Theorem prjspnval 42589

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 7361	. . . 4 ⊢ (𝑛 = 𝑁 → (0...𝑛) = (0...𝑁))
2	1	oveq2d 7369	. . 3 ⊢ (𝑛 = 𝑁 → (𝑘 freeLMod (0...𝑛)) = (𝑘 freeLMod (0...𝑁)))
3	2	fveq2d 6830	. 2 ⊢ (𝑛 = 𝑁 → (ℙ𝕣𝕠𝕛‘(𝑘 freeLMod (0...𝑛))) = (ℙ𝕣𝕠𝕛‘(𝑘 freeLMod (0...𝑁))))
4		fvoveq1 7376	. 2 ⊢ (𝑘 = 𝐾 → (ℙ𝕣𝕠𝕛‘(𝑘 freeLMod (0...𝑁))) = (ℙ𝕣𝕠𝕛‘(𝐾 freeLMod (0...𝑁))))
5		df-prjspn 42588	. 2 ⊢ ℙ𝕣𝕠𝕛n = (𝑛 ∈ ℕ0, 𝑘 ∈ DivRing ↦ (ℙ𝕣𝕠𝕛‘(𝑘 freeLMod (0...𝑛))))
6		fvex 6839	. 2 ⊢ (ℙ𝕣𝕠𝕛‘(𝐾 freeLMod (0...𝑁))) ∈ V
7	3, 4, 5, 6	ovmpo 7513	1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ DivRing) → (𝑁ℙ𝕣𝕠𝕛n𝐾) = (ℙ𝕣𝕠𝕛‘(𝐾 freeLMod (0...𝑁))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ‘cfv 6486  (class class class)co 7353  0cc0 11028  ℕ₀cn0 12402  ...cfz 13428  DivRingcdr 20632   freeLMod cfrlm 21671  ℙ𝕣𝕠𝕛cprjsp 42574  ℙ𝕣𝕠𝕛_ncprjspn 42587

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-sbc 3745  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-iota 6442  df-fun 6488  df-fv 6494  df-ov 7356  df-oprab 7357  df-mpo 7358  df-prjspn 42588

This theorem is referenced by:  prjspnval2  42591