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.

Step	Hyp	Ref	Expression
1		nmf2.x	. . . . . 6 ⊢ 𝑋 = (Base‘𝑊)
2		eqid 2610	. . . . . 6 ⊢ (0g‘𝑊) = (0g‘𝑊)
3	1, 2	grpidcl 17273	. . . . 5 ⊢ (𝑊 ∈ Grp → (0g‘𝑊) ∈ 𝑋)
4		metcl 21947	. . . . . 6 ⊢ ((𝐸 ∈ (Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ (0g‘𝑊) ∈ 𝑋) → (𝑥𝐸(0g‘𝑊)) ∈ ℝ)
5	4	3comr 1265	. . . . 5 ⊢ (((0g‘𝑊) ∈ 𝑋 ∧ 𝐸 ∈ (Met‘𝑋) ∧ 𝑥 ∈ 𝑋) → (𝑥𝐸(0g‘𝑊)) ∈ ℝ)
6	3, 5	syl3an1 1351	. . . 4 ⊢ ((𝑊 ∈ Grp ∧ 𝐸 ∈ (Met‘𝑋) ∧ 𝑥 ∈ 𝑋) → (𝑥𝐸(0g‘𝑊)) ∈ ℝ)
7	6	3expa 1257	. . 3 ⊢ (((𝑊 ∈ Grp ∧ 𝐸 ∈ (Met‘𝑋)) ∧ 𝑥 ∈ 𝑋) → (𝑥𝐸(0g‘𝑊)) ∈ ℝ)
8		eqid 2610	. . 3 ⊢ (𝑥 ∈ 𝑋 ↦ (𝑥𝐸(0g‘𝑊))) = (𝑥 ∈ 𝑋 ↦ (𝑥𝐸(0g‘𝑊)))
9	7, 8	fmptd 6292	. 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 22205	. . . 4 ⊢ (𝑊 ∈ Grp → 𝑁 = (𝑥 ∈ 𝑋 ↦ (𝑥𝐸(0g‘𝑊))))
14	13	adantr 480	. . 3 ⊢ ((𝑊 ∈ Grp ∧ 𝐸 ∈ (Met‘𝑋)) → 𝑁 = (𝑥 ∈ 𝑋 ↦ (𝑥𝐸(0g‘𝑊))))
15	14	feq1d 5943	. 2 ⊢ ((𝑊 ∈ Grp ∧ 𝐸 ∈ (Met‘𝑋)) → (𝑁:𝑋⟶ℝ ↔ (𝑥 ∈ 𝑋 ↦ (𝑥𝐸(0g‘𝑊))):𝑋⟶ℝ))
16	9, 15	mpbird 246	1 ⊢ ((𝑊 ∈ Grp ∧ 𝐸 ∈ (Met‘𝑋)) → 𝑁:𝑋⟶ℝ)

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  0_gc0g 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