prjspnval - Mathbox for Steven Nguyen

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Theorem prjspnval 43753

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

**Assertion**
Ref	Expression
prjspnval	⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐾 ∈ DivRing) → (𝑁 ℙroj_n 𝐾) = (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 7432	. . . 4 ⊢ (𝑛 = 𝑁 → (0...𝑛) = (0...𝑁))
2	1	oveq2d 7440	. . 3 ⊢ (𝑛 = 𝑁 → (𝑘 freeLMod (0...𝑛)) = (𝑘 freeLMod (0...𝑁)))
3	2	fveq2d 6894	. 2 ⊢ (𝑛 = 𝑁 → (ℙroj‘(𝑘 freeLMod (0...𝑛))) = (ℙroj‘(𝑘 freeLMod (0...𝑁))))
4		fvoveq1 7447	. 2 ⊢ (𝑘 = 𝐾 → (ℙroj‘(𝑘 freeLMod (0...𝑁))) = (ℙroj‘(𝐾 freeLMod (0...𝑁))))
5		df-prjspn 43752	. 2 ⊢ ℙroj_n = (𝑛 ∈ ℕ₀, 𝑘 ∈ DivRing ↦ (ℙroj‘(𝑘 freeLMod (0...𝑛))))
6		fvex 6903	. 2 ⊢ (ℙroj‘(𝐾 freeLMod (0...𝑁))) ∈ V
7	3, 4, 5, 6	ovmpo 7584	1 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐾 ∈ DivRing) → (𝑁 ℙroj_n 𝐾) = (ℙroj‘(𝐾 freeLMod (0...𝑁))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1569 ∈ wcel 2144 ‘cfv 6543 (class class class)co 7424 0cc0 11207 ℕ₀cn0 12613 ...cfz 13646 DivRingcdr 20989 freeLMod cfrlm 22061 ℙrojcprjsp 43738 ℙroj_n cprjspn 43751

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 5391

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-sbc 3739 df-ss 3897 df-dif 4046 df-in 4090 df-un 4150 df-nul 4278 df-if 4482 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-br 5103 df-opab 5167 df-id 5546 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-iota 6497 df-fun 6545 df-fv 6551 df-ov 7427 df-oprab 7428 df-mpo 7429 df-prjspn 43752

This theorem is used by: prjspnval2 43755

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