Theorem prjspnnormval 43639

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 6883	. . . . 5 ⊢ (𝑣 = 𝑉 → (𝐽‘𝑣) = (𝐽‘𝑉))
4	2, 3	fveq12d 6890	. . . 4 ⊢ (𝑣 = 𝑉 → (𝑣‘(𝐽‘𝑣)) = (𝑉‘(𝐽‘𝑉)))
5	4	fveq2d 6887	. . 3 ⊢ (𝑣 = 𝑉 → (𝐼‘(𝑣‘(𝐽‘𝑣))) = (𝐼‘(𝑉‘(𝐽‘𝑉))))
6	5, 2	oveq12d 7436	. 2 ⊢ (𝑣 = 𝑉 → ((𝐼‘(𝑣‘(𝐽‘𝑣))) · 𝑣) = ((𝐼‘(𝑉‘(𝐽‘𝑉))) · 𝑉))
7		prjspnnormval.v	. 2 ⊢ (𝜑 → 𝑉 ∈ 𝐵)
8		ovexd 7453	. 2 ⊢ (𝜑 → ((𝐼‘(𝑉‘(𝐽‘𝑉))) · 𝑉) ∈ V)
9	1, 6, 7, 8	fvmptd3 7015	1 ⊢ (𝜑 → (𝐹‘𝑉) = ((𝐼‘(𝑉‘(𝐽‘𝑉))) · 𝑉))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  Vcvv 3451   ↦ cmpt 5186  ‘cfv 6537  (class class class)co 7418

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

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 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-iota 6493  df-fun 6539  df-fv 6545  df-ov 7421

This theorem is used by:  prjspnequivnorm  43640  prjspnnorm  43641