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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 46868 | . 2 ⊢ FermatNo = (𝑛 ∈ ℕ0 ↦ ((2↑(2↑𝑛)) + 1)) | |
2 | oveq2 7428 | . . . 4 ⊢ (𝑛 = 𝑁 → (2↑𝑛) = (2↑𝑁)) | |
3 | 2 | oveq2d 7436 | . . 3 ⊢ (𝑛 = 𝑁 → (2↑(2↑𝑛)) = (2↑(2↑𝑁))) |
4 | 3 | oveq1d 7435 | . 2 ⊢ (𝑛 = 𝑁 → ((2↑(2↑𝑛)) + 1) = ((2↑(2↑𝑁)) + 1)) |
5 | id 22 | . 2 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℕ0) | |
6 | ovexd 7455 | . 2 ⊢ (𝑁 ∈ ℕ0 → ((2↑(2↑𝑁)) + 1) ∈ V) | |
7 | 1, 4, 5, 6 | fvmptd3 7028 | 1 ⊢ (𝑁 ∈ ℕ0 → (FermatNo‘𝑁) = ((2↑(2↑𝑁)) + 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 Vcvv 3471 ‘cfv 6548 (class class class)co 7420 1c1 11140 + caddc 11142 2c2 12298 ℕ0cn0 12503 ↑cexp 14059 FermatNocfmtno 46867 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-iota 6500 df-fun 6550 df-fv 6556 df-ov 7423 df-fmtno 46868 |
This theorem is referenced by: fmtnoge3 46870 fmtnom1nn 46872 fmtnoodd 46873 fmtnof1 46875 fmtnorec1 46877 fmtnosqrt 46879 fmtno0 46880 fmtno1 46881 fmtnorec2lem 46882 fmtnorec3 46888 fmtnorec4 46889 fmtno2 46890 fmtno3 46891 fmtno4 46892 fmtnoprmfac1lem 46904 fmtno4prm 46915 2pwp1prmfmtno 46930 |
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