Theorem fmtnorec2 48901

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 7474	. . 3 ⊢ (𝑥 = 0 → (FermatNo‘(𝑥 + 1)) = (FermatNo‘(0 + 1)))
2		oveq2 7459	. . . . 5 ⊢ (𝑥 = 0 → (0...𝑥) = (0...0))
3	2	prodeq1d 16132	. . . 4 ⊢ (𝑥 = 0 → ∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) = ∏𝑛 ∈ (0...0)(FermatNo‘𝑛))
4	3	oveq1d 7466	. . 3 ⊢ (𝑥 = 0 → (∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) + 2) = (∏𝑛 ∈ (0...0)(FermatNo‘𝑛) + 2))
5	1, 4	eqeq12d 2779	. 2 ⊢ (𝑥 = 0 → ((FermatNo‘(𝑥 + 1)) = (∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) + 2) ↔ (FermatNo‘(0 + 1)) = (∏𝑛 ∈ (0...0)(FermatNo‘𝑛) + 2)))
6		fvoveq1 7474	. . 3 ⊢ (𝑥 = 𝑦 → (FermatNo‘(𝑥 + 1)) = (FermatNo‘(𝑦 + 1)))
7		oveq2 7459	. . . . 5 ⊢ (𝑥 = 𝑦 → (0...𝑥) = (0...𝑦))
8	7	prodeq1d 16132	. . . 4 ⊢ (𝑥 = 𝑦 → ∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) = ∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛))
9	8	oveq1d 7466	. . 3 ⊢ (𝑥 = 𝑦 → (∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) + 2) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2))
10	6, 9	eqeq12d 2779	. 2 ⊢ (𝑥 = 𝑦 → ((FermatNo‘(𝑥 + 1)) = (∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) + 2) ↔ (FermatNo‘(𝑦 + 1)) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2)))
11		fvoveq1 7474	. . 3 ⊢ (𝑥 = (𝑦 + 1) → (FermatNo‘(𝑥 + 1)) = (FermatNo‘((𝑦 + 1) + 1)))
12		oveq2 7459	. . . . 5 ⊢ (𝑥 = (𝑦 + 1) → (0...𝑥) = (0...(𝑦 + 1)))
13	12	prodeq1d 16132	. . . 4 ⊢ (𝑥 = (𝑦 + 1) → ∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) = ∏𝑛 ∈ (0...(𝑦 + 1))(FermatNo‘𝑛))
14	13	oveq1d 7466	. . 3 ⊢ (𝑥 = (𝑦 + 1) → (∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) + 2) = (∏𝑛 ∈ (0...(𝑦 + 1))(FermatNo‘𝑛) + 2))
15	11, 14	eqeq12d 2779	. 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 7474	. . 3 ⊢ (𝑥 = 𝑁 → (FermatNo‘(𝑥 + 1)) = (FermatNo‘(𝑁 + 1)))
17		oveq2 7459	. . . 4 ⊢ (𝑥 = 𝑁 → (0...𝑥) = (0...𝑁))
18		prodeq1 16120	. . . . 5 ⊢ ((0...𝑥) = (0...𝑁) → ∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) = ∏𝑛 ∈ (0...𝑁)(FermatNo‘𝑛))
19	18	oveq1d 7466	. . . 4 ⊢ ((0...𝑥) = (0...𝑁) → (∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) + 2) = (∏𝑛 ∈ (0...𝑁)(FermatNo‘𝑛) + 2))
20	17, 19	syl 18	. . 3 ⊢ (𝑥 = 𝑁 → (∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) + 2) = (∏𝑛 ∈ (0...𝑁)(FermatNo‘𝑛) + 2))
21	16, 20	eqeq12d 2779	. 2 ⊢ (𝑥 = 𝑁 → ((FermatNo‘(𝑥 + 1)) = (∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) + 2) ↔ (FermatNo‘(𝑁 + 1)) = (∏𝑛 ∈ (0...𝑁)(FermatNo‘𝑛) + 2)))
22		fmtno0 48898	. . . . 5 ⊢ (FermatNo‘0) = 3
23	22	oveq1i 7461	. . . 4 ⊢ ((FermatNo‘0) + 2) = (3 + 2)
24		3p2e5 12533	. . . 4 ⊢ (3 + 2) = 5
25	23, 24	eqtri 2786	. . 3 ⊢ ((FermatNo‘0) + 2) = 5
26		fz0sn 13801	. . . . . 6 ⊢ (0...0) = {0}
27	26	prodeq1i 16129	. . . . 5 ⊢ ∏𝑛 ∈ (0...0)(FermatNo‘𝑛) = ∏𝑛 ∈ {0} (FermatNo‘𝑛)
28		0z 12744	. . . . . 6 ⊢ 0 ∈ ℤ
29		0nn0 12661	. . . . . . 7 ⊢ 0 ∈ ℕ0
30		fmtnonn 48889	. . . . . . . 8 ⊢ (0 ∈ ℕ0 → (FermatNo‘0) ∈ ℕ)
31	30	nncnd 12391	. . . . . . 7 ⊢ (0 ∈ ℕ0 → (FermatNo‘0) ∈ ℂ)
32	29, 31	ax-mp 5	. . . . . 6 ⊢ (FermatNo‘0) ∈ ℂ
33		fveq2 6916	. . . . . . 7 ⊢ (𝑛 = 0 → (FermatNo‘𝑛) = (FermatNo‘0))
34	33	prodsn 16173	. . . . . 6 ⊢ ((0 ∈ ℤ ∧ (FermatNo‘0) ∈ ℂ) → ∏𝑛 ∈ {0} (FermatNo‘𝑛) = (FermatNo‘0))
35	28, 32, 34	mp2an 705	. . . . 5 ⊢ ∏𝑛 ∈ {0} (FermatNo‘𝑛) = (FermatNo‘0)
36	27, 35	eqtri 2786	. . . 4 ⊢ ∏𝑛 ∈ (0...0)(FermatNo‘𝑛) = (FermatNo‘0)
37	36	oveq1i 7461	. . 3 ⊢ (∏𝑛 ∈ (0...0)(FermatNo‘𝑛) + 2) = ((FermatNo‘0) + 2)
38		0p1e1 12503	. . . . 5 ⊢ (0 + 1) = 1
39	38	fveq2i 6919	. . . 4 ⊢ (FermatNo‘(0 + 1)) = (FermatNo‘1)
40		fmtno1 48899	. . . 4 ⊢ (FermatNo‘1) = 5
41	39, 40	eqtri 2786	. . 3 ⊢ (FermatNo‘(0 + 1)) = 5
42	25, 37, 41	3eqtr4ri 2797	. 2 ⊢ (FermatNo‘(0 + 1)) = (∏𝑛 ∈ (0...0)(FermatNo‘𝑛) + 2)
43		fmtnorec2lem 48900	. 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 12834	1 ⊢ (𝑁 ∈ ℕ0 → (FermatNo‘(𝑁 + 1)) = (∏𝑛 ∈ (0...𝑁)(FermatNo‘𝑛) + 2))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  {csn 4588  ‘cfv 6566  (class class class)co 7451  ℂcc 11237  0cc0 11239  1c1 11240   + caddc 11242  2c2 12437  3c3 12438  5c5 12440  ℕ₀cn0 12646  ℤcz 12733  ...cfz 13679  ∏cprod 16116  FermatNocfmtno 48885

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-inf2 9679  ax-cnex 11295  ax-resscn 11296  ax-1cn 11297  ax-icn 11298  ax-addcl 11299  ax-addrcl 11300  ax-mulcl 11301  ax-mulrcl 11302  ax-mulcom 11303  ax-addass 11304  ax-mulass 11305  ax-distr 11306  ax-i2m1 11307  ax-1ne0 11308  ax-1rid 11309  ax-rnegex 11310  ax-rrecex 11311  ax-cnre 11312  ax-pre-lttri 11313  ax-pre-lttrn 11314  ax-pre-ltadd 11315  ax-pre-mulgt0 11316  ax-pre-sup 11317

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-isom 6575  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-1o 8514  df-er 8755  df-en 9012  df-dom 9013  df-sdom 9014  df-fin 9015  df-sup 9471  df-oi 9541  df-card 10057  df-pnf 11384  df-mnf 11385  df-xr 11386  df-ltxr 11387  df-le 11388  df-sub 11582  df-neg 11583  df-div 12014  df-nn 12376  df-2 12445  df-3 12446  df-4 12447  df-5 12448  df-n0 12647  df-z 12734  df-uz 13006  df-rp 13161  df-fz 13680  df-fzo 13829  df-seq 14185  df-exp 14245  df-hash 14516  df-cj 15309  df-re 15310  df-im 15311  df-sqrt 15445  df-abs 15446  df-clim 15699  df-prod 16117  df-fmtno 48886

This theorem is used by:  fmtnodvds  48902  fmtnorec3  48906