Theorem fmtnorec2 47527

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.

Step	Hyp	Ref	Expression
1		fvoveq1 7372	. . 3 ⊢ (𝑥 = 0 → (FermatNo‘(𝑥 + 1)) = (FermatNo‘(0 + 1)))
2		oveq2 7357	. . . . 5 ⊢ (𝑥 = 0 → (0...𝑥) = (0...0))
3	2	prodeq1d 15827	. . . 4 ⊢ (𝑥 = 0 → ∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) = ∏𝑛 ∈ (0...0)(FermatNo‘𝑛))
4	3	oveq1d 7364	. . 3 ⊢ (𝑥 = 0 → (∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) + 2) = (∏𝑛 ∈ (0...0)(FermatNo‘𝑛) + 2))
5	1, 4	eqeq12d 2745	. 2 ⊢ (𝑥 = 0 → ((FermatNo‘(𝑥 + 1)) = (∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) + 2) ↔ (FermatNo‘(0 + 1)) = (∏𝑛 ∈ (0...0)(FermatNo‘𝑛) + 2)))
6		fvoveq1 7372	. . 3 ⊢ (𝑥 = 𝑦 → (FermatNo‘(𝑥 + 1)) = (FermatNo‘(𝑦 + 1)))
7		oveq2 7357	. . . . 5 ⊢ (𝑥 = 𝑦 → (0...𝑥) = (0...𝑦))
8	7	prodeq1d 15827	. . . 4 ⊢ (𝑥 = 𝑦 → ∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) = ∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛))
9	8	oveq1d 7364	. . 3 ⊢ (𝑥 = 𝑦 → (∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) + 2) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2))
10	6, 9	eqeq12d 2745	. 2 ⊢ (𝑥 = 𝑦 → ((FermatNo‘(𝑥 + 1)) = (∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) + 2) ↔ (FermatNo‘(𝑦 + 1)) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2)))
11		fvoveq1 7372	. . 3 ⊢ (𝑥 = (𝑦 + 1) → (FermatNo‘(𝑥 + 1)) = (FermatNo‘((𝑦 + 1) + 1)))
12		oveq2 7357	. . . . 5 ⊢ (𝑥 = (𝑦 + 1) → (0...𝑥) = (0...(𝑦 + 1)))
13	12	prodeq1d 15827	. . . 4 ⊢ (𝑥 = (𝑦 + 1) → ∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) = ∏𝑛 ∈ (0...(𝑦 + 1))(FermatNo‘𝑛))
14	13	oveq1d 7364	. . 3 ⊢ (𝑥 = (𝑦 + 1) → (∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) + 2) = (∏𝑛 ∈ (0...(𝑦 + 1))(FermatNo‘𝑛) + 2))
15	11, 14	eqeq12d 2745	. 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 7372	. . 3 ⊢ (𝑥 = 𝑁 → (FermatNo‘(𝑥 + 1)) = (FermatNo‘(𝑁 + 1)))
17		oveq2 7357	. . . 4 ⊢ (𝑥 = 𝑁 → (0...𝑥) = (0...𝑁))
18		prodeq1 15814	. . . . 5 ⊢ ((0...𝑥) = (0...𝑁) → ∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) = ∏𝑛 ∈ (0...𝑁)(FermatNo‘𝑛))
19	18	oveq1d 7364	. . . 4 ⊢ ((0...𝑥) = (0...𝑁) → (∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) + 2) = (∏𝑛 ∈ (0...𝑁)(FermatNo‘𝑛) + 2))
20	17, 19	syl 17	. . 3 ⊢ (𝑥 = 𝑁 → (∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) + 2) = (∏𝑛 ∈ (0...𝑁)(FermatNo‘𝑛) + 2))
21	16, 20	eqeq12d 2745	. 2 ⊢ (𝑥 = 𝑁 → ((FermatNo‘(𝑥 + 1)) = (∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) + 2) ↔ (FermatNo‘(𝑁 + 1)) = (∏𝑛 ∈ (0...𝑁)(FermatNo‘𝑛) + 2)))
22		fmtno0 47524	. . . . 5 ⊢ (FermatNo‘0) = 3
23	22	oveq1i 7359	. . . 4 ⊢ ((FermatNo‘0) + 2) = (3 + 2)
24		3p2e5 12274	. . . 4 ⊢ (3 + 2) = 5
25	23, 24	eqtri 2752	. . 3 ⊢ ((FermatNo‘0) + 2) = 5
26		fz0sn 13530	. . . . . 6 ⊢ (0...0) = {0}
27	26	prodeq1i 15823	. . . . 5 ⊢ ∏𝑛 ∈ (0...0)(FermatNo‘𝑛) = ∏𝑛 ∈ {0} (FermatNo‘𝑛)
28		0z 12482	. . . . . 6 ⊢ 0 ∈ ℤ
29		0nn0 12399	. . . . . . 7 ⊢ 0 ∈ ℕ0
30		fmtnonn 47515	. . . . . . . 8 ⊢ (0 ∈ ℕ0 → (FermatNo‘0) ∈ ℕ)
31	30	nncnd 12144	. . . . . . 7 ⊢ (0 ∈ ℕ0 → (FermatNo‘0) ∈ ℂ)
32	29, 31	ax-mp 5	. . . . . 6 ⊢ (FermatNo‘0) ∈ ℂ
33		fveq2 6822	. . . . . . 7 ⊢ (𝑛 = 0 → (FermatNo‘𝑛) = (FermatNo‘0))
34	33	prodsn 15869	. . . . . 6 ⊢ ((0 ∈ ℤ ∧ (FermatNo‘0) ∈ ℂ) → ∏𝑛 ∈ {0} (FermatNo‘𝑛) = (FermatNo‘0))
35	28, 32, 34	mp2an 692	. . . . 5 ⊢ ∏𝑛 ∈ {0} (FermatNo‘𝑛) = (FermatNo‘0)
36	27, 35	eqtri 2752	. . . 4 ⊢ ∏𝑛 ∈ (0...0)(FermatNo‘𝑛) = (FermatNo‘0)
37	36	oveq1i 7359	. . 3 ⊢ (∏𝑛 ∈ (0...0)(FermatNo‘𝑛) + 2) = ((FermatNo‘0) + 2)
38		0p1e1 12245	. . . . 5 ⊢ (0 + 1) = 1
39	38	fveq2i 6825	. . . 4 ⊢ (FermatNo‘(0 + 1)) = (FermatNo‘1)
40		fmtno1 47525	. . . 4 ⊢ (FermatNo‘1) = 5
41	39, 40	eqtri 2752	. . 3 ⊢ (FermatNo‘(0 + 1)) = 5
42	25, 37, 41	3eqtr4ri 2763	. 2 ⊢ (FermatNo‘(0 + 1)) = (∏𝑛 ∈ (0...0)(FermatNo‘𝑛) + 2)
43		fmtnorec2lem 47526	. 2 ⊢ (𝑦 ∈ ℕ0 → ((FermatNo‘(𝑦 + 1)) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2) → (FermatNo‘((𝑦 + 1) + 1)) = (∏𝑛 ∈ (0...(𝑦 + 1))(FermatNo‘𝑛) + 2)))
44	5, 10, 15, 21, 42, 43	nn0ind 12571	1 ⊢ (𝑁 ∈ ℕ0 → (FermatNo‘(𝑁 + 1)) = (∏𝑛 ∈ (0...𝑁)(FermatNo‘𝑛) + 2))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  {csn 4577  ‘cfv 6482  (class class class)co 7349  ℂcc 11007  0cc0 11009  1c1 11010   + caddc 11012  2c2 12183  3c3 12184  5c5 12186  ℕ₀cn0 12384  ℤcz 12471  ...cfz 13410  ∏cprod 15810  FermatNocfmtno 47511

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-inf2 9537  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086  ax-pre-sup 11087

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-isom 6491  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-er 8625  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-sup 9332  df-oi 9402  df-card 9835  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-div 11778  df-nn 12129  df-2 12191  df-3 12192  df-4 12193  df-5 12194  df-n0 12385  df-z 12472  df-uz 12736  df-rp 12894  df-fz 13411  df-fzo 13558  df-seq 13909  df-exp 13969  df-hash 14238  df-cj 15006  df-re 15007  df-im 15008  df-sqrt 15142  df-abs 15143  df-clim 15395  df-prod 15811  df-fmtno 47512

This theorem is referenced by:  fmtnodvds  47528  fmtnorec3  47532