Theorem fmtnorec2 41455

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		oveq1 6657	. . . 4 ⊢ (𝑥 = 0 → (𝑥 + 1) = (0 + 1))
2	1	fveq2d 6195	. . 3 ⊢ (𝑥 = 0 → (FermatNo‘(𝑥 + 1)) = (FermatNo‘(0 + 1)))
3		oveq2 6658	. . . . 5 ⊢ (𝑥 = 0 → (0...𝑥) = (0...0))
4	3	prodeq1d 14651	. . . 4 ⊢ (𝑥 = 0 → ∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) = ∏𝑛 ∈ (0...0)(FermatNo‘𝑛))
5	4	oveq1d 6665	. . 3 ⊢ (𝑥 = 0 → (∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) + 2) = (∏𝑛 ∈ (0...0)(FermatNo‘𝑛) + 2))
6	2, 5	eqeq12d 2637	. 2 ⊢ (𝑥 = 0 → ((FermatNo‘(𝑥 + 1)) = (∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) + 2) ↔ (FermatNo‘(0 + 1)) = (∏𝑛 ∈ (0...0)(FermatNo‘𝑛) + 2)))
7		oveq1 6657	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 + 1) = (𝑦 + 1))
8	7	fveq2d 6195	. . 3 ⊢ (𝑥 = 𝑦 → (FermatNo‘(𝑥 + 1)) = (FermatNo‘(𝑦 + 1)))
9		oveq2 6658	. . . . 5 ⊢ (𝑥 = 𝑦 → (0...𝑥) = (0...𝑦))
10	9	prodeq1d 14651	. . . 4 ⊢ (𝑥 = 𝑦 → ∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) = ∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛))
11	10	oveq1d 6665	. . 3 ⊢ (𝑥 = 𝑦 → (∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) + 2) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2))
12	8, 11	eqeq12d 2637	. 2 ⊢ (𝑥 = 𝑦 → ((FermatNo‘(𝑥 + 1)) = (∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) + 2) ↔ (FermatNo‘(𝑦 + 1)) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2)))
13		oveq1 6657	. . . 4 ⊢ (𝑥 = (𝑦 + 1) → (𝑥 + 1) = ((𝑦 + 1) + 1))
14	13	fveq2d 6195	. . 3 ⊢ (𝑥 = (𝑦 + 1) → (FermatNo‘(𝑥 + 1)) = (FermatNo‘((𝑦 + 1) + 1)))
15		oveq2 6658	. . . . 5 ⊢ (𝑥 = (𝑦 + 1) → (0...𝑥) = (0...(𝑦 + 1)))
16	15	prodeq1d 14651	. . . 4 ⊢ (𝑥 = (𝑦 + 1) → ∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) = ∏𝑛 ∈ (0...(𝑦 + 1))(FermatNo‘𝑛))
17	16	oveq1d 6665	. . 3 ⊢ (𝑥 = (𝑦 + 1) → (∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) + 2) = (∏𝑛 ∈ (0...(𝑦 + 1))(FermatNo‘𝑛) + 2))
18	14, 17	eqeq12d 2637	. 2 ⊢ (𝑥 = (𝑦 + 1) → ((FermatNo‘(𝑥 + 1)) = (∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) + 2) ↔ (FermatNo‘((𝑦 + 1) + 1)) = (∏𝑛 ∈ (0...(𝑦 + 1))(FermatNo‘𝑛) + 2)))
19		oveq1 6657	. . . 4 ⊢ (𝑥 = 𝑁 → (𝑥 + 1) = (𝑁 + 1))
20	19	fveq2d 6195	. . 3 ⊢ (𝑥 = 𝑁 → (FermatNo‘(𝑥 + 1)) = (FermatNo‘(𝑁 + 1)))
21		oveq2 6658	. . . 4 ⊢ (𝑥 = 𝑁 → (0...𝑥) = (0...𝑁))
22		prodeq1 14639	. . . . 5 ⊢ ((0...𝑥) = (0...𝑁) → ∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) = ∏𝑛 ∈ (0...𝑁)(FermatNo‘𝑛))
23	22	oveq1d 6665	. . . 4 ⊢ ((0...𝑥) = (0...𝑁) → (∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) + 2) = (∏𝑛 ∈ (0...𝑁)(FermatNo‘𝑛) + 2))
24	21, 23	syl 17	. . 3 ⊢ (𝑥 = 𝑁 → (∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) + 2) = (∏𝑛 ∈ (0...𝑁)(FermatNo‘𝑛) + 2))
25	20, 24	eqeq12d 2637	. 2 ⊢ (𝑥 = 𝑁 → ((FermatNo‘(𝑥 + 1)) = (∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) + 2) ↔ (FermatNo‘(𝑁 + 1)) = (∏𝑛 ∈ (0...𝑁)(FermatNo‘𝑛) + 2)))
26		fmtno0 41452	. . . . 5 ⊢ (FermatNo‘0) = 3
27	26	oveq1i 6660	. . . 4 ⊢ ((FermatNo‘0) + 2) = (3 + 2)
28		3p2e5 11160	. . . 4 ⊢ (3 + 2) = 5
29	27, 28	eqtri 2644	. . 3 ⊢ ((FermatNo‘0) + 2) = 5
30		fz0sn 12439	. . . . . 6 ⊢ (0...0) = {0}
31	30	prodeq1i 14648	. . . . 5 ⊢ ∏𝑛 ∈ (0...0)(FermatNo‘𝑛) = ∏𝑛 ∈ {0} (FermatNo‘𝑛)
32		0z 11388	. . . . . 6 ⊢ 0 ∈ ℤ
33		0nn0 11307	. . . . . . 7 ⊢ 0 ∈ ℕ0
34		fmtnonn 41443	. . . . . . . 8 ⊢ (0 ∈ ℕ0 → (FermatNo‘0) ∈ ℕ)
35	34	nncnd 11036	. . . . . . 7 ⊢ (0 ∈ ℕ0 → (FermatNo‘0) ∈ ℂ)
36	33, 35	ax-mp 5	. . . . . 6 ⊢ (FermatNo‘0) ∈ ℂ
37		fveq2 6191	. . . . . . 7 ⊢ (𝑛 = 0 → (FermatNo‘𝑛) = (FermatNo‘0))
38	37	prodsn 14692	. . . . . 6 ⊢ ((0 ∈ ℤ ∧ (FermatNo‘0) ∈ ℂ) → ∏𝑛 ∈ {0} (FermatNo‘𝑛) = (FermatNo‘0))
39	32, 36, 38	mp2an 708	. . . . 5 ⊢ ∏𝑛 ∈ {0} (FermatNo‘𝑛) = (FermatNo‘0)
40	31, 39	eqtri 2644	. . . 4 ⊢ ∏𝑛 ∈ (0...0)(FermatNo‘𝑛) = (FermatNo‘0)
41	40	oveq1i 6660	. . 3 ⊢ (∏𝑛 ∈ (0...0)(FermatNo‘𝑛) + 2) = ((FermatNo‘0) + 2)
42		0p1e1 11132	. . . . 5 ⊢ (0 + 1) = 1
43	42	fveq2i 6194	. . . 4 ⊢ (FermatNo‘(0 + 1)) = (FermatNo‘1)
44		fmtno1 41453	. . . 4 ⊢ (FermatNo‘1) = 5
45	43, 44	eqtri 2644	. . 3 ⊢ (FermatNo‘(0 + 1)) = 5
46	29, 41, 45	3eqtr4ri 2655	. 2 ⊢ (FermatNo‘(0 + 1)) = (∏𝑛 ∈ (0...0)(FermatNo‘𝑛) + 2)
47		fmtnorec2lem 41454	. 2 ⊢ (𝑦 ∈ ℕ0 → ((FermatNo‘(𝑦 + 1)) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2) → (FermatNo‘((𝑦 + 1) + 1)) = (∏𝑛 ∈ (0...(𝑦 + 1))(FermatNo‘𝑛) + 2)))
48	6, 12, 18, 25, 46, 47	nn0ind 11472	1 ⊢ (𝑁 ∈ ℕ0 → (FermatNo‘(𝑁 + 1)) = (∏𝑛 ∈ (0...𝑁)(FermatNo‘𝑛) + 2))

Syntax hints:   → wi 4   = wceq 1483   ∈ wcel 1990  {csn 4177  ‘cfv 5888  (class class class)co 6650  ℂcc 9934  0cc0 9936  1c1 9937   + caddc 9939  2c2 11070  3c3 11071  5c5 11073  ℕ₀cn0 11292  ℤcz 11377  ...cfz 12326  ∏cprod 14635  FermatNocfmtno 41439

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-prod 14636  df-fmtno 41440

This theorem is referenced by:  fmtnodvds  41456  fmtnorec3  41460