Theorem prjspnnormval 43925

Description: Value of 𝐹, a function that 'normalizes' a vector 𝑉 by scaling the first nonzero coordinate to 1. (Contributed by SN, 24-Sep-2026.)

Hypotheses

Ref	Expression
prjspnnormval.f	⊢ 𝐹 = (𝑣 ∈ 𝐵 ↦ ((𝐼‘(𝑣‘(𝐽‘𝑣))) · 𝑣))
prjspnnormval.v	⊢ (𝜑 → 𝑉 ∈ 𝐵)

Assertion

Ref	Expression
prjspnnormval	⊢ (𝜑 → (𝐹‘𝑉) = ((𝐼‘(𝑉‘(𝐽‘𝑉))) · 𝑉))

Distinct variable groups:   𝑣,𝐵   𝑣,𝐼   𝑣,𝐽   𝑣,𝑉   𝑣, ·

Allowed substitution hints:   𝜑(𝑣)   𝐹(𝑣)

Proof of Theorem prjspnnormval

Step	Hyp	Ref	Expression
1		prjspnnormval.f	. 2 ⊢ 𝐹 = (𝑣 ∈ 𝐵 ↦ ((𝐼‘(𝑣‘(𝐽‘𝑣))) · 𝑣))
2		id 23	. . . . 5 ⊢ (𝑣 = 𝑉 → 𝑣 = 𝑉)
3		fveq2 6916	. . . . 5 ⊢ (𝑣 = 𝑉 → (𝐽‘𝑣) = (𝐽‘𝑉))
4	2, 3	fveq12d 6923	. . . 4 ⊢ (𝑣 = 𝑉 → (𝑣‘(𝐽‘𝑣)) = (𝑉‘(𝐽‘𝑉)))
5	4	fveq2d 6920	. . 3 ⊢ (𝑣 = 𝑉 → (𝐼‘(𝑣‘(𝐽‘𝑣))) = (𝐼‘(𝑉‘(𝐽‘𝑉))))
6	5, 2	oveq12d 7469	. 2 ⊢ (𝑣 = 𝑉 → ((𝐼‘(𝑣‘(𝐽‘𝑣))) · 𝑣) = ((𝐼‘(𝑉‘(𝐽‘𝑉))) · 𝑉))
7		prjspnnormval.v	. 2 ⊢ (𝜑 → 𝑉 ∈ 𝐵)
8		ovexd 7486	. 2 ⊢ (𝜑 → ((𝐼‘(𝑉‘(𝐽‘𝑉))) · 𝑉) ∈ V)
9	1, 6, 7, 8	fvmptd3 7048	1 ⊢ (𝜑 → (𝐹‘𝑉) = ((𝐼‘(𝑉‘(𝐽‘𝑉))) · 𝑉))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  Vcvv 3454   ↦ cmpt 5192  ‘cfv 6566  (class class class)co 7451

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-iota 6520  df-fun 6568  df-fv 6574  df-ov 7454

This theorem is used by:  prjspnequivnorm  43926  prjspnnorm  43927