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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 6910 | . . 3 ⊢ ((2↑(2↑𝑛)) + 1) ∈ V | |
2 | df-fmtno 42222 | . . 3 ⊢ FermatNo = (𝑛 ∈ ℕ0 ↦ ((2↑(2↑𝑛)) + 1)) | |
3 | 1, 2 | fnmpti 6233 | . 2 ⊢ FermatNo Fn ℕ0 |
4 | fvelrnb 6468 | . 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 198 = wceq 1653 ∈ wcel 2157 ∃wrex 3090 ran crn 5313 Fn wfn 6096 ‘cfv 6101 (class class class)co 6878 1c1 10225 + caddc 10227 2c2 11368 ℕ0cn0 11580 ↑cexp 13114 FermatNocfmtno 42221 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pr 5097 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3387 df-sbc 3634 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-br 4844 df-opab 4906 df-mpt 4923 df-id 5220 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-iota 6064 df-fun 6103 df-fn 6104 df-fv 6109 df-ov 6881 df-fmtno 42222 |
This theorem is referenced by: prmdvdsfmtnof1lem2 42279 prmdvdsfmtnof 42280 prmdvdsfmtnof1 42281 |
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