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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fmtno | Structured version Visualization version GIF version |
Description: The 𝑁 th Fermat number. (Contributed by AV, 13-Jun-2021.) |
Ref | Expression |
---|---|
fmtno | ⊢ (𝑁 ∈ ℕ0 → (FermatNo‘𝑁) = ((2↑(2↑𝑁)) + 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-fmtno 47452 | . 2 ⊢ FermatNo = (𝑛 ∈ ℕ0 ↦ ((2↑(2↑𝑛)) + 1)) | |
2 | oveq2 7438 | . . . 4 ⊢ (𝑛 = 𝑁 → (2↑𝑛) = (2↑𝑁)) | |
3 | 2 | oveq2d 7446 | . . 3 ⊢ (𝑛 = 𝑁 → (2↑(2↑𝑛)) = (2↑(2↑𝑁))) |
4 | 3 | oveq1d 7445 | . 2 ⊢ (𝑛 = 𝑁 → ((2↑(2↑𝑛)) + 1) = ((2↑(2↑𝑁)) + 1)) |
5 | id 22 | . 2 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℕ0) | |
6 | ovexd 7465 | . 2 ⊢ (𝑁 ∈ ℕ0 → ((2↑(2↑𝑁)) + 1) ∈ V) | |
7 | 1, 4, 5, 6 | fvmptd3 7038 | 1 ⊢ (𝑁 ∈ ℕ0 → (FermatNo‘𝑁) = ((2↑(2↑𝑁)) + 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2105 Vcvv 3477 ‘cfv 6562 (class class class)co 7430 1c1 11153 + caddc 11155 2c2 12318 ℕ0cn0 12523 ↑cexp 14098 FermatNocfmtno 47451 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-iota 6515 df-fun 6564 df-fv 6570 df-ov 7433 df-fmtno 47452 |
This theorem is referenced by: fmtnoge3 47454 fmtnom1nn 47456 fmtnoodd 47457 fmtnof1 47459 fmtnorec1 47461 fmtnosqrt 47463 fmtno0 47464 fmtno1 47465 fmtnorec2lem 47466 fmtnorec3 47472 fmtnorec4 47473 fmtno2 47474 fmtno3 47475 fmtno4 47476 fmtnoprmfac1lem 47488 fmtno4prm 47499 2pwp1prmfmtno 47514 |
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