Theorem prjspnval 42974

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 7376	. . . 4 ⊢ (𝑛 = 𝑁 → (0...𝑛) = (0...𝑁))
2	1	oveq2d 7384	. . 3 ⊢ (𝑛 = 𝑁 → (𝑘 freeLMod (0...𝑛)) = (𝑘 freeLMod (0...𝑁)))
3	2	fveq2d 6846	. 2 ⊢ (𝑛 = 𝑁 → (ℙ𝕣𝕠𝕛‘(𝑘 freeLMod (0...𝑛))) = (ℙ𝕣𝕠𝕛‘(𝑘 freeLMod (0...𝑁))))
4		fvoveq1 7391	. 2 ⊢ (𝑘 = 𝐾 → (ℙ𝕣𝕠𝕛‘(𝑘 freeLMod (0...𝑁))) = (ℙ𝕣𝕠𝕛‘(𝐾 freeLMod (0...𝑁))))
5		df-prjspn 42973	. 2 ⊢ ℙ𝕣𝕠𝕛n = (𝑛 ∈ ℕ0, 𝑘 ∈ DivRing ↦ (ℙ𝕣𝕠𝕛‘(𝑘 freeLMod (0...𝑛))))
6		fvex 6855	. 2 ⊢ (ℙ𝕣𝕠𝕛‘(𝐾 freeLMod (0...𝑁))) ∈ V
7	3, 4, 5, 6	ovmpo 7528	1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ DivRing) → (𝑁ℙ𝕣𝕠𝕛n𝐾) = (ℙ𝕣𝕠𝕛‘(𝐾 freeLMod (0...𝑁))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ‘cfv 6500  (class class class)co 7368  0cc0 11038  ℕ₀cn0 12413  ...cfz 13435  DivRingcdr 20674   freeLMod cfrlm 21713  ℙ𝕣𝕠𝕛cprjsp 42959  ℙ𝕣𝕠𝕛_ncprjspn 42972

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-iota 6456  df-fun 6502  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-prjspn 42973

This theorem is referenced by:  prjspnval2  42976