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Theorem nmfval0 24443
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 24444 proved from this theorem and grpidcl 18891) or more generally monoids (see mndidcl 18678), or pointed sets). (Contributed by Mario Carneiro, 2-Oct-2015.) Extract this result from the proof of nmfval2 24444. (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 24441 . 2 𝑁 = (π‘₯ ∈ 𝑋 ↦ (π‘₯𝐷 0 ))
6 nmfval0.e . . . . 5 𝐸 = (𝐷 β†Ύ (𝑋 Γ— 𝑋))
76oveqi 7415 . . . 4 (π‘₯𝐸 0 ) = (π‘₯(𝐷 β†Ύ (𝑋 Γ— 𝑋)) 0 )
8 ovres 7567 . . . . 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 2777 . . 3 (( 0 ∈ 𝑋 ∧ π‘₯ ∈ 𝑋) β†’ (π‘₯𝐷 0 ) = (π‘₯𝐸 0 ))
1110mpteq2dva 5239 . 2 ( 0 ∈ 𝑋 β†’ (π‘₯ ∈ 𝑋 ↦ (π‘₯𝐷 0 )) = (π‘₯ ∈ 𝑋 ↦ (π‘₯𝐸 0 )))
125, 11eqtrid 2776 1 ( 0 ∈ 𝑋 β†’ 𝑁 = (π‘₯ ∈ 𝑋 ↦ (π‘₯𝐸 0 )))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098   ↦ cmpt 5222   Γ— cxp 5665   β†Ύ cres 5669  β€˜cfv 6534  (class class class)co 7402  Basecbs 17149  distcds 17211  0gc0g 17390  normcnm 24429
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-fv 6542  df-ov 7405  df-nm 24435
This theorem is referenced by:  nmfval2  24444
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