Theorem fmtnorec2 46211

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 7432	. . 3 ⊢ (𝑥 = 0 → (FermatNo‘(𝑥 + 1)) = (FermatNo‘(0 + 1)))
2		oveq2 7417	. . . . 5 ⊢ (𝑥 = 0 → (0...𝑥) = (0...0))
3	2	prodeq1d 15865	. . . 4 ⊢ (𝑥 = 0 → ∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) = ∏𝑛 ∈ (0...0)(FermatNo‘𝑛))
4	3	oveq1d 7424	. . 3 ⊢ (𝑥 = 0 → (∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) + 2) = (∏𝑛 ∈ (0...0)(FermatNo‘𝑛) + 2))
5	1, 4	eqeq12d 2749	. 2 ⊢ (𝑥 = 0 → ((FermatNo‘(𝑥 + 1)) = (∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) + 2) ↔ (FermatNo‘(0 + 1)) = (∏𝑛 ∈ (0...0)(FermatNo‘𝑛) + 2)))
6		fvoveq1 7432	. . 3 ⊢ (𝑥 = 𝑦 → (FermatNo‘(𝑥 + 1)) = (FermatNo‘(𝑦 + 1)))
7		oveq2 7417	. . . . 5 ⊢ (𝑥 = 𝑦 → (0...𝑥) = (0...𝑦))
8	7	prodeq1d 15865	. . . 4 ⊢ (𝑥 = 𝑦 → ∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) = ∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛))
9	8	oveq1d 7424	. . 3 ⊢ (𝑥 = 𝑦 → (∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) + 2) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2))
10	6, 9	eqeq12d 2749	. 2 ⊢ (𝑥 = 𝑦 → ((FermatNo‘(𝑥 + 1)) = (∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) + 2) ↔ (FermatNo‘(𝑦 + 1)) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2)))
11		fvoveq1 7432	. . 3 ⊢ (𝑥 = (𝑦 + 1) → (FermatNo‘(𝑥 + 1)) = (FermatNo‘((𝑦 + 1) + 1)))
12		oveq2 7417	. . . . 5 ⊢ (𝑥 = (𝑦 + 1) → (0...𝑥) = (0...(𝑦 + 1)))
13	12	prodeq1d 15865	. . . 4 ⊢ (𝑥 = (𝑦 + 1) → ∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) = ∏𝑛 ∈ (0...(𝑦 + 1))(FermatNo‘𝑛))
14	13	oveq1d 7424	. . 3 ⊢ (𝑥 = (𝑦 + 1) → (∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) + 2) = (∏𝑛 ∈ (0...(𝑦 + 1))(FermatNo‘𝑛) + 2))
15	11, 14	eqeq12d 2749	. 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 7432	. . 3 ⊢ (𝑥 = 𝑁 → (FermatNo‘(𝑥 + 1)) = (FermatNo‘(𝑁 + 1)))
17		oveq2 7417	. . . 4 ⊢ (𝑥 = 𝑁 → (0...𝑥) = (0...𝑁))
18		prodeq1 15853	. . . . 5 ⊢ ((0...𝑥) = (0...𝑁) → ∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) = ∏𝑛 ∈ (0...𝑁)(FermatNo‘𝑛))
19	18	oveq1d 7424	. . . 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 2749	. 2 ⊢ (𝑥 = 𝑁 → ((FermatNo‘(𝑥 + 1)) = (∏𝑛 ∈ (0...𝑥)(FermatNo‘𝑛) + 2) ↔ (FermatNo‘(𝑁 + 1)) = (∏𝑛 ∈ (0...𝑁)(FermatNo‘𝑛) + 2)))
22		fmtno0 46208	. . . . 5 ⊢ (FermatNo‘0) = 3
23	22	oveq1i 7419	. . . 4 ⊢ ((FermatNo‘0) + 2) = (3 + 2)
24		3p2e5 12363	. . . 4 ⊢ (3 + 2) = 5
25	23, 24	eqtri 2761	. . 3 ⊢ ((FermatNo‘0) + 2) = 5
26		fz0sn 13601	. . . . . 6 ⊢ (0...0) = {0}
27	26	prodeq1i 15862	. . . . 5 ⊢ ∏𝑛 ∈ (0...0)(FermatNo‘𝑛) = ∏𝑛 ∈ {0} (FermatNo‘𝑛)
28		0z 12569	. . . . . 6 ⊢ 0 ∈ ℤ
29		0nn0 12487	. . . . . . 7 ⊢ 0 ∈ ℕ0
30		fmtnonn 46199	. . . . . . . 8 ⊢ (0 ∈ ℕ0 → (FermatNo‘0) ∈ ℕ)
31	30	nncnd 12228	. . . . . . 7 ⊢ (0 ∈ ℕ0 → (FermatNo‘0) ∈ ℂ)
32	29, 31	ax-mp 5	. . . . . 6 ⊢ (FermatNo‘0) ∈ ℂ
33		fveq2 6892	. . . . . . 7 ⊢ (𝑛 = 0 → (FermatNo‘𝑛) = (FermatNo‘0))
34	33	prodsn 15906	. . . . . 6 ⊢ ((0 ∈ ℤ ∧ (FermatNo‘0) ∈ ℂ) → ∏𝑛 ∈ {0} (FermatNo‘𝑛) = (FermatNo‘0))
35	28, 32, 34	mp2an 691	. . . . 5 ⊢ ∏𝑛 ∈ {0} (FermatNo‘𝑛) = (FermatNo‘0)
36	27, 35	eqtri 2761	. . . 4 ⊢ ∏𝑛 ∈ (0...0)(FermatNo‘𝑛) = (FermatNo‘0)
37	36	oveq1i 7419	. . 3 ⊢ (∏𝑛 ∈ (0...0)(FermatNo‘𝑛) + 2) = ((FermatNo‘0) + 2)
38		0p1e1 12334	. . . . 5 ⊢ (0 + 1) = 1
39	38	fveq2i 6895	. . . 4 ⊢ (FermatNo‘(0 + 1)) = (FermatNo‘1)
40		fmtno1 46209	. . . 4 ⊢ (FermatNo‘1) = 5
41	39, 40	eqtri 2761	. . 3 ⊢ (FermatNo‘(0 + 1)) = 5
42	25, 37, 41	3eqtr4ri 2772	. 2 ⊢ (FermatNo‘(0 + 1)) = (∏𝑛 ∈ (0...0)(FermatNo‘𝑛) + 2)
43		fmtnorec2lem 46210	. 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 12657	1 ⊢ (𝑁 ∈ ℕ0 → (FermatNo‘(𝑁 + 1)) = (∏𝑛 ∈ (0...𝑁)(FermatNo‘𝑛) + 2))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2107  {csn 4629  ‘cfv 6544  (class class class)co 7409  ℂcc 11108  0cc0 11110  1c1 11111   + caddc 11113  2c2 12267  3c3 12268  5c5 12270  ℕ₀cn0 12472  ℤcz 12558  ...cfz 13484  ∏cprod 15849  FermatNocfmtno 46195

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-inf2 9636  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187  ax-pre-sup 11188

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-sup 9437  df-oi 9505  df-card 9934  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-div 11872  df-nn 12213  df-2 12275  df-3 12276  df-4 12277  df-5 12278  df-n0 12473  df-z 12559  df-uz 12823  df-rp 12975  df-fz 13485  df-fzo 13628  df-seq 13967  df-exp 14028  df-hash 14291  df-cj 15046  df-re 15047  df-im 15048  df-sqrt 15182  df-abs 15183  df-clim 15432  df-prod 15850  df-fmtno 46196

This theorem is referenced by:  fmtnodvds  46212  fmtnorec3  46216