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Theorem nmfval2 22300
Description: The value of the norm function using a restricted metric. (Contributed by Mario Carneiro, 2-Oct-2015.)
Hypotheses
Ref Expression
nmfval.n 𝑁 = (norm‘𝑊)
nmfval.x 𝑋 = (Base‘𝑊)
nmfval.z 0 = (0g𝑊)
nmfval.d 𝐷 = (dist‘𝑊)
nmfval.e 𝐸 = (𝐷 ↾ (𝑋 × 𝑋))
Assertion
Ref Expression
nmfval2 (𝑊 ∈ Grp → 𝑁 = (𝑥𝑋 ↦ (𝑥𝐸 0 )))
Distinct variable groups:   𝑥,𝐷   𝑥,𝑊   𝑥,𝑋   𝑥, 0
Allowed substitution hints:   𝐸(𝑥)   𝑁(𝑥)

Proof of Theorem nmfval2
StepHypRef Expression
1 nmfval.n . . 3 𝑁 = (norm‘𝑊)
2 nmfval.x . . 3 𝑋 = (Base‘𝑊)
3 nmfval.z . . 3 0 = (0g𝑊)
4 nmfval.d . . 3 𝐷 = (dist‘𝑊)
51, 2, 3, 4nmfval 22298 . 2 𝑁 = (𝑥𝑋 ↦ (𝑥𝐷 0 ))
6 nmfval.e . . . . 5 𝐸 = (𝐷 ↾ (𝑋 × 𝑋))
76oveqi 6618 . . . 4 (𝑥𝐸 0 ) = (𝑥(𝐷 ↾ (𝑋 × 𝑋)) 0 )
8 id 22 . . . . 5 (𝑥𝑋𝑥𝑋)
92, 3grpidcl 17366 . . . . 5 (𝑊 ∈ Grp → 0𝑋)
10 ovres 6754 . . . . 5 ((𝑥𝑋0𝑋) → (𝑥(𝐷 ↾ (𝑋 × 𝑋)) 0 ) = (𝑥𝐷 0 ))
118, 9, 10syl2anr 495 . . . 4 ((𝑊 ∈ Grp ∧ 𝑥𝑋) → (𝑥(𝐷 ↾ (𝑋 × 𝑋)) 0 ) = (𝑥𝐷 0 ))
127, 11syl5req 2673 . . 3 ((𝑊 ∈ Grp ∧ 𝑥𝑋) → (𝑥𝐷 0 ) = (𝑥𝐸 0 ))
1312mpteq2dva 4709 . 2 (𝑊 ∈ Grp → (𝑥𝑋 ↦ (𝑥𝐷 0 )) = (𝑥𝑋 ↦ (𝑥𝐸 0 )))
145, 13syl5eq 2672 1 (𝑊 ∈ Grp → 𝑁 = (𝑥𝑋 ↦ (𝑥𝐸 0 )))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1992  cmpt 4678   × cxp 5077  cres 5081  cfv 5850  (class class class)co 6605  Basecbs 15776  distcds 15866  0gc0g 16016  Grpcgrp 17338  normcnm 22286
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-fv 5858  df-riota 6566  df-ov 6608  df-0g 16018  df-mgm 17158  df-sgrp 17200  df-mnd 17211  df-grp 17341  df-nm 22292
This theorem is referenced by:  nmf2  22302  nmpropd2  22304
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