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Theorem nmval2 22206
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
nmval2 ((𝑊 ∈ Grp ∧ 𝐴𝑋) → (𝑁𝐴) = (𝐴𝐸 0 ))

Proof of Theorem nmval2
StepHypRef Expression
1 nmfval.n . . . 4 𝑁 = (norm‘𝑊)
2 nmfval.x . . . 4 𝑋 = (Base‘𝑊)
3 nmfval.z . . . 4 0 = (0g𝑊)
4 nmfval.d . . . 4 𝐷 = (dist‘𝑊)
51, 2, 3, 4nmval 22204 . . 3 (𝐴𝑋 → (𝑁𝐴) = (𝐴𝐷 0 ))
65adantl 481 . 2 ((𝑊 ∈ Grp ∧ 𝐴𝑋) → (𝑁𝐴) = (𝐴𝐷 0 ))
7 nmfval.e . . . 4 𝐸 = (𝐷 ↾ (𝑋 × 𝑋))
87oveqi 6562 . . 3 (𝐴𝐸 0 ) = (𝐴(𝐷 ↾ (𝑋 × 𝑋)) 0 )
9 id 22 . . . 4 (𝐴𝑋𝐴𝑋)
102, 3grpidcl 17273 . . . 4 (𝑊 ∈ Grp → 0𝑋)
11 ovres 6698 . . . 4 ((𝐴𝑋0𝑋) → (𝐴(𝐷 ↾ (𝑋 × 𝑋)) 0 ) = (𝐴𝐷 0 ))
129, 10, 11syl2anr 494 . . 3 ((𝑊 ∈ Grp ∧ 𝐴𝑋) → (𝐴(𝐷 ↾ (𝑋 × 𝑋)) 0 ) = (𝐴𝐷 0 ))
138, 12syl5req 2657 . 2 ((𝑊 ∈ Grp ∧ 𝐴𝑋) → (𝐴𝐷 0 ) = (𝐴𝐸 0 ))
146, 13eqtrd 2644 1 ((𝑊 ∈ Grp ∧ 𝐴𝑋) → (𝑁𝐴) = (𝐴𝐸 0 ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977   × cxp 5036  cres 5040  cfv 5804  (class class class)co 6549  Basecbs 15695  distcds 15777  0gc0g 15923  Grpcgrp 17245  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
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-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-nm 22197
This theorem is referenced by:  nmhmcn  22728  nglmle  22908
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