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Theorem fmtnorec2 40742
Description: The second recurrence relation for Fermat numbers, see ProofWiki "Product of Sequence of Fermat Numbers plus 2", 29-Jul-2021, https://proofwiki.org/wiki/Product_of_Sequence_of_Fermat_Numbers_plus_2 or Wikipedia "Fermat number", 29-Jul-2021, https://en.wikipedia.org/wiki/Fermat_number#Basic_properties. (Contributed by AV, 29-Jul-2021.)
Assertion
Ref Expression
fmtnorec2 (𝑁 ∈ ℕ0 → (FermatNo‘(𝑁 + 1)) = (∏𝑛 ∈ (0...𝑁)(FermatNo‘𝑛) + 2))
Distinct variable group:   𝑛,𝑁

Proof of Theorem fmtnorec2
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 6612 . . . 4 (𝑥 = 0 → (𝑥 + 1) = (0 + 1))
21fveq2d 6154 . . 3 (𝑥 = 0 → (FermatNo‘(𝑥 + 1)) = (FermatNo‘(0 + 1)))
3 oveq2 6613 . . . . 5 (𝑥 = 0 → (0...𝑥) = (0...0))
43prodeq1d 14571 . . . 4 (𝑥 = 0 → ∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) = ∏𝑛 ∈ (0...0)(FermatNo‘𝑛))
54oveq1d 6620 . . 3 (𝑥 = 0 → (∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) + 2) = (∏𝑛 ∈ (0...0)(FermatNo‘𝑛) + 2))
62, 5eqeq12d 2641 . 2 (𝑥 = 0 → ((FermatNo‘(𝑥 + 1)) = (∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) + 2) ↔ (FermatNo‘(0 + 1)) = (∏𝑛 ∈ (0...0)(FermatNo‘𝑛) + 2)))
7 oveq1 6612 . . . 4 (𝑥 = 𝑦 → (𝑥 + 1) = (𝑦 + 1))
87fveq2d 6154 . . 3 (𝑥 = 𝑦 → (FermatNo‘(𝑥 + 1)) = (FermatNo‘(𝑦 + 1)))
9 oveq2 6613 . . . . 5 (𝑥 = 𝑦 → (0...𝑥) = (0...𝑦))
109prodeq1d 14571 . . . 4 (𝑥 = 𝑦 → ∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) = ∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛))
1110oveq1d 6620 . . 3 (𝑥 = 𝑦 → (∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) + 2) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2))
128, 11eqeq12d 2641 . 2 (𝑥 = 𝑦 → ((FermatNo‘(𝑥 + 1)) = (∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) + 2) ↔ (FermatNo‘(𝑦 + 1)) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2)))
13 oveq1 6612 . . . 4 (𝑥 = (𝑦 + 1) → (𝑥 + 1) = ((𝑦 + 1) + 1))
1413fveq2d 6154 . . 3 (𝑥 = (𝑦 + 1) → (FermatNo‘(𝑥 + 1)) = (FermatNo‘((𝑦 + 1) + 1)))
15 oveq2 6613 . . . . 5 (𝑥 = (𝑦 + 1) → (0...𝑥) = (0...(𝑦 + 1)))
1615prodeq1d 14571 . . . 4 (𝑥 = (𝑦 + 1) → ∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) = ∏𝑛 ∈ (0...(𝑦 + 1))(FermatNo‘𝑛))
1716oveq1d 6620 . . 3 (𝑥 = (𝑦 + 1) → (∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) + 2) = (∏𝑛 ∈ (0...(𝑦 + 1))(FermatNo‘𝑛) + 2))
1814, 17eqeq12d 2641 . 2 (𝑥 = (𝑦 + 1) → ((FermatNo‘(𝑥 + 1)) = (∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) + 2) ↔ (FermatNo‘((𝑦 + 1) + 1)) = (∏𝑛 ∈ (0...(𝑦 + 1))(FermatNo‘𝑛) + 2)))
19 oveq1 6612 . . . 4 (𝑥 = 𝑁 → (𝑥 + 1) = (𝑁 + 1))
2019fveq2d 6154 . . 3 (𝑥 = 𝑁 → (FermatNo‘(𝑥 + 1)) = (FermatNo‘(𝑁 + 1)))
21 oveq2 6613 . . . 4 (𝑥 = 𝑁 → (0...𝑥) = (0...𝑁))
22 prodeq1 14559 . . . . 5 ((0...𝑥) = (0...𝑁) → ∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) = ∏𝑛 ∈ (0...𝑁)(FermatNo‘𝑛))
2322oveq1d 6620 . . . 4 ((0...𝑥) = (0...𝑁) → (∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) + 2) = (∏𝑛 ∈ (0...𝑁)(FermatNo‘𝑛) + 2))
2421, 23syl 17 . . 3 (𝑥 = 𝑁 → (∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) + 2) = (∏𝑛 ∈ (0...𝑁)(FermatNo‘𝑛) + 2))
2520, 24eqeq12d 2641 . 2 (𝑥 = 𝑁 → ((FermatNo‘(𝑥 + 1)) = (∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) + 2) ↔ (FermatNo‘(𝑁 + 1)) = (∏𝑛 ∈ (0...𝑁)(FermatNo‘𝑛) + 2)))
26 fmtno0 40739 . . . . 5 (FermatNo‘0) = 3
2726oveq1i 6615 . . . 4 ((FermatNo‘0) + 2) = (3 + 2)
28 3p2e5 11105 . . . 4 (3 + 2) = 5
2927, 28eqtri 2648 . . 3 ((FermatNo‘0) + 2) = 5
30 fz0sn 12377 . . . . . 6 (0...0) = {0}
3130prodeq1i 14568 . . . . 5 𝑛 ∈ (0...0)(FermatNo‘𝑛) = ∏𝑛 ∈ {0} (FermatNo‘𝑛)
32 0z 11333 . . . . . 6 0 ∈ ℤ
33 0nn0 11252 . . . . . . 7 0 ∈ ℕ0
34 fmtnonn 40730 . . . . . . . 8 (0 ∈ ℕ0 → (FermatNo‘0) ∈ ℕ)
3534nncnd 10981 . . . . . . 7 (0 ∈ ℕ0 → (FermatNo‘0) ∈ ℂ)
3633, 35ax-mp 5 . . . . . 6 (FermatNo‘0) ∈ ℂ
37 fveq2 6150 . . . . . . 7 (𝑛 = 0 → (FermatNo‘𝑛) = (FermatNo‘0))
3837prodsn 14612 . . . . . 6 ((0 ∈ ℤ ∧ (FermatNo‘0) ∈ ℂ) → ∏𝑛 ∈ {0} (FermatNo‘𝑛) = (FermatNo‘0))
3932, 36, 38mp2an 707 . . . . 5 𝑛 ∈ {0} (FermatNo‘𝑛) = (FermatNo‘0)
4031, 39eqtri 2648 . . . 4 𝑛 ∈ (0...0)(FermatNo‘𝑛) = (FermatNo‘0)
4140oveq1i 6615 . . 3 (∏𝑛 ∈ (0...0)(FermatNo‘𝑛) + 2) = ((FermatNo‘0) + 2)
42 0p1e1 11077 . . . . 5 (0 + 1) = 1
4342fveq2i 6153 . . . 4 (FermatNo‘(0 + 1)) = (FermatNo‘1)
44 fmtno1 40740 . . . 4 (FermatNo‘1) = 5
4543, 44eqtri 2648 . . 3 (FermatNo‘(0 + 1)) = 5
4629, 41, 453eqtr4ri 2659 . 2 (FermatNo‘(0 + 1)) = (∏𝑛 ∈ (0...0)(FermatNo‘𝑛) + 2)
47 fmtnorec2lem 40741 . 2 (𝑦 ∈ ℕ0 → ((FermatNo‘(𝑦 + 1)) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2) → (FermatNo‘((𝑦 + 1) + 1)) = (∏𝑛 ∈ (0...(𝑦 + 1))(FermatNo‘𝑛) + 2)))
486, 12, 18, 25, 46, 47nn0ind 11416 1 (𝑁 ∈ ℕ0 → (FermatNo‘(𝑁 + 1)) = (∏𝑛 ∈ (0...𝑁)(FermatNo‘𝑛) + 2))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1992  {csn 4153  cfv 5850  (class class class)co 6605  cc 9879  0cc0 9881  1c1 9882   + caddc 9884  2c2 11015  3c3 11016  5c5 11018  0cn0 11237  cz 11322  ...cfz 12265  cprod 14555  FermatNocfmtno 40726
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-inf2 8483  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958  ax-pre-sup 9959
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-isom 5859  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-oadd 7510  df-er 7688  df-en 7901  df-dom 7902  df-sdom 7903  df-fin 7904  df-sup 8293  df-oi 8360  df-card 8710  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-div 10630  df-nn 10966  df-2 11024  df-3 11025  df-4 11026  df-5 11027  df-n0 11238  df-z 11323  df-uz 11632  df-rp 11777  df-fz 12266  df-fzo 12404  df-seq 12739  df-exp 12798  df-hash 13055  df-cj 13768  df-re 13769  df-im 13770  df-sqrt 13904  df-abs 13905  df-clim 14148  df-prod 14556  df-fmtno 40727
This theorem is referenced by:  fmtnodvds  40743  fmtnorec3  40747
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