Theorem fmtnorec2 48446

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 7436	. . 3 ⊢ (𝑥 = 0 → (FermatNo‘(𝑥 + 1)) = (FermatNo‘(0 + 1)))
2		oveq2 7421	. . . . 5 ⊢ (𝑥 = 0 → (0...𝑥) = (0...0))
3	2	prodeq1d 16008	. . . 4 ⊢ (𝑥 = 0 → ∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) = ∏𝑛 ∈ (0...0)(FermatNo‘𝑛))
4	3	oveq1d 7428	. . 3 ⊢ (𝑥 = 0 → (∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) + 2) = (∏𝑛 ∈ (0...0)(FermatNo‘𝑛) + 2))
5	1, 4	eqeq12d 2776	. 2 ⊢ (𝑥 = 0 → ((FermatNo‘(𝑥 + 1)) = (∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) + 2) ↔ (FermatNo‘(0 + 1)) = (∏𝑛 ∈ (0...0)(FermatNo‘𝑛) + 2)))
6		fvoveq1 7436	. . 3 ⊢ (𝑥 = 𝑦 → (FermatNo‘(𝑥 + 1)) = (FermatNo‘(𝑦 + 1)))
7		oveq2 7421	. . . . 5 ⊢ (𝑥 = 𝑦 → (0...𝑥) = (0...𝑦))
8	7	prodeq1d 16008	. . . 4 ⊢ (𝑥 = 𝑦 → ∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) = ∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛))
9	8	oveq1d 7428	. . 3 ⊢ (𝑥 = 𝑦 → (∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) + 2) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2))
10	6, 9	eqeq12d 2776	. 2 ⊢ (𝑥 = 𝑦 → ((FermatNo‘(𝑥 + 1)) = (∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) + 2) ↔ (FermatNo‘(𝑦 + 1)) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2)))
11		fvoveq1 7436	. . 3 ⊢ (𝑥 = (𝑦 + 1) → (FermatNo‘(𝑥 + 1)) = (FermatNo‘((𝑦 + 1) + 1)))
12		oveq2 7421	. . . . 5 ⊢ (𝑥 = (𝑦 + 1) → (0...𝑥) = (0...(𝑦 + 1)))
13	12	prodeq1d 16008	. . . 4 ⊢ (𝑥 = (𝑦 + 1) → ∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) = ∏𝑛 ∈ (0...(𝑦 + 1))(FermatNo‘𝑛))
14	13	oveq1d 7428	. . 3 ⊢ (𝑥 = (𝑦 + 1) → (∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) + 2) = (∏𝑛 ∈ (0...(𝑦 + 1))(FermatNo‘𝑛) + 2))
15	11, 14	eqeq12d 2776	. 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 7436	. . 3 ⊢ (𝑥 = 𝑁 → (FermatNo‘(𝑥 + 1)) = (FermatNo‘(𝑁 + 1)))
17		oveq2 7421	. . . 4 ⊢ (𝑥 = 𝑁 → (0...𝑥) = (0...𝑁))
18		prodeq1 15996	. . . . 5 ⊢ ((0...𝑥) = (0...𝑁) → ∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) = ∏𝑛 ∈ (0...𝑁)(FermatNo‘𝑛))
19	18	oveq1d 7428	. . . 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 2776	. 2 ⊢ (𝑥 = 𝑁 → ((FermatNo‘(𝑥 + 1)) = (∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) + 2) ↔ (FermatNo‘(𝑁 + 1)) = (∏𝑛 ∈ (0...𝑁)(FermatNo‘𝑛) + 2)))
22		fmtno0 48443	. . . . 5 ⊢ (FermatNo‘0) = 3
23	22	oveq1i 7423	. . . 4 ⊢ ((FermatNo‘0) + 2) = (3 + 2)
24		3p2e5 12415	. . . 4 ⊢ (3 + 2) = 5
25	23, 24	eqtri 2783	. . 3 ⊢ ((FermatNo‘0) + 2) = 5
26		fz0sn 13682	. . . . . 6 ⊢ (0...0) = {0}
27	26	prodeq1i 16005	. . . . 5 ⊢ ∏𝑛 ∈ (0...0)(FermatNo‘𝑛) = ∏𝑛 ∈ {0} (FermatNo‘𝑛)
28		0z 12626	. . . . . 6 ⊢ 0 ∈ ℤ
29		0nn0 12543	. . . . . . 7 ⊢ 0 ∈ ℕ0
30		fmtnonn 48434	. . . . . . . 8 ⊢ (0 ∈ ℕ0 → (FermatNo‘0) ∈ ℕ)
31	30	nncnd 12273	. . . . . . 7 ⊢ (0 ∈ ℕ0 → (FermatNo‘0) ∈ ℂ)
32	29, 31	ax-mp 5	. . . . . 6 ⊢ (FermatNo‘0) ∈ ℂ
33		fveq2 6878	. . . . . . 7 ⊢ (𝑛 = 0 → (FermatNo‘𝑛) = (FermatNo‘0))
34	33	prodsn 16049	. . . . . 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 2783	. . . 4 ⊢ ∏𝑛 ∈ (0...0)(FermatNo‘𝑛) = (FermatNo‘0)
37	36	oveq1i 7423	. . 3 ⊢ (∏𝑛 ∈ (0...0)(FermatNo‘𝑛) + 2) = ((FermatNo‘0) + 2)
38		0p1e1 12385	. . . . 5 ⊢ (0 + 1) = 1
39	38	fveq2i 6881	. . . 4 ⊢ (FermatNo‘(0 + 1)) = (FermatNo‘1)
40		fmtno1 48444	. . . 4 ⊢ (FermatNo‘1) = 5
41	39, 40	eqtri 2783	. . 3 ⊢ (FermatNo‘(0 + 1)) = 5
42	25, 37, 41	3eqtr4ri 2794	. 2 ⊢ (FermatNo‘(0 + 1)) = (∏𝑛 ∈ (0...0)(FermatNo‘𝑛) + 2)
43		fmtnorec2lem 48445	. 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 12716	1 ⊢ (𝑁 ∈ ℕ0 → (FermatNo‘(𝑁 + 1)) = (∏𝑛 ∈ (0...𝑁)(FermatNo‘𝑛) + 2))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  {csn 4584  ‘cfv 6533  (class class class)co 7413  ℂcc 11122  0cc0 11124  1c1 11125   + caddc 11127  2c2 12319  3c3 12320  5c5 12322  ℕ₀cn0 12528  ℤcz 12615  ...cfz 13561  ∏cprod 15992  FermatNocfmtno 48430

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736  ax-inf2 9620  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201  ax-pre-sup 11202

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 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-er 8696  df-en 8953  df-dom 8954  df-sdom 8955  df-fin 8956  df-sup 9412  df-oi 9482  df-card 9944  df-pnf 11269  df-mnf 11270  df-xr 11271  df-ltxr 11272  df-le 11273  df-sub 11467  df-neg 11468  df-div 11896  df-nn 12258  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-n0 12529  df-z 12616  df-uz 12888  df-rp 13043  df-fz 13562  df-fzo 13710  df-seq 14066  df-exp 14126  df-hash 14395  df-cj 15186  df-re 15187  df-im 15188  df-sqrt 15322  df-abs 15323  df-clim 15575  df-prod 15993  df-fmtno 48431

This theorem is used by:  fmtnodvds  48447  fmtnorec3  48451