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Theorem fmtnorec2 45725
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 7380 . . 3 (𝑥 = 0 → (FermatNo‘(𝑥 + 1)) = (FermatNo‘(0 + 1)))
2 oveq2 7365 . . . . 5 (𝑥 = 0 → (0...𝑥) = (0...0))
32prodeq1d 15804 . . . 4 (𝑥 = 0 → ∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) = ∏𝑛 ∈ (0...0)(FermatNo‘𝑛))
43oveq1d 7372 . . 3 (𝑥 = 0 → (∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) + 2) = (∏𝑛 ∈ (0...0)(FermatNo‘𝑛) + 2))
51, 4eqeq12d 2752 . 2 (𝑥 = 0 → ((FermatNo‘(𝑥 + 1)) = (∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) + 2) ↔ (FermatNo‘(0 + 1)) = (∏𝑛 ∈ (0...0)(FermatNo‘𝑛) + 2)))
6 fvoveq1 7380 . . 3 (𝑥 = 𝑦 → (FermatNo‘(𝑥 + 1)) = (FermatNo‘(𝑦 + 1)))
7 oveq2 7365 . . . . 5 (𝑥 = 𝑦 → (0...𝑥) = (0...𝑦))
87prodeq1d 15804 . . . 4 (𝑥 = 𝑦 → ∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) = ∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛))
98oveq1d 7372 . . 3 (𝑥 = 𝑦 → (∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) + 2) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2))
106, 9eqeq12d 2752 . 2 (𝑥 = 𝑦 → ((FermatNo‘(𝑥 + 1)) = (∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) + 2) ↔ (FermatNo‘(𝑦 + 1)) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2)))
11 fvoveq1 7380 . . 3 (𝑥 = (𝑦 + 1) → (FermatNo‘(𝑥 + 1)) = (FermatNo‘((𝑦 + 1) + 1)))
12 oveq2 7365 . . . . 5 (𝑥 = (𝑦 + 1) → (0...𝑥) = (0...(𝑦 + 1)))
1312prodeq1d 15804 . . . 4 (𝑥 = (𝑦 + 1) → ∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) = ∏𝑛 ∈ (0...(𝑦 + 1))(FermatNo‘𝑛))
1413oveq1d 7372 . . 3 (𝑥 = (𝑦 + 1) → (∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) + 2) = (∏𝑛 ∈ (0...(𝑦 + 1))(FermatNo‘𝑛) + 2))
1511, 14eqeq12d 2752 . 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 7380 . . 3 (𝑥 = 𝑁 → (FermatNo‘(𝑥 + 1)) = (FermatNo‘(𝑁 + 1)))
17 oveq2 7365 . . . 4 (𝑥 = 𝑁 → (0...𝑥) = (0...𝑁))
18 prodeq1 15792 . . . . 5 ((0...𝑥) = (0...𝑁) → ∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) = ∏𝑛 ∈ (0...𝑁)(FermatNo‘𝑛))
1918oveq1d 7372 . . . 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 2752 . 2 (𝑥 = 𝑁 → ((FermatNo‘(𝑥 + 1)) = (∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) + 2) ↔ (FermatNo‘(𝑁 + 1)) = (∏𝑛 ∈ (0...𝑁)(FermatNo‘𝑛) + 2)))
22 fmtno0 45722 . . . . 5 (FermatNo‘0) = 3
2322oveq1i 7367 . . . 4 ((FermatNo‘0) + 2) = (3 + 2)
24 3p2e5 12304 . . . 4 (3 + 2) = 5
2523, 24eqtri 2764 . . 3 ((FermatNo‘0) + 2) = 5
26 fz0sn 13541 . . . . . 6 (0...0) = {0}
2726prodeq1i 15801 . . . . 5 𝑛 ∈ (0...0)(FermatNo‘𝑛) = ∏𝑛 ∈ {0} (FermatNo‘𝑛)
28 0z 12510 . . . . . 6 0 ∈ ℤ
29 0nn0 12428 . . . . . . 7 0 ∈ ℕ0
30 fmtnonn 45713 . . . . . . . 8 (0 ∈ ℕ0 → (FermatNo‘0) ∈ ℕ)
3130nncnd 12169 . . . . . . 7 (0 ∈ ℕ0 → (FermatNo‘0) ∈ ℂ)
3229, 31ax-mp 5 . . . . . 6 (FermatNo‘0) ∈ ℂ
33 fveq2 6842 . . . . . . 7 (𝑛 = 0 → (FermatNo‘𝑛) = (FermatNo‘0))
3433prodsn 15845 . . . . . 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 2764 . . . 4 𝑛 ∈ (0...0)(FermatNo‘𝑛) = (FermatNo‘0)
3736oveq1i 7367 . . 3 (∏𝑛 ∈ (0...0)(FermatNo‘𝑛) + 2) = ((FermatNo‘0) + 2)
38 0p1e1 12275 . . . . 5 (0 + 1) = 1
3938fveq2i 6845 . . . 4 (FermatNo‘(0 + 1)) = (FermatNo‘1)
40 fmtno1 45723 . . . 4 (FermatNo‘1) = 5
4139, 40eqtri 2764 . . 3 (FermatNo‘(0 + 1)) = 5
4225, 37, 413eqtr4ri 2775 . 2 (FermatNo‘(0 + 1)) = (∏𝑛 ∈ (0...0)(FermatNo‘𝑛) + 2)
43 fmtnorec2lem 45724 . 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 12598 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 4586  cfv 6496  (class class class)co 7357  cc 11049  0cc0 11051  1c1 11052   + caddc 11054  2c2 12208  3c3 12209  5c5 12211  0cn0 12413  cz 12499  ...cfz 13424  cprod 15788  FermatNocfmtno 45709
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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-n0 12414  df-z 12500  df-uz 12764  df-rp 12916  df-fz 13425  df-fzo 13568  df-seq 13907  df-exp 13968  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-clim 15370  df-prod 15789  df-fmtno 45710
This theorem is referenced by:  fmtnodvds  45726  fmtnorec3  45730
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