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Theorem fmtnorec2 47151
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 fvoveq1 7439 . . 3 (𝑥 = 0 → (FermatNo‘(𝑥 + 1)) = (FermatNo‘(0 + 1)))
2 oveq2 7424 . . . . 5 (𝑥 = 0 → (0...𝑥) = (0...0))
32prodeq1d 15918 . . . 4 (𝑥 = 0 → ∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) = ∏𝑛 ∈ (0...0)(FermatNo‘𝑛))
43oveq1d 7431 . . 3 (𝑥 = 0 → (∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) + 2) = (∏𝑛 ∈ (0...0)(FermatNo‘𝑛) + 2))
51, 4eqeq12d 2742 . 2 (𝑥 = 0 → ((FermatNo‘(𝑥 + 1)) = (∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) + 2) ↔ (FermatNo‘(0 + 1)) = (∏𝑛 ∈ (0...0)(FermatNo‘𝑛) + 2)))
6 fvoveq1 7439 . . 3 (𝑥 = 𝑦 → (FermatNo‘(𝑥 + 1)) = (FermatNo‘(𝑦 + 1)))
7 oveq2 7424 . . . . 5 (𝑥 = 𝑦 → (0...𝑥) = (0...𝑦))
87prodeq1d 15918 . . . 4 (𝑥 = 𝑦 → ∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) = ∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛))
98oveq1d 7431 . . 3 (𝑥 = 𝑦 → (∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) + 2) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2))
106, 9eqeq12d 2742 . 2 (𝑥 = 𝑦 → ((FermatNo‘(𝑥 + 1)) = (∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) + 2) ↔ (FermatNo‘(𝑦 + 1)) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2)))
11 fvoveq1 7439 . . 3 (𝑥 = (𝑦 + 1) → (FermatNo‘(𝑥 + 1)) = (FermatNo‘((𝑦 + 1) + 1)))
12 oveq2 7424 . . . . 5 (𝑥 = (𝑦 + 1) → (0...𝑥) = (0...(𝑦 + 1)))
1312prodeq1d 15918 . . . 4 (𝑥 = (𝑦 + 1) → ∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) = ∏𝑛 ∈ (0...(𝑦 + 1))(FermatNo‘𝑛))
1413oveq1d 7431 . . 3 (𝑥 = (𝑦 + 1) → (∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) + 2) = (∏𝑛 ∈ (0...(𝑦 + 1))(FermatNo‘𝑛) + 2))
1511, 14eqeq12d 2742 . 2 (𝑥 = (𝑦 + 1) → ((FermatNo‘(𝑥 + 1)) = (∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) + 2) ↔ (FermatNo‘((𝑦 + 1) + 1)) = (∏𝑛 ∈ (0...(𝑦 + 1))(FermatNo‘𝑛) + 2)))
16 fvoveq1 7439 . . 3 (𝑥 = 𝑁 → (FermatNo‘(𝑥 + 1)) = (FermatNo‘(𝑁 + 1)))
17 oveq2 7424 . . . 4 (𝑥 = 𝑁 → (0...𝑥) = (0...𝑁))
18 prodeq1 15906 . . . . 5 ((0...𝑥) = (0...𝑁) → ∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) = ∏𝑛 ∈ (0...𝑁)(FermatNo‘𝑛))
1918oveq1d 7431 . . . 4 ((0...𝑥) = (0...𝑁) → (∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) + 2) = (∏𝑛 ∈ (0...𝑁)(FermatNo‘𝑛) + 2))
2017, 19syl 17 . . 3 (𝑥 = 𝑁 → (∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) + 2) = (∏𝑛 ∈ (0...𝑁)(FermatNo‘𝑛) + 2))
2116, 20eqeq12d 2742 . 2 (𝑥 = 𝑁 → ((FermatNo‘(𝑥 + 1)) = (∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) + 2) ↔ (FermatNo‘(𝑁 + 1)) = (∏𝑛 ∈ (0...𝑁)(FermatNo‘𝑛) + 2)))
22 fmtno0 47148 . . . . 5 (FermatNo‘0) = 3
2322oveq1i 7426 . . . 4 ((FermatNo‘0) + 2) = (3 + 2)
24 3p2e5 12409 . . . 4 (3 + 2) = 5
2523, 24eqtri 2754 . . 3 ((FermatNo‘0) + 2) = 5
26 fz0sn 13649 . . . . . 6 (0...0) = {0}
2726prodeq1i 15915 . . . . 5 𝑛 ∈ (0...0)(FermatNo‘𝑛) = ∏𝑛 ∈ {0} (FermatNo‘𝑛)
28 0z 12615 . . . . . 6 0 ∈ ℤ
29 0nn0 12533 . . . . . . 7 0 ∈ ℕ0
30 fmtnonn 47139 . . . . . . . 8 (0 ∈ ℕ0 → (FermatNo‘0) ∈ ℕ)
3130nncnd 12274 . . . . . . 7 (0 ∈ ℕ0 → (FermatNo‘0) ∈ ℂ)
3229, 31ax-mp 5 . . . . . 6 (FermatNo‘0) ∈ ℂ
33 fveq2 6893 . . . . . . 7 (𝑛 = 0 → (FermatNo‘𝑛) = (FermatNo‘0))
3433prodsn 15959 . . . . . 6 ((0 ∈ ℤ ∧ (FermatNo‘0) ∈ ℂ) → ∏𝑛 ∈ {0} (FermatNo‘𝑛) = (FermatNo‘0))
3528, 32, 34mp2an 690 . . . . 5 𝑛 ∈ {0} (FermatNo‘𝑛) = (FermatNo‘0)
3627, 35eqtri 2754 . . . 4 𝑛 ∈ (0...0)(FermatNo‘𝑛) = (FermatNo‘0)
3736oveq1i 7426 . . 3 (∏𝑛 ∈ (0...0)(FermatNo‘𝑛) + 2) = ((FermatNo‘0) + 2)
38 0p1e1 12380 . . . . 5 (0 + 1) = 1
3938fveq2i 6896 . . . 4 (FermatNo‘(0 + 1)) = (FermatNo‘1)
40 fmtno1 47149 . . . 4 (FermatNo‘1) = 5
4139, 40eqtri 2754 . . 3 (FermatNo‘(0 + 1)) = 5
4225, 37, 413eqtr4ri 2765 . 2 (FermatNo‘(0 + 1)) = (∏𝑛 ∈ (0...0)(FermatNo‘𝑛) + 2)
43 fmtnorec2lem 47150 . 2 (𝑦 ∈ ℕ0 → ((FermatNo‘(𝑦 + 1)) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2) → (FermatNo‘((𝑦 + 1) + 1)) = (∏𝑛 ∈ (0...(𝑦 + 1))(FermatNo‘𝑛) + 2)))
445, 10, 15, 21, 42, 43nn0ind 12703 1 (𝑁 ∈ ℕ0 → (FermatNo‘(𝑁 + 1)) = (∏𝑛 ∈ (0...𝑁)(FermatNo‘𝑛) + 2))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wcel 2099  {csn 4623  cfv 6546  (class class class)co 7416  cc 11147  0cc0 11149  1c1 11150   + caddc 11152  2c2 12313  3c3 12314  5c5 12316  0cn0 12518  cz 12604  ...cfz 13532  cprod 15902  FermatNocfmtno 47135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5282  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7738  ax-inf2 9677  ax-cnex 11205  ax-resscn 11206  ax-1cn 11207  ax-icn 11208  ax-addcl 11209  ax-addrcl 11210  ax-mulcl 11211  ax-mulrcl 11212  ax-mulcom 11213  ax-addass 11214  ax-mulass 11215  ax-distr 11216  ax-i2m1 11217  ax-1ne0 11218  ax-1rid 11219  ax-rnegex 11220  ax-rrecex 11221  ax-cnre 11222  ax-pre-lttri 11223  ax-pre-lttrn 11224  ax-pre-ltadd 11225  ax-pre-mulgt0 11226  ax-pre-sup 11227
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3966  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-int 4947  df-iun 4995  df-br 5146  df-opab 5208  df-mpt 5229  df-tr 5263  df-id 5572  df-eprel 5578  df-po 5586  df-so 5587  df-fr 5629  df-se 5630  df-we 5631  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-pred 6304  df-ord 6371  df-on 6372  df-lim 6373  df-suc 6374  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-isom 6555  df-riota 7372  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7869  df-1st 7995  df-2nd 7996  df-frecs 8288  df-wrecs 8319  df-recs 8393  df-rdg 8432  df-1o 8488  df-er 8726  df-en 8967  df-dom 8968  df-sdom 8969  df-fin 8970  df-sup 9478  df-oi 9546  df-card 9975  df-pnf 11291  df-mnf 11292  df-xr 11293  df-ltxr 11294  df-le 11295  df-sub 11487  df-neg 11488  df-div 11913  df-nn 12259  df-2 12321  df-3 12322  df-4 12323  df-5 12324  df-n0 12519  df-z 12605  df-uz 12869  df-rp 13023  df-fz 13533  df-fzo 13676  df-seq 14016  df-exp 14076  df-hash 14343  df-cj 15099  df-re 15100  df-im 15101  df-sqrt 15235  df-abs 15236  df-clim 15485  df-prod 15903  df-fmtno 47136
This theorem is referenced by:  fmtnodvds  47152  fmtnorec3  47156
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