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Theorem nmfval2 24029
Description: The value of the norm function 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 = (0g𝑊)
nmfval2.d 𝐷 = (dist‘𝑊)
nmfval2.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 nmfval2.x . . 3 𝑋 = (Base‘𝑊)
2 nmfval2.z . . 3 0 = (0g𝑊)
31, 2grpidcl 18825 . 2 (𝑊 ∈ Grp → 0𝑋)
4 nmfval2.n . . 3 𝑁 = (norm‘𝑊)
5 nmfval2.d . . 3 𝐷 = (dist‘𝑊)
6 nmfval2.e . . 3 𝐸 = (𝐷 ↾ (𝑋 × 𝑋))
74, 1, 2, 5, 6nmfval0 24028 . 2 ( 0𝑋𝑁 = (𝑥𝑋 ↦ (𝑥𝐸 0 )))
83, 7syl 17 1 (𝑊 ∈ Grp → 𝑁 = (𝑥𝑋 ↦ (𝑥𝐸 0 )))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  cmpt 5224   × cxp 5667  cres 5671  cfv 6532  (class class class)co 7393  Basecbs 17126  distcds 17188  0gc0g 17367  Grpcgrp 18794  normcnm 24014
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 2702  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708
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 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-fv 6540  df-riota 7349  df-ov 7396  df-0g 17369  df-mgm 18543  df-sgrp 18592  df-mnd 18603  df-grp 18797  df-nm 24020
This theorem is referenced by:  nmf2  24031  nmpropd2  24033
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