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Theorem nmval2 23193
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 23191 . . 3 (𝐴𝑋 → (𝑁𝐴) = (𝐴𝐷 0 ))
65adantl 484 . 2 ((𝑊 ∈ Grp ∧ 𝐴𝑋) → (𝑁𝐴) = (𝐴𝐷 0 ))
7 nmfval.e . . . 4 𝐸 = (𝐷 ↾ (𝑋 × 𝑋))
87oveqi 7161 . . 3 (𝐴𝐸 0 ) = (𝐴(𝐷 ↾ (𝑋 × 𝑋)) 0 )
9 id 22 . . . 4 (𝐴𝑋𝐴𝑋)
102, 3grpidcl 18123 . . . 4 (𝑊 ∈ Grp → 0𝑋)
11 ovres 7306 . . . 4 ((𝐴𝑋0𝑋) → (𝐴(𝐷 ↾ (𝑋 × 𝑋)) 0 ) = (𝐴𝐷 0 ))
129, 10, 11syl2anr 598 . . 3 ((𝑊 ∈ Grp ∧ 𝐴𝑋) → (𝐴(𝐷 ↾ (𝑋 × 𝑋)) 0 ) = (𝐴𝐷 0 ))
138, 12syl5req 2867 . 2 ((𝑊 ∈ Grp ∧ 𝐴𝑋) → (𝐴𝐷 0 ) = (𝐴𝐸 0 ))
146, 13eqtrd 2854 1 ((𝑊 ∈ Grp ∧ 𝐴𝑋) → (𝑁𝐴) = (𝐴𝐸 0 ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1531  wcel 2108   × cxp 5546  cres 5550  cfv 6348  (class class class)co 7148  Basecbs 16475  distcds 16566  0gc0g 16705  Grpcgrp 18095  normcnm 23178
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-riota 7106  df-ov 7151  df-0g 16707  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-grp 18098  df-nm 23184
This theorem is referenced by:  nmhmcn  23716  nglmle  23897
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