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Theorem nmfval2 22442
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 22440 . 2 𝑁 = (𝑥𝑋 ↦ (𝑥𝐷 0 ))
6 nmfval.e . . . . 5 𝐸 = (𝐷 ↾ (𝑋 × 𝑋))
76oveqi 6703 . . . 4 (𝑥𝐸 0 ) = (𝑥(𝐷 ↾ (𝑋 × 𝑋)) 0 )
8 id 22 . . . . 5 (𝑥𝑋𝑥𝑋)
92, 3grpidcl 17497 . . . . 5 (𝑊 ∈ Grp → 0𝑋)
10 ovres 6842 . . . . 5 ((𝑥𝑋0𝑋) → (𝑥(𝐷 ↾ (𝑋 × 𝑋)) 0 ) = (𝑥𝐷 0 ))
118, 9, 10syl2anr 494 . . . 4 ((𝑊 ∈ Grp ∧ 𝑥𝑋) → (𝑥(𝐷 ↾ (𝑋 × 𝑋)) 0 ) = (𝑥𝐷 0 ))
127, 11syl5req 2698 . . 3 ((𝑊 ∈ Grp ∧ 𝑥𝑋) → (𝑥𝐷 0 ) = (𝑥𝐸 0 ))
1312mpteq2dva 4777 . 2 (𝑊 ∈ Grp → (𝑥𝑋 ↦ (𝑥𝐷 0 )) = (𝑥𝑋 ↦ (𝑥𝐸 0 )))
145, 13syl5eq 2697 1 (𝑊 ∈ Grp → 𝑁 = (𝑥𝑋 ↦ (𝑥𝐸 0 )))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  cmpt 4762   × cxp 5141  cres 5145  cfv 5926  (class class class)co 6690  Basecbs 15904  distcds 15997  0gc0g 16147  Grpcgrp 17469  normcnm 22428
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-riota 6651  df-ov 6693  df-0g 16149  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-grp 17472  df-nm 22434
This theorem is referenced by:  nmf2  22444  nmpropd2  22446
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