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Theorem nmval2 22377
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 22375 . . 3 (𝐴𝑋 → (𝑁𝐴) = (𝐴𝐷 0 ))
65adantl 482 . 2 ((𝑊 ∈ Grp ∧ 𝐴𝑋) → (𝑁𝐴) = (𝐴𝐷 0 ))
7 nmfval.e . . . 4 𝐸 = (𝐷 ↾ (𝑋 × 𝑋))
87oveqi 6648 . . 3 (𝐴𝐸 0 ) = (𝐴(𝐷 ↾ (𝑋 × 𝑋)) 0 )
9 id 22 . . . 4 (𝐴𝑋𝐴𝑋)
102, 3grpidcl 17431 . . . 4 (𝑊 ∈ Grp → 0𝑋)
11 ovres 6785 . . . 4 ((𝐴𝑋0𝑋) → (𝐴(𝐷 ↾ (𝑋 × 𝑋)) 0 ) = (𝐴𝐷 0 ))
129, 10, 11syl2anr 495 . . 3 ((𝑊 ∈ Grp ∧ 𝐴𝑋) → (𝐴(𝐷 ↾ (𝑋 × 𝑋)) 0 ) = (𝐴𝐷 0 ))
138, 12syl5req 2667 . 2 ((𝑊 ∈ Grp ∧ 𝐴𝑋) → (𝐴𝐷 0 ) = (𝐴𝐸 0 ))
146, 13eqtrd 2654 1 ((𝑊 ∈ Grp ∧ 𝐴𝑋) → (𝑁𝐴) = (𝐴𝐸 0 ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1481  wcel 1988   × cxp 5102  cres 5106  cfv 5876  (class class class)co 6635  Basecbs 15838  distcds 15931  0gc0g 16081  Grpcgrp 17403  normcnm 22362
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-fv 5884  df-riota 6596  df-ov 6638  df-0g 16083  df-mgm 17223  df-sgrp 17265  df-mnd 17276  df-grp 17406  df-nm 22368
This theorem is referenced by:  nmhmcn  22901  nglmle  23081
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