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Theorem nmfval0 23969
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 23970 proved from this theorem and grpidcl 18786) or more generally monoids (see mndidcl 18579), or pointed sets). (Contributed by Mario Carneiro, 2-Oct-2015.) Extract this result from the proof of nmfval2 23970. (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 23967 . 2 𝑁 = (π‘₯ ∈ 𝑋 ↦ (π‘₯𝐷 0 ))
6 nmfval0.e . . . . 5 𝐸 = (𝐷 β†Ύ (𝑋 Γ— 𝑋))
76oveqi 7374 . . . 4 (π‘₯𝐸 0 ) = (π‘₯(𝐷 β†Ύ (𝑋 Γ— 𝑋)) 0 )
8 ovres 7524 . . . . 5 ((π‘₯ ∈ 𝑋 ∧ 0 ∈ 𝑋) β†’ (π‘₯(𝐷 β†Ύ (𝑋 Γ— 𝑋)) 0 ) = (π‘₯𝐷 0 ))
98ancoms 460 . . . 4 (( 0 ∈ 𝑋 ∧ π‘₯ ∈ 𝑋) β†’ (π‘₯(𝐷 β†Ύ (𝑋 Γ— 𝑋)) 0 ) = (π‘₯𝐷 0 ))
107, 9eqtr2id 2786 . . 3 (( 0 ∈ 𝑋 ∧ π‘₯ ∈ 𝑋) β†’ (π‘₯𝐷 0 ) = (π‘₯𝐸 0 ))
1110mpteq2dva 5209 . 2 ( 0 ∈ 𝑋 β†’ (π‘₯ ∈ 𝑋 ↦ (π‘₯𝐷 0 )) = (π‘₯ ∈ 𝑋 ↦ (π‘₯𝐸 0 )))
125, 11eqtrid 2785 1 ( 0 ∈ 𝑋 β†’ 𝑁 = (π‘₯ ∈ 𝑋 ↦ (π‘₯𝐸 0 )))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107   ↦ cmpt 5192   Γ— cxp 5635   β†Ύ cres 5639  β€˜cfv 6500  (class class class)co 7361  Basecbs 17091  distcds 17150  0gc0g 17329  normcnm 23955
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 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-ov 7364  df-nm 23961
This theorem is referenced by:  nmfval2  23970
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