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Theorem nmval 23961
Description: The value of the norm as the distance to zero. Problem 1 of [Kreyszig] p. 63. (Contributed by NM, 4-Dec-2006.) (Revised by Mario Carneiro, 2-Oct-2015.)
Hypotheses
Ref Expression
nmfval.n 𝑁 = (normβ€˜π‘Š)
nmfval.x 𝑋 = (Baseβ€˜π‘Š)
nmfval.z 0 = (0gβ€˜π‘Š)
nmfval.d 𝐷 = (distβ€˜π‘Š)
Assertion
Ref Expression
nmval (𝐴 ∈ 𝑋 β†’ (π‘β€˜π΄) = (𝐴𝐷 0 ))

Proof of Theorem nmval
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 oveq1 7369 . 2 (π‘₯ = 𝐴 β†’ (π‘₯𝐷 0 ) = (𝐴𝐷 0 ))
2 nmfval.n . . 3 𝑁 = (normβ€˜π‘Š)
3 nmfval.x . . 3 𝑋 = (Baseβ€˜π‘Š)
4 nmfval.z . . 3 0 = (0gβ€˜π‘Š)
5 nmfval.d . . 3 𝐷 = (distβ€˜π‘Š)
62, 3, 4, 5nmfval 23960 . 2 𝑁 = (π‘₯ ∈ 𝑋 ↦ (π‘₯𝐷 0 ))
7 ovex 7395 . 2 (𝐴𝐷 0 ) ∈ V
81, 6, 7fvmpt 6953 1 (𝐴 ∈ 𝑋 β†’ (π‘β€˜π΄) = (𝐴𝐷 0 ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107  β€˜cfv 6501  (class class class)co 7362  Basecbs 17090  distcds 17149  0gc0g 17328  normcnm 23948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-fv 6509  df-ov 7365  df-nm 23954
This theorem is referenced by:  nmval2  23964  ngpds2  23978  isngp4  23984  nmge0  23989  nmeq0  23990  nminv  23993  nmmtri  23994  nmrtri  23996
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