	3comr 1273	. . . . 5 ⊢ (((0g‘𝑊) ∈ 𝑋 ∧ 𝐸 ∈ (Met‘𝑋) ∧ 𝑥 ∈ 𝑋) → (𝑥𝐸(0g‘𝑊)) ∈ ℝ)
6	3, 5	syl3an1 1359	. . . 4 ⊢ ((𝑊 ∈ Grp ∧ 𝐸 ∈ (Met‘𝑋) ∧ 𝑥 ∈ 𝑋) → (𝑥𝐸(0g‘𝑊)) ∈ ℝ)
7	6	3expa 1265	. . 3 ⊢ (((𝑊 ∈ Grp ∧ 𝐸 ∈ (Met‘𝑋)) ∧ 𝑥 ∈ 𝑋) → (𝑥𝐸(0g‘𝑊)) ∈ ℝ)
8		eqid 2622	. . 3 ⊢ (𝑥 ∈ 𝑋 ↦ (𝑥𝐸(0g‘𝑊))) = (𝑥 ∈ 𝑋 ↦ (𝑥𝐸(0g‘𝑊)))
9	7, 8	fmptd 6385	. 2 ⊢ ((𝑊 ∈ Grp ∧ 𝐸 ∈ (Met‘𝑋)) → (𝑥 ∈ 𝑋 ↦ (𝑥𝐸(0g‘𝑊))):𝑋⟶ℝ)
10		nmf2.n	. . . . 5 ⊢ 𝑁 = (norm‘𝑊)
11		nmf2.d	. . . . 5 ⊢ 𝐷 = (dist‘𝑊)
12		nmf2.e	. . . . 5 ⊢ 𝐸 = (𝐷 ↾ (𝑋 × 𝑋))
13	10, 1, 2, 11, 12	nmfval2 22395	. . . 4 ⊢ (𝑊 ∈ Grp → 𝑁 = (𝑥 ∈ 𝑋 ↦ (𝑥𝐸(0g‘𝑊))))
14	13	adantr 481	. . 3 ⊢ ((𝑊 ∈ Grp ∧ 𝐸 ∈ (Met‘𝑋)) → 𝑁 = (𝑥 ∈ 𝑋 ↦ (𝑥𝐸(0g‘𝑊))))
15	14	feq1d 6030	. 2 ⊢ ((𝑊 ∈ Grp ∧ 𝐸 ∈ (Met‘𝑋)) → (𝑁:𝑋⟶ℝ ↔ (𝑥 ∈ 𝑋 ↦ (𝑥𝐸(0g‘𝑊))):𝑋⟶ℝ))
16	9, 15	mpbird 247	1 ⊢ ((𝑊 ∈ Grp ∧ 𝐸 ∈ (Met‘𝑋)) → 𝑁:𝑋⟶ℝ)

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ↦ cmpt 4729   × cxp 5112   ↾ cres 5116  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650  ℝcr 9935  Basecbs 15857  distcds 15950  0_gc0g 16100  Grpcgrp 17422  Metcme 19732  normcnm 22381

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-met 19740  df-nm 22387

This theorem is referenced by:  isngp2  22401  isngp3  22402  nmf  22419