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Theorem nmfval0 24498
Description: The value of the norm function on a structure containing a zero as the distance restricted to the elements of the base set to zero. Examples of structures containing a "zero" are groups (see nmfval2 24499 proved from this theorem and grpidcl 18921) or more generally monoids (see mndidcl 18708), or pointed sets). (Contributed by Mario Carneiro, 2-Oct-2015.) Extract this result from the proof of nmfval2 24499. (Revised by BJ, 27-Aug-2024.)
Hypotheses
Ref Expression
nmfval0.n 𝑁 = (normβ€˜π‘Š)
nmfval0.x 𝑋 = (Baseβ€˜π‘Š)
nmfval0.z 0 = (0gβ€˜π‘Š)
nmfval0.d 𝐷 = (distβ€˜π‘Š)
nmfval0.e 𝐸 = (𝐷 β†Ύ (𝑋 Γ— 𝑋))
Assertion
Ref Expression
nmfval0 ( 0 ∈ 𝑋 β†’ 𝑁 = (π‘₯ ∈ 𝑋 ↦ (π‘₯𝐸 0 )))
Distinct variable groups:   π‘₯,𝐷   π‘₯,π‘Š   π‘₯,𝑋   π‘₯, 0
Allowed substitution hints:   𝐸(π‘₯)   𝑁(π‘₯)

Proof of Theorem nmfval0
StepHypRef Expression
1 nmfval0.n . . 3 𝑁 = (normβ€˜π‘Š)
2 nmfval0.x . . 3 𝑋 = (Baseβ€˜π‘Š)
3 nmfval0.z . . 3 0 = (0gβ€˜π‘Š)
4 nmfval0.d . . 3 𝐷 = (distβ€˜π‘Š)
51, 2, 3, 4nmfval 24496 . 2 𝑁 = (π‘₯ ∈ 𝑋 ↦ (π‘₯𝐷 0 ))
6 nmfval0.e . . . . 5 𝐸 = (𝐷 β†Ύ (𝑋 Γ— 𝑋))
76oveqi 7433 . . . 4 (π‘₯𝐸 0 ) = (π‘₯(𝐷 β†Ύ (𝑋 Γ— 𝑋)) 0 )
8 ovres 7587 . . . . 5 ((π‘₯ ∈ 𝑋 ∧ 0 ∈ 𝑋) β†’ (π‘₯(𝐷 β†Ύ (𝑋 Γ— 𝑋)) 0 ) = (π‘₯𝐷 0 ))
98ancoms 458 . . . 4 (( 0 ∈ 𝑋 ∧ π‘₯ ∈ 𝑋) β†’ (π‘₯(𝐷 β†Ύ (𝑋 Γ— 𝑋)) 0 ) = (π‘₯𝐷 0 ))
107, 9eqtr2id 2781 . . 3 (( 0 ∈ 𝑋 ∧ π‘₯ ∈ 𝑋) β†’ (π‘₯𝐷 0 ) = (π‘₯𝐸 0 ))
1110mpteq2dva 5248 . 2 ( 0 ∈ 𝑋 β†’ (π‘₯ ∈ 𝑋 ↦ (π‘₯𝐷 0 )) = (π‘₯ ∈ 𝑋 ↦ (π‘₯𝐸 0 )))
125, 11eqtrid 2780 1 ( 0 ∈ 𝑋 β†’ 𝑁 = (π‘₯ ∈ 𝑋 ↦ (π‘₯𝐸 0 )))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099   ↦ cmpt 5231   Γ— cxp 5676   β†Ύ cres 5680  β€˜cfv 6548  (class class class)co 7420  Basecbs 17179  distcds 17241  0gc0g 17420  normcnm 24484
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 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-fv 6556  df-ov 7423  df-nm 24490
This theorem is referenced by:  nmfval2  24499
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