prjspnval - Mathbox for Steven Nguyen

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Theorem prjspnval 43910

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 7459	. . . 4 ⊢ (𝑛 = 𝑁 → (0...𝑛) = (0...𝑁))
2	1	oveq2d 7467	. . 3 ⊢ (𝑛 = 𝑁 → (𝑘 freeLMod (0...𝑛)) = (𝑘 freeLMod (0...𝑁)))
3	2	fveq2d 6920	. 2 ⊢ (𝑛 = 𝑁 → (ℙroj‘(𝑘 freeLMod (0...𝑛))) = (ℙroj‘(𝑘 freeLMod (0...𝑁))))
4		fvoveq1 7474	. 2 ⊢ (𝑘 = 𝐾 → (ℙroj‘(𝑘 freeLMod (0...𝑁))) = (ℙroj‘(𝐾 freeLMod (0...𝑁))))
5		df-prjspn 43909	. 2 ⊢ ℙroj_n = (𝑛 ∈ ℕ₀, 𝑘 ∈ DivRing ↦ (ℙroj‘(𝑘 freeLMod (0...𝑛))))
6		fvex 6929	. 2 ⊢ (ℙroj‘(𝐾 freeLMod (0...𝑁))) ∈ V
7	3, 4, 5, 6	ovmpo 7612	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 1570 ∈ wcel 2147 ‘cfv 6566 (class class class)co 7451 0cc0 11239 ℕ₀cn0 12646 ...cfz 13679 DivRingcdr 21031 freeLMod cfrlm 22107 ℙrojcprjsp 43895 ℙroj_n cprjspn 43908

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-sep 5255 ax-nul 5266 ax-pr 5398

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-ral 3081 df-rex 3091 df-rab 3417 df-v 3456 df-sbc 3743 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-sn 4589 df-pr 4591 df-op 4841 df-uni 4873 df-br 5109 df-opab 5174 df-id 5553 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-iota 6520 df-fun 6568 df-fv 6574 df-ov 7454 df-oprab 7455 df-mpo 7456 df-prjspn 43909

This theorem is used by: prjspnval2 43912

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