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Mirrors > Home > MPE Home > Th. List > Mathboxes > prjspnval | Structured version Visualization version GIF version |
Description: Value of the n-dimensional projective space function. (Contributed by Steven Nguyen, 1-May-2023.) |
Ref | Expression |
---|---|
prjspnval | ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ DivRing) → (𝑁ℙ𝕣𝕠𝕛n𝐾) = (ℙ𝕣𝕠𝕛‘(𝐾 freeLMod (0...𝑁)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 7157 | . . . 4 ⊢ (𝑛 = 𝑁 → (0...𝑛) = (0...𝑁)) | |
2 | 1 | oveq2d 7165 | . . 3 ⊢ (𝑛 = 𝑁 → (𝑘 freeLMod (0...𝑛)) = (𝑘 freeLMod (0...𝑁))) |
3 | 2 | fveq2d 6667 | . 2 ⊢ (𝑛 = 𝑁 → (ℙ𝕣𝕠𝕛‘(𝑘 freeLMod (0...𝑛))) = (ℙ𝕣𝕠𝕛‘(𝑘 freeLMod (0...𝑁)))) |
4 | fvoveq1 7172 | . 2 ⊢ (𝑘 = 𝐾 → (ℙ𝕣𝕠𝕛‘(𝑘 freeLMod (0...𝑁))) = (ℙ𝕣𝕠𝕛‘(𝐾 freeLMod (0...𝑁)))) | |
5 | df-prjspn 39341 | . 2 ⊢ ℙ𝕣𝕠𝕛n = (𝑛 ∈ ℕ0, 𝑘 ∈ DivRing ↦ (ℙ𝕣𝕠𝕛‘(𝑘 freeLMod (0...𝑛)))) | |
6 | fvex 6676 | . 2 ⊢ (ℙ𝕣𝕠𝕛‘(𝐾 freeLMod (0...𝑁))) ∈ V | |
7 | 3, 4, 5, 6 | ovmpo 7303 | 1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ DivRing) → (𝑁ℙ𝕣𝕠𝕛n𝐾) = (ℙ𝕣𝕠𝕛‘(𝐾 freeLMod (0...𝑁)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ‘cfv 6348 (class class class)co 7149 0cc0 10530 ℕ0cn0 11891 ...cfz 12889 DivRingcdr 19497 freeLMod cfrlm 20885 ℙ𝕣𝕠𝕛cprjsp 39327 ℙ𝕣𝕠𝕛ncprjspn 39340 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pr 5323 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-sbc 3769 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5060 df-opab 5122 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-ov 7152 df-oprab 7153 df-mpo 7154 df-prjspn 39341 |
This theorem is referenced by: prjspnval2 39343 |
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