Theorem prjspnnormval 43753

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 6886	. . . . 5 ⊢ (𝑣 = 𝑉 → (𝐽‘𝑣) = (𝐽‘𝑉))
4	2, 3	fveq12d 6893	. . . 4 ⊢ (𝑣 = 𝑉 → (𝑣‘(𝐽‘𝑣)) = (𝑉‘(𝐽‘𝑉)))
5	4	fveq2d 6890	. . 3 ⊢ (𝑣 = 𝑉 → (𝐼‘(𝑣‘(𝐽‘𝑣))) = (𝐼‘(𝑉‘(𝐽‘𝑉))))
6	5, 2	oveq12d 7439	. 2 ⊢ (𝑣 = 𝑉 → ((𝐼‘(𝑣‘(𝐽‘𝑣))) · 𝑣) = ((𝐼‘(𝑉‘(𝐽‘𝑉))) · 𝑉))
7		prjspnnormval.v	. 2 ⊢ (𝜑 → 𝑉 ∈ 𝐵)
8		ovexd 7456	. 2 ⊢ (𝜑 → ((𝐼‘(𝑉‘(𝐽‘𝑉))) · 𝑉) ∈ V)
9	1, 6, 7, 8	fvmptd3 7018	1 ⊢ (𝜑 → (𝐹‘𝑉) = ((𝐼‘(𝑉‘(𝐽‘𝑉))) · 𝑉))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144  Vcvv 3450   ↦ cmpt 5185  ‘cfv 6539  (class class class)co 7421

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-iota 6494  df-fun 6541  df-fv 6547  df-ov 7424

This theorem is used by:  prjspnequivnorm  43754  prjspnnorm  43755