Theorem prjspnval 40463

Description: Value of the n-dimensional projective space function. (Contributed by Steven Nguyen, 1-May-2023.)

Assertion

Ref	Expression
prjspnval	⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ DivRing) → (𝑁ℙ𝕣𝕠𝕛n𝐾) = (ℙ𝕣𝕠𝕛‘(𝐾 freeLMod (0...𝑁))))

Proof of Theorem prjspnval

Dummy variables 𝑘 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 7275	. . . 4 ⊢ (𝑛 = 𝑁 → (0...𝑛) = (0...𝑁))
2	1	oveq2d 7283	. . 3 ⊢ (𝑛 = 𝑁 → (𝑘 freeLMod (0...𝑛)) = (𝑘 freeLMod (0...𝑁)))
3	2	fveq2d 6770	. 2 ⊢ (𝑛 = 𝑁 → (ℙ𝕣𝕠𝕛‘(𝑘 freeLMod (0...𝑛))) = (ℙ𝕣𝕠𝕛‘(𝑘 freeLMod (0...𝑁))))
4		fvoveq1 7290	. 2 ⊢ (𝑘 = 𝐾 → (ℙ𝕣𝕠𝕛‘(𝑘 freeLMod (0...𝑁))) = (ℙ𝕣𝕠𝕛‘(𝐾 freeLMod (0...𝑁))))
5		df-prjspn 40462	. 2 ⊢ ℙ𝕣𝕠𝕛n = (𝑛 ∈ ℕ0, 𝑘 ∈ DivRing ↦ (ℙ𝕣𝕠𝕛‘(𝑘 freeLMod (0...𝑛))))
6		fvex 6779	. 2 ⊢ (ℙ𝕣𝕠𝕛‘(𝐾 freeLMod (0...𝑁))) ∈ V
7	3, 4, 5, 6	ovmpo 7423	1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ DivRing) → (𝑁ℙ𝕣𝕠𝕛n𝐾) = (ℙ𝕣𝕠𝕛‘(𝐾 freeLMod (0...𝑁))))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ‘cfv 6426  (class class class)co 7267  0cc0 10881  ℕ₀cn0 12243  ...cfz 13249  DivRingcdr 20001   freeLMod cfrlm 20963  ℙ𝕣𝕠𝕛cprjsp 40448  ℙ𝕣𝕠𝕛_ncprjspn 40461

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5221  ax-nul 5228  ax-pr 5350

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3431  df-sbc 3716  df-dif 3889  df-un 3891  df-in 3893  df-ss 3903  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5074  df-opab 5136  df-id 5484  df-xp 5590  df-rel 5591  df-cnv 5592  df-co 5593  df-dm 5594  df-iota 6384  df-fun 6428  df-fv 6434  df-ov 7270  df-oprab 7271  df-mpo 7272  df-prjspn 40462

This theorem is referenced by:  prjspnval2  40465  prjcrv0  40478