Theorem nmval2 24486

Description: The value of the norm on a group as the distance restricted to the elements of the base set to zero. (Contributed by Mario Carneiro, 2-Oct-2015.)

Hypotheses

Ref	Expression
nmfval2.n	⊢ 𝑁 = (norm‘𝑊)
nmfval2.x	⊢ 𝑋 = (Base‘𝑊)
nmfval2.z	⊢ 0 = (0g‘𝑊)
nmfval2.d	⊢ 𝐷 = (dist‘𝑊)
nmfval2.e	⊢ 𝐸 = (𝐷 ↾ (𝑋 × 𝑋))

Assertion

Ref	Expression
nmval2	⊢ ((𝑊 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) = (𝐴𝐸 0 ))

Proof of Theorem nmval2

Step	Hyp	Ref	Expression
1		nmfval2.n	. . . 4 ⊢ 𝑁 = (norm‘𝑊)
2		nmfval2.x	. . . 4 ⊢ 𝑋 = (Base‘𝑊)
3		nmfval2.z	. . . 4 ⊢ 0 = (0g‘𝑊)
4		nmfval2.d	. . . 4 ⊢ 𝐷 = (dist‘𝑊)
5	1, 2, 3, 4	nmval 24483	. . 3 ⊢ (𝐴 ∈ 𝑋 → (𝑁‘𝐴) = (𝐴𝐷 0 ))
6	5	adantl 481	. 2 ⊢ ((𝑊 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) = (𝐴𝐷 0 ))
7		nmfval2.e	. . . 4 ⊢ 𝐸 = (𝐷 ↾ (𝑋 × 𝑋))
8	7	oveqi 7402	. . 3 ⊢ (𝐴𝐸 0 ) = (𝐴(𝐷 ↾ (𝑋 × 𝑋)) 0 )
9		id 22	. . . 4 ⊢ (𝐴 ∈ 𝑋 → 𝐴 ∈ 𝑋)
10	2, 3	grpidcl 18903	. . . 4 ⊢ (𝑊 ∈ Grp → 0 ∈ 𝑋)
11		ovres 7557	. . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 0 ∈ 𝑋) → (𝐴(𝐷 ↾ (𝑋 × 𝑋)) 0 ) = (𝐴𝐷 0 ))
12	9, 10, 11	syl2anr 597	. . 3 ⊢ ((𝑊 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝐴(𝐷 ↾ (𝑋 × 𝑋)) 0 ) = (𝐴𝐷 0 ))
13	8, 12	eqtr2id 2778	. 2 ⊢ ((𝑊 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝐴𝐷 0 ) = (𝐴𝐸 0 ))
14	6, 13	eqtrd 2765	1 ⊢ ((𝑊 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) = (𝐴𝐸 0 ))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   × cxp 5638   ↾ cres 5642  ‘cfv 6513  (class class class)co 7389  Basecbs 17185  distcds 17235  0_gc0g 17408  Grpcgrp 18871  normcnm 24470

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5253  ax-nul 5263  ax-pow 5322  ax-pr 5389  ax-un 7713

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3756  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5110  df-opab 5172  df-mpt 5191  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-fv 6521  df-riota 7346  df-ov 7392  df-0g 17410  df-mgm 18573  df-sgrp 18652  df-mnd 18668  df-grp 18874  df-nm 24476

This theorem is referenced by:  nmhmcn  25026  nglmle  25208