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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 43697 | . 2 ⊢ FermatNo = (𝑛 ∈ ℕ0 ↦ ((2↑(2↑𝑛)) + 1)) | |
2 | oveq2 7166 | . . . 4 ⊢ (𝑛 = 𝑁 → (2↑𝑛) = (2↑𝑁)) | |
3 | 2 | oveq2d 7174 | . . 3 ⊢ (𝑛 = 𝑁 → (2↑(2↑𝑛)) = (2↑(2↑𝑁))) |
4 | 3 | oveq1d 7173 | . 2 ⊢ (𝑛 = 𝑁 → ((2↑(2↑𝑛)) + 1) = ((2↑(2↑𝑁)) + 1)) |
5 | id 22 | . 2 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℕ0) | |
6 | ovexd 7193 | . 2 ⊢ (𝑁 ∈ ℕ0 → ((2↑(2↑𝑁)) + 1) ∈ V) | |
7 | 1, 4, 5, 6 | fvmptd3 6793 | 1 ⊢ (𝑁 ∈ ℕ0 → (FermatNo‘𝑁) = ((2↑(2↑𝑁)) + 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ‘cfv 6357 (class class class)co 7158 1c1 10540 + caddc 10542 2c2 11695 ℕ0cn0 11900 ↑cexp 13432 FermatNocfmtno 43696 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-iota 6316 df-fun 6359 df-fv 6365 df-ov 7161 df-fmtno 43697 |
This theorem is referenced by: fmtnoge3 43699 fmtnom1nn 43701 fmtnoodd 43702 fmtnof1 43704 fmtnorec1 43706 fmtnosqrt 43708 fmtno0 43709 fmtno1 43710 fmtnorec2lem 43711 fmtnorec3 43717 fmtnorec4 43718 fmtno2 43719 fmtno3 43720 fmtno4 43721 fmtnoprmfac1lem 43733 fmtno4prm 43744 2pwp1prmfmtno 43759 |
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