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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 7464 | . . 3 ⊢ ((2↑(2↑𝑛)) + 1) ∈ V | |
| 2 | df-fmtno 47515 | . . 3 ⊢ FermatNo = (𝑛 ∈ ℕ0 ↦ ((2↑(2↑𝑛)) + 1)) | |
| 3 | 1, 2 | fnmpti 6711 | . 2 ⊢ FermatNo Fn ℕ0 |
| 4 | fvelrnb 6969 | . 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 1540 ∈ wcel 2108 ∃wrex 3070 ran crn 5686 Fn wfn 6556 ‘cfv 6561 (class class class)co 7431 1c1 11156 + caddc 11158 2c2 12321 ℕ0cn0 12526 ↑cexp 14102 FermatNocfmtno 47514 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fn 6564 df-fv 6569 df-ov 7434 df-fmtno 47515 |
| This theorem is referenced by: prmdvdsfmtnof1lem2 47572 prmdvdsfmtnof 47573 prmdvdsfmtnof1 47574 |
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