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Theorem fmtnorec2 46201
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 7431 . . 3 (𝑥 = 0 → (FermatNo‘(𝑥 + 1)) = (FermatNo‘(0 + 1)))
2 oveq2 7416 . . . . 5 (𝑥 = 0 → (0...𝑥) = (0...0))
32prodeq1d 15864 . . . 4 (𝑥 = 0 → ∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) = ∏𝑛 ∈ (0...0)(FermatNo‘𝑛))
43oveq1d 7423 . . 3 (𝑥 = 0 → (∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) + 2) = (∏𝑛 ∈ (0...0)(FermatNo‘𝑛) + 2))
51, 4eqeq12d 2748 . 2 (𝑥 = 0 → ((FermatNo‘(𝑥 + 1)) = (∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) + 2) ↔ (FermatNo‘(0 + 1)) = (∏𝑛 ∈ (0...0)(FermatNo‘𝑛) + 2)))
6 fvoveq1 7431 . . 3 (𝑥 = 𝑦 → (FermatNo‘(𝑥 + 1)) = (FermatNo‘(𝑦 + 1)))
7 oveq2 7416 . . . . 5 (𝑥 = 𝑦 → (0...𝑥) = (0...𝑦))
87prodeq1d 15864 . . . 4 (𝑥 = 𝑦 → ∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) = ∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛))
98oveq1d 7423 . . 3 (𝑥 = 𝑦 → (∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) + 2) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2))
106, 9eqeq12d 2748 . 2 (𝑥 = 𝑦 → ((FermatNo‘(𝑥 + 1)) = (∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) + 2) ↔ (FermatNo‘(𝑦 + 1)) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2)))
11 fvoveq1 7431 . . 3 (𝑥 = (𝑦 + 1) → (FermatNo‘(𝑥 + 1)) = (FermatNo‘((𝑦 + 1) + 1)))
12 oveq2 7416 . . . . 5 (𝑥 = (𝑦 + 1) → (0...𝑥) = (0...(𝑦 + 1)))
1312prodeq1d 15864 . . . 4 (𝑥 = (𝑦 + 1) → ∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) = ∏𝑛 ∈ (0...(𝑦 + 1))(FermatNo‘𝑛))
1413oveq1d 7423 . . 3 (𝑥 = (𝑦 + 1) → (∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) + 2) = (∏𝑛 ∈ (0...(𝑦 + 1))(FermatNo‘𝑛) + 2))
1511, 14eqeq12d 2748 . 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 7431 . . 3 (𝑥 = 𝑁 → (FermatNo‘(𝑥 + 1)) = (FermatNo‘(𝑁 + 1)))
17 oveq2 7416 . . . 4 (𝑥 = 𝑁 → (0...𝑥) = (0...𝑁))
18 prodeq1 15852 . . . . 5 ((0...𝑥) = (0...𝑁) → ∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) = ∏𝑛 ∈ (0...𝑁)(FermatNo‘𝑛))
1918oveq1d 7423 . . . 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 2748 . 2 (𝑥 = 𝑁 → ((FermatNo‘(𝑥 + 1)) = (∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) + 2) ↔ (FermatNo‘(𝑁 + 1)) = (∏𝑛 ∈ (0...𝑁)(FermatNo‘𝑛) + 2)))
22 fmtno0 46198 . . . . 5 (FermatNo‘0) = 3
2322oveq1i 7418 . . . 4 ((FermatNo‘0) + 2) = (3 + 2)
24 3p2e5 12362 . . . 4 (3 + 2) = 5
2523, 24eqtri 2760 . . 3 ((FermatNo‘0) + 2) = 5
26 fz0sn 13600 . . . . . 6 (0...0) = {0}
2726prodeq1i 15861 . . . . 5 𝑛 ∈ (0...0)(FermatNo‘𝑛) = ∏𝑛 ∈ {0} (FermatNo‘𝑛)
28 0z 12568 . . . . . 6 0 ∈ ℤ
29 0nn0 12486 . . . . . . 7 0 ∈ ℕ0
30 fmtnonn 46189 . . . . . . . 8 (0 ∈ ℕ0 → (FermatNo‘0) ∈ ℕ)
3130nncnd 12227 . . . . . . 7 (0 ∈ ℕ0 → (FermatNo‘0) ∈ ℂ)
3229, 31ax-mp 5 . . . . . 6 (FermatNo‘0) ∈ ℂ
33 fveq2 6891 . . . . . . 7 (𝑛 = 0 → (FermatNo‘𝑛) = (FermatNo‘0))
3433prodsn 15905 . . . . . 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 2760 . . . 4 𝑛 ∈ (0...0)(FermatNo‘𝑛) = (FermatNo‘0)
3736oveq1i 7418 . . 3 (∏𝑛 ∈ (0...0)(FermatNo‘𝑛) + 2) = ((FermatNo‘0) + 2)
38 0p1e1 12333 . . . . 5 (0 + 1) = 1
3938fveq2i 6894 . . . 4 (FermatNo‘(0 + 1)) = (FermatNo‘1)
40 fmtno1 46199 . . . 4 (FermatNo‘1) = 5
4139, 40eqtri 2760 . . 3 (FermatNo‘(0 + 1)) = 5
4225, 37, 413eqtr4ri 2771 . 2 (FermatNo‘(0 + 1)) = (∏𝑛 ∈ (0...0)(FermatNo‘𝑛) + 2)
43 fmtnorec2lem 46200 . 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 12656 1 (𝑁 ∈ ℕ0 → (FermatNo‘(𝑁 + 1)) = (∏𝑛 ∈ (0...𝑁)(FermatNo‘𝑛) + 2))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  {csn 4628  cfv 6543  (class class class)co 7408  cc 11107  0cc0 11109  1c1 11110   + caddc 11112  2c2 12266  3c3 12267  5c5 12269  0cn0 12471  cz 12557  ...cfz 13483  cprod 15848  FermatNocfmtno 46185
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-inf2 9635  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186  ax-pre-sup 11187
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-1st 7974  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-1o 8465  df-er 8702  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-sup 9436  df-oi 9504  df-card 9933  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-div 11871  df-nn 12212  df-2 12274  df-3 12275  df-4 12276  df-5 12277  df-n0 12472  df-z 12558  df-uz 12822  df-rp 12974  df-fz 13484  df-fzo 13627  df-seq 13966  df-exp 14027  df-hash 14290  df-cj 15045  df-re 15046  df-im 15047  df-sqrt 15181  df-abs 15182  df-clim 15431  df-prod 15849  df-fmtno 46186
This theorem is referenced by:  fmtnodvds  46202  fmtnorec3  46206
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