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Theorem fmtno 42016
Description: The 𝑁 th Fermat number. (Contributed by AV, 13-Jun-2021.)
Assertion
Ref Expression
fmtno (𝑁 ∈ ℕ0 → (FermatNo‘𝑁) = ((2↑(2↑𝑁)) + 1))

Proof of Theorem fmtno
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 df-fmtno 42015 . . 3 FermatNo = (𝑛 ∈ ℕ0 ↦ ((2↑(2↑𝑛)) + 1))
21a1i 11 . 2 (𝑁 ∈ ℕ0 → FermatNo = (𝑛 ∈ ℕ0 ↦ ((2↑(2↑𝑛)) + 1)))
3 oveq2 6882 . . . . 5 (𝑛 = 𝑁 → (2↑𝑛) = (2↑𝑁))
43oveq2d 6890 . . . 4 (𝑛 = 𝑁 → (2↑(2↑𝑛)) = (2↑(2↑𝑁)))
54oveq1d 6889 . . 3 (𝑛 = 𝑁 → ((2↑(2↑𝑛)) + 1) = ((2↑(2↑𝑁)) + 1))
65adantl 469 . 2 ((𝑁 ∈ ℕ0𝑛 = 𝑁) → ((2↑(2↑𝑛)) + 1) = ((2↑(2↑𝑁)) + 1))
7 id 22 . 2 (𝑁 ∈ ℕ0𝑁 ∈ ℕ0)
8 ovexd 6908 . 2 (𝑁 ∈ ℕ0 → ((2↑(2↑𝑁)) + 1) ∈ V)
92, 6, 7, 8fvmptd 6509 1 (𝑁 ∈ ℕ0 → (FermatNo‘𝑁) = ((2↑(2↑𝑁)) + 1))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1637  wcel 2156  Vcvv 3391  cmpt 4923  cfv 6101  (class class class)co 6874  1c1 10222   + caddc 10224  2c2 11356  0cn0 11559  cexp 13083  FermatNocfmtno 42014
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pr 5096
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-iota 6064  df-fun 6103  df-fv 6109  df-ov 6877  df-fmtno 42015
This theorem is referenced by:  fmtnoge3  42017  fmtnom1nn  42019  fmtnoodd  42020  fmtnof1  42022  fmtnorec1  42024  fmtnosqrt  42026  fmtno0  42027  fmtno1  42028  fmtnorec2lem  42029  fmtnorec3  42035  fmtnorec4  42036  fmtno2  42037  fmtno3  42038  fmtno4  42039  fmtnoprmfac1lem  42051  fmtno4prm  42062  2pwp1prmfmtno  42079
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