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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fmtnorn | Structured version Visualization version GIF version |
Description: A Fermat number is a function value of the enumeration of the Fermat numbers. (Contributed by AV, 3-Aug-2021.) |
Ref | Expression |
---|---|
fmtnorn | ⊢ (𝐹 ∈ ran FermatNo ↔ ∃𝑛 ∈ ℕ0 (FermatNo‘𝑛) = 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovex 7463 | . . 3 ⊢ ((2↑(2↑𝑛)) + 1) ∈ V | |
2 | df-fmtno 47452 | . . 3 ⊢ FermatNo = (𝑛 ∈ ℕ0 ↦ ((2↑(2↑𝑛)) + 1)) | |
3 | 1, 2 | fnmpti 6711 | . 2 ⊢ FermatNo Fn ℕ0 |
4 | fvelrnb 6968 | . 2 ⊢ (FermatNo Fn ℕ0 → (𝐹 ∈ ran FermatNo ↔ ∃𝑛 ∈ ℕ0 (FermatNo‘𝑛) = 𝐹)) | |
5 | 3, 4 | ax-mp 5 | 1 ⊢ (𝐹 ∈ ran FermatNo ↔ ∃𝑛 ∈ ℕ0 (FermatNo‘𝑛) = 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 = wceq 1536 ∈ wcel 2105 ∃wrex 3067 ran crn 5689 Fn wfn 6557 ‘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-rn 5699 df-iota 6515 df-fun 6564 df-fn 6565 df-fv 6570 df-ov 7433 df-fmtno 47452 |
This theorem is referenced by: prmdvdsfmtnof1lem2 47509 prmdvdsfmtnof 47510 prmdvdsfmtnof1 47511 |
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