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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 7440 | . . . 4 ⊢ (𝑛 = 𝑁 → (0...𝑛) = (0...𝑁)) | |
| 2 | 1 | oveq2d 7448 | . . 3 ⊢ (𝑛 = 𝑁 → (𝑘 freeLMod (0...𝑛)) = (𝑘 freeLMod (0...𝑁))) | 
| 3 | 2 | fveq2d 6909 | . 2 ⊢ (𝑛 = 𝑁 → (ℙ𝕣𝕠𝕛‘(𝑘 freeLMod (0...𝑛))) = (ℙ𝕣𝕠𝕛‘(𝑘 freeLMod (0...𝑁)))) | 
| 4 | fvoveq1 7455 | . 2 ⊢ (𝑘 = 𝐾 → (ℙ𝕣𝕠𝕛‘(𝑘 freeLMod (0...𝑁))) = (ℙ𝕣𝕠𝕛‘(𝐾 freeLMod (0...𝑁)))) | |
| 5 | df-prjspn 42630 | . 2 ⊢ ℙ𝕣𝕠𝕛n = (𝑛 ∈ ℕ0, 𝑘 ∈ DivRing ↦ (ℙ𝕣𝕠𝕛‘(𝑘 freeLMod (0...𝑛)))) | |
| 6 | fvex 6918 | . 2 ⊢ (ℙ𝕣𝕠𝕛‘(𝐾 freeLMod (0...𝑁))) ∈ V | |
| 7 | 3, 4, 5, 6 | ovmpo 7594 | 1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ DivRing) → (𝑁ℙ𝕣𝕠𝕛n𝐾) = (ℙ𝕣𝕠𝕛‘(𝐾 freeLMod (0...𝑁)))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ‘cfv 6560 (class class class)co 7432 0cc0 11156 ℕ0cn0 12528 ...cfz 13548 DivRingcdr 20730 freeLMod cfrlm 21767 ℙ𝕣𝕠𝕛cprjsp 42616 ℙ𝕣𝕠𝕛ncprjspn 42629 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-iota 6513 df-fun 6562 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-prjspn 42630 | 
| This theorem is referenced by: prjspnval2 42633 | 
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