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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 43567 | . 2 ⊢ FermatNo = (𝑛 ∈ ℕ0 ↦ ((2↑(2↑𝑛)) + 1)) | |
2 | oveq2 7153 | . . . 4 ⊢ (𝑛 = 𝑁 → (2↑𝑛) = (2↑𝑁)) | |
3 | 2 | oveq2d 7161 | . . 3 ⊢ (𝑛 = 𝑁 → (2↑(2↑𝑛)) = (2↑(2↑𝑁))) |
4 | 3 | oveq1d 7160 | . 2 ⊢ (𝑛 = 𝑁 → ((2↑(2↑𝑛)) + 1) = ((2↑(2↑𝑁)) + 1)) |
5 | id 22 | . 2 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℕ0) | |
6 | ovexd 7180 | . 2 ⊢ (𝑁 ∈ ℕ0 → ((2↑(2↑𝑁)) + 1) ∈ V) | |
7 | 1, 4, 5, 6 | fvmptd3 6783 | 1 ⊢ (𝑁 ∈ ℕ0 → (FermatNo‘𝑁) = ((2↑(2↑𝑁)) + 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ‘cfv 6348 (class class class)co 7145 1c1 10526 + caddc 10528 2c2 11680 ℕ0cn0 11885 ↑cexp 13417 FermatNocfmtno 43566 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 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 7148 df-fmtno 43567 |
This theorem is referenced by: fmtnoge3 43569 fmtnom1nn 43571 fmtnoodd 43572 fmtnof1 43574 fmtnorec1 43576 fmtnosqrt 43578 fmtno0 43579 fmtno1 43580 fmtnorec2lem 43581 fmtnorec3 43587 fmtnorec4 43588 fmtno2 43589 fmtno3 43590 fmtno4 43591 fmtnoprmfac1lem 43603 fmtno4prm 43614 2pwp1prmfmtno 43629 |
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