fmtnorec2lem - Mathbox for Alexander van der Vekens

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Theorem fmtnorec2lem 42485

Description: Lemma for fmtnorec2 42486 (induction step). (Contributed by AV, 29-Jul-2021.)

**Assertion**
Ref	Expression
fmtnorec2lem	⊢ (𝑦 ∈ ℕ₀ → ((FermatNo‘(𝑦 + 1)) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2) → (FermatNo‘((𝑦 + 1) + 1)) = (∏𝑛 ∈ (0...(𝑦 + 1))(FermatNo‘𝑛) + 2)))

Distinct variable group: 𝑦,𝑛

**Proof of Theorem fmtnorec2lem**
Step	Hyp	Ref	Expression
1		peano2nn0 11688	. . . . . 6 ⊢ (𝑦 ∈ ℕ₀ → (𝑦 + 1) ∈ ℕ₀)
2		peano2nn0 11688	. . . . . 6 ⊢ ((𝑦 + 1) ∈ ℕ₀ → ((𝑦 + 1) + 1) ∈ ℕ₀)
3		fmtno 42472	. . . . . 6 ⊢ (((𝑦 + 1) + 1) ∈ ℕ₀ → (FermatNo‘((𝑦 + 1) + 1)) = ((2↑(2↑((𝑦 + 1) + 1))) + 1))
4	1, 2, 3	3syl 18	. . . . 5 ⊢ (𝑦 ∈ ℕ₀ → (FermatNo‘((𝑦 + 1) + 1)) = ((2↑(2↑((𝑦 + 1) + 1))) + 1))
5		2cnd 11457	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ₀ → 2 ∈ ℂ)
6	5, 1	expp1d 13332	. . . . . . . 8 ⊢ (𝑦 ∈ ℕ₀ → (2↑((𝑦 + 1) + 1)) = ((2↑(𝑦 + 1)) · 2))
7	6	oveq2d 6940	. . . . . . 7 ⊢ (𝑦 ∈ ℕ₀ → (2↑(2↑((𝑦 + 1) + 1))) = (2↑((2↑(𝑦 + 1)) · 2)))
8		2nn0 11665	. . . . . . . . . . . 12 ⊢ 2 ∈ ℕ₀
9	8	a1i 11	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ₀ → 2 ∈ ℕ₀)
10	9, 1	nn0expcld 13356	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ₀ → (2↑(𝑦 + 1)) ∈ ℕ₀)
11	9, 10	nn0expcld 13356	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ₀ → (2↑(2↑(𝑦 + 1))) ∈ ℕ₀)
12	11	nn0cnd 11708	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ₀ → (2↑(2↑(𝑦 + 1))) ∈ ℂ)
13	12	sqvald 13328	. . . . . . . 8 ⊢ (𝑦 ∈ ℕ₀ → ((2↑(2↑(𝑦 + 1)))↑2) = ((2↑(2↑(𝑦 + 1))) · (2↑(2↑(𝑦 + 1)))))
14	5, 9, 10	expmuld 13334	. . . . . . . 8 ⊢ (𝑦 ∈ ℕ₀ → (2↑((2↑(𝑦 + 1)) · 2)) = ((2↑(2↑(𝑦 + 1)))↑2))
15		fmtnom1nn 42475	. . . . . . . . . 10 ⊢ ((𝑦 + 1) ∈ ℕ₀ → ((FermatNo‘(𝑦 + 1)) − 1) = (2↑(2↑(𝑦 + 1))))
16	1, 15	syl 17	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ₀ → ((FermatNo‘(𝑦 + 1)) − 1) = (2↑(2↑(𝑦 + 1))))
17	16, 16	oveq12d 6942	. . . . . . . 8 ⊢ (𝑦 ∈ ℕ₀ → (((FermatNo‘(𝑦 + 1)) − 1) · ((FermatNo‘(𝑦 + 1)) − 1)) = ((2↑(2↑(𝑦 + 1))) · (2↑(2↑(𝑦 + 1)))))
18	13, 14, 17	3eqtr4d 2824	. . . . . . 7 ⊢ (𝑦 ∈ ℕ₀ → (2↑((2↑(𝑦 + 1)) · 2)) = (((FermatNo‘(𝑦 + 1)) − 1) · ((FermatNo‘(𝑦 + 1)) − 1)))
19	7, 18	eqtrd 2814	. . . . . 6 ⊢ (𝑦 ∈ ℕ₀ → (2↑(2↑((𝑦 + 1) + 1))) = (((FermatNo‘(𝑦 + 1)) − 1) · ((FermatNo‘(𝑦 + 1)) − 1)))
20	19	oveq1d 6939	. . . . 5 ⊢ (𝑦 ∈ ℕ₀ → ((2↑(2↑((𝑦 + 1) + 1))) + 1) = ((((FermatNo‘(𝑦 + 1)) − 1) · ((FermatNo‘(𝑦 + 1)) − 1)) + 1))
21	4, 20	eqtrd 2814	. . . 4 ⊢ (𝑦 ∈ ℕ₀ → (FermatNo‘((𝑦 + 1) + 1)) = ((((FermatNo‘(𝑦 + 1)) − 1) · ((FermatNo‘(𝑦 + 1)) − 1)) + 1))
22	21	adantr 474	. . 3 ⊢ ((𝑦 ∈ ℕ₀ ∧ (FermatNo‘(𝑦 + 1)) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2)) → (FermatNo‘((𝑦 + 1) + 1)) = ((((FermatNo‘(𝑦 + 1)) − 1) · ((FermatNo‘(𝑦 + 1)) − 1)) + 1))
23		oveq1 6931	. . . . . 6 ⊢ ((FermatNo‘(𝑦 + 1)) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2) → ((FermatNo‘(𝑦 + 1)) − 1) = ((∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2) − 1))
24	23	oveq1d 6939	. . . . 5 ⊢ ((FermatNo‘(𝑦 + 1)) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2) → (((FermatNo‘(𝑦 + 1)) − 1) · ((FermatNo‘(𝑦 + 1)) − 1)) = (((∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2) − 1) · ((FermatNo‘(𝑦 + 1)) − 1)))
25	24	oveq1d 6939	. . . 4 ⊢ ((FermatNo‘(𝑦 + 1)) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2) → ((((FermatNo‘(𝑦 + 1)) − 1) · ((FermatNo‘(𝑦 + 1)) − 1)) + 1) = ((((∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2) − 1) · ((FermatNo‘(𝑦 + 1)) − 1)) + 1))
26	25	adantl 475	. . 3 ⊢ ((𝑦 ∈ ℕ₀ ∧ (FermatNo‘(𝑦 + 1)) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2)) → ((((FermatNo‘(𝑦 + 1)) − 1) · ((FermatNo‘(𝑦 + 1)) − 1)) + 1) = ((((∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2) − 1) · ((FermatNo‘(𝑦 + 1)) − 1)) + 1))
27		fzfid 13095	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ₀ → (0...𝑦) ∈ Fin)
28		elfznn0 12755	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (0...𝑦) → 𝑛 ∈ ℕ₀)
29		fmtnonn 42474	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ₀ → (FermatNo‘𝑛) ∈ ℕ)
30	29	nncnd 11396	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ₀ → (FermatNo‘𝑛) ∈ ℂ)
31	28, 30	syl 17	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (0...𝑦) → (FermatNo‘𝑛) ∈ ℂ)
32	31	adantl 475	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ₀ ∧ 𝑛 ∈ (0...𝑦)) → (FermatNo‘𝑛) ∈ ℂ)
33	27, 32	fprodcl 15089	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ₀ → ∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) ∈ ℂ)
34		1cnd 10373	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ₀ → 1 ∈ ℂ)
35	33, 5, 34	addsubassd 10756	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ₀ → ((∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2) − 1) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + (2 − 1)))
36		2m1e1 11512	. . . . . . . . . 10 ⊢ (2 − 1) = 1
37	36	oveq2i 6935	. . . . . . . . 9 ⊢ (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + (2 − 1)) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 1)
38	35, 37	syl6eq 2830	. . . . . . . 8 ⊢ (𝑦 ∈ ℕ₀ → ((∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2) − 1) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 1))
39	38	oveq1d 6939	. . . . . . 7 ⊢ (𝑦 ∈ ℕ₀ → (((∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2) − 1) · ((FermatNo‘(𝑦 + 1)) − 1)) = ((∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 1) · ((FermatNo‘(𝑦 + 1)) − 1)))
40		fmtnonn 42474	. . . . . . . . . . 11 ⊢ ((𝑦 + 1) ∈ ℕ₀ → (FermatNo‘(𝑦 + 1)) ∈ ℕ)
41	1, 40	syl 17	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ₀ → (FermatNo‘(𝑦 + 1)) ∈ ℕ)
42	41	nncnd 11396	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ₀ → (FermatNo‘(𝑦 + 1)) ∈ ℂ)
43	42, 34	subcld 10736	. . . . . . . 8 ⊢ (𝑦 ∈ ℕ₀ → ((FermatNo‘(𝑦 + 1)) − 1) ∈ ℂ)
44	33, 42	muls1d 10837	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ₀ → (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) · ((FermatNo‘(𝑦 + 1)) − 1)) = ((∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) · (FermatNo‘(𝑦 + 1))) − ∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛)))
45	43	mulid2d 10397	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ₀ → (1 · ((FermatNo‘(𝑦 + 1)) − 1)) = ((FermatNo‘(𝑦 + 1)) − 1))
46	44, 45	oveq12d 6942	. . . . . . . 8 ⊢ (𝑦 ∈ ℕ₀ → ((∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) · ((FermatNo‘(𝑦 + 1)) − 1)) + (1 · ((FermatNo‘(𝑦 + 1)) − 1))) = (((∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) · (FermatNo‘(𝑦 + 1))) − ∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛)) + ((FermatNo‘(𝑦 + 1)) − 1)))
47	33, 43, 34, 46	joinlmuladdmuld 10406	. . . . . . 7 ⊢ (𝑦 ∈ ℕ₀ → ((∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 1) · ((FermatNo‘(𝑦 + 1)) − 1)) = (((∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) · (FermatNo‘(𝑦 + 1))) − ∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛)) + ((FermatNo‘(𝑦 + 1)) − 1)))
48	39, 47	eqtrd 2814	. . . . . 6 ⊢ (𝑦 ∈ ℕ₀ → (((∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2) − 1) · ((FermatNo‘(𝑦 + 1)) − 1)) = (((∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) · (FermatNo‘(𝑦 + 1))) − ∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛)) + ((FermatNo‘(𝑦 + 1)) − 1)))
49	48	adantr 474	. . . . 5 ⊢ ((𝑦 ∈ ℕ₀ ∧ (FermatNo‘(𝑦 + 1)) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2)) → (((∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2) − 1) · ((FermatNo‘(𝑦 + 1)) − 1)) = (((∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) · (FermatNo‘(𝑦 + 1))) − ∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛)) + ((FermatNo‘(𝑦 + 1)) − 1)))
50	49	oveq1d 6939	. . . 4 ⊢ ((𝑦 ∈ ℕ₀ ∧ (FermatNo‘(𝑦 + 1)) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2)) → ((((∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2) − 1) · ((FermatNo‘(𝑦 + 1)) − 1)) + 1) = ((((∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) · (FermatNo‘(𝑦 + 1))) − ∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛)) + ((FermatNo‘(𝑦 + 1)) − 1)) + 1))
51	42, 5, 33	subadd2d 10755	. . . . . . 7 ⊢ (𝑦 ∈ ℕ₀ → (((FermatNo‘(𝑦 + 1)) − 2) = ∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) ↔ (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2) = (FermatNo‘(𝑦 + 1))))
52		eqcom 2785	. . . . . . 7 ⊢ ((FermatNo‘(𝑦 + 1)) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2) ↔ (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2) = (FermatNo‘(𝑦 + 1)))
53	51, 52	syl6rbbr 282	. . . . . 6 ⊢ (𝑦 ∈ ℕ₀ → ((FermatNo‘(𝑦 + 1)) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2) ↔ ((FermatNo‘(𝑦 + 1)) − 2) = ∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛)))
54		oveq2 6932	. . . . . . . . . . 11 ⊢ (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) = ((FermatNo‘(𝑦 + 1)) − 2) → ((∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) · (FermatNo‘(𝑦 + 1))) − ∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛)) = ((∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) · (FermatNo‘(𝑦 + 1))) − ((FermatNo‘(𝑦 + 1)) − 2)))
55	54	oveq1d 6939	. . . . . . . . . 10 ⊢ (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) = ((FermatNo‘(𝑦 + 1)) − 2) → (((∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) · (FermatNo‘(𝑦 + 1))) − ∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛)) + ((FermatNo‘(𝑦 + 1)) − 1)) = (((∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) · (FermatNo‘(𝑦 + 1))) − ((FermatNo‘(𝑦 + 1)) − 2)) + ((FermatNo‘(𝑦 + 1)) − 1)))
56	55	oveq1d 6939	. . . . . . . . 9 ⊢ (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) = ((FermatNo‘(𝑦 + 1)) − 2) → ((((∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) · (FermatNo‘(𝑦 + 1))) − ∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛)) + ((FermatNo‘(𝑦 + 1)) − 1)) + 1) = ((((∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) · (FermatNo‘(𝑦 + 1))) − ((FermatNo‘(𝑦 + 1)) − 2)) + ((FermatNo‘(𝑦 + 1)) − 1)) + 1))
57	56	eqcoms 2786	. . . . . . . 8 ⊢ (((FermatNo‘(𝑦 + 1)) − 2) = ∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) → ((((∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) · (FermatNo‘(𝑦 + 1))) − ∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛)) + ((FermatNo‘(𝑦 + 1)) − 1)) + 1) = ((((∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) · (FermatNo‘(𝑦 + 1))) − ((FermatNo‘(𝑦 + 1)) − 2)) + ((FermatNo‘(𝑦 + 1)) − 1)) + 1))
58	33, 42	mulcld 10399	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ₀ → (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) · (FermatNo‘(𝑦 + 1))) ∈ ℂ)
59	42, 5	subcld 10736	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ₀ → ((FermatNo‘(𝑦 + 1)) − 2) ∈ ℂ)
60	58, 59	subcld 10736	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ₀ → ((∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) · (FermatNo‘(𝑦 + 1))) − ((FermatNo‘(𝑦 + 1)) − 2)) ∈ ℂ)
61	60, 43, 34	addassd 10401	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ₀ → ((((∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) · (FermatNo‘(𝑦 + 1))) − ((FermatNo‘(𝑦 + 1)) − 2)) + ((FermatNo‘(𝑦 + 1)) − 1)) + 1) = (((∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) · (FermatNo‘(𝑦 + 1))) − ((FermatNo‘(𝑦 + 1)) − 2)) + (((FermatNo‘(𝑦 + 1)) − 1) + 1)))
62		elnn0uz 12035	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℕ₀ ↔ 𝑦 ∈ (ℤ_≥‘0))
63	62	biimpi 208	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℕ₀ → 𝑦 ∈ (ℤ_≥‘0))
64		elfznn0 12755	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ (0...(𝑦 + 1)) → 𝑛 ∈ ℕ₀)
65	64, 30	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ (0...(𝑦 + 1)) → (FermatNo‘𝑛) ∈ ℂ)
66	65	adantl 475	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℕ₀ ∧ 𝑛 ∈ (0...(𝑦 + 1))) → (FermatNo‘𝑛) ∈ ℂ)
67		fveq2 6448	. . . . . . . . . . . . . 14 ⊢ (𝑛 = (𝑦 + 1) → (FermatNo‘𝑛) = (FermatNo‘(𝑦 + 1)))
68	63, 66, 67	fprodp1 15106	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℕ₀ → ∏𝑛 ∈ (0...(𝑦 + 1))(FermatNo‘𝑛) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) · (FermatNo‘(𝑦 + 1))))
69	68	eqcomd 2784	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℕ₀ → (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) · (FermatNo‘(𝑦 + 1))) = ∏𝑛 ∈ (0...(𝑦 + 1))(FermatNo‘𝑛))
70	69	oveq1d 6939	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ₀ → ((∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) · (FermatNo‘(𝑦 + 1))) − ((FermatNo‘(𝑦 + 1)) − 2)) = (∏𝑛 ∈ (0...(𝑦 + 1))(FermatNo‘𝑛) − ((FermatNo‘(𝑦 + 1)) − 2)))
71		npcan1 10802	. . . . . . . . . . . 12 ⊢ ((FermatNo‘(𝑦 + 1)) ∈ ℂ → (((FermatNo‘(𝑦 + 1)) − 1) + 1) = (FermatNo‘(𝑦 + 1)))
72	42, 71	syl 17	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ₀ → (((FermatNo‘(𝑦 + 1)) − 1) + 1) = (FermatNo‘(𝑦 + 1)))
73	70, 72	oveq12d 6942	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ₀ → (((∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) · (FermatNo‘(𝑦 + 1))) − ((FermatNo‘(𝑦 + 1)) − 2)) + (((FermatNo‘(𝑦 + 1)) − 1) + 1)) = ((∏𝑛 ∈ (0...(𝑦 + 1))(FermatNo‘𝑛) − ((FermatNo‘(𝑦 + 1)) − 2)) + (FermatNo‘(𝑦 + 1))))
74		fzfid 13095	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℕ₀ → (0...(𝑦 + 1)) ∈ Fin)
75	74, 66	fprodcl 15089	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ₀ → ∏𝑛 ∈ (0...(𝑦 + 1))(FermatNo‘𝑛) ∈ ℂ)
76	75, 59, 42	subadd23d 10758	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ₀ → ((∏𝑛 ∈ (0...(𝑦 + 1))(FermatNo‘𝑛) − ((FermatNo‘(𝑦 + 1)) − 2)) + (FermatNo‘(𝑦 + 1))) = (∏𝑛 ∈ (0...(𝑦 + 1))(FermatNo‘𝑛) + ((FermatNo‘(𝑦 + 1)) − ((FermatNo‘(𝑦 + 1)) − 2))))
77	73, 76	eqtrd 2814	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ₀ → (((∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) · (FermatNo‘(𝑦 + 1))) − ((FermatNo‘(𝑦 + 1)) − 2)) + (((FermatNo‘(𝑦 + 1)) − 1) + 1)) = (∏𝑛 ∈ (0...(𝑦 + 1))(FermatNo‘𝑛) + ((FermatNo‘(𝑦 + 1)) − ((FermatNo‘(𝑦 + 1)) − 2))))
78	42, 5	nncand 10741	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ₀ → ((FermatNo‘(𝑦 + 1)) − ((FermatNo‘(𝑦 + 1)) − 2)) = 2)
79	78	oveq2d 6940	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ₀ → (∏𝑛 ∈ (0...(𝑦 + 1))(FermatNo‘𝑛) + ((FermatNo‘(𝑦 + 1)) − ((FermatNo‘(𝑦 + 1)) − 2))) = (∏𝑛 ∈ (0...(𝑦 + 1))(FermatNo‘𝑛) + 2))
80	61, 77, 79	3eqtrd 2818	. . . . . . . 8 ⊢ (𝑦 ∈ ℕ₀ → ((((∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) · (FermatNo‘(𝑦 + 1))) − ((FermatNo‘(𝑦 + 1)) − 2)) + ((FermatNo‘(𝑦 + 1)) − 1)) + 1) = (∏𝑛 ∈ (0...(𝑦 + 1))(FermatNo‘𝑛) + 2))
81	57, 80	sylan9eqr 2836	. . . . . . 7 ⊢ ((𝑦 ∈ ℕ₀ ∧ ((FermatNo‘(𝑦 + 1)) − 2) = ∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛)) → ((((∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) · (FermatNo‘(𝑦 + 1))) − ∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛)) + ((FermatNo‘(𝑦 + 1)) − 1)) + 1) = (∏𝑛 ∈ (0...(𝑦 + 1))(FermatNo‘𝑛) + 2))
82	81	ex 403	. . . . . 6 ⊢ (𝑦 ∈ ℕ₀ → (((FermatNo‘(𝑦 + 1)) − 2) = ∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) → ((((∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) · (FermatNo‘(𝑦 + 1))) − ∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛)) + ((FermatNo‘(𝑦 + 1)) − 1)) + 1) = (∏𝑛 ∈ (0...(𝑦 + 1))(FermatNo‘𝑛) + 2)))
83	53, 82	sylbid 232	. . . . 5 ⊢ (𝑦 ∈ ℕ₀ → ((FermatNo‘(𝑦 + 1)) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2) → ((((∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) · (FermatNo‘(𝑦 + 1))) − ∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛)) + ((FermatNo‘(𝑦 + 1)) − 1)) + 1) = (∏𝑛 ∈ (0...(𝑦 + 1))(FermatNo‘𝑛) + 2)))
84	83	imp 397	. . . 4 ⊢ ((𝑦 ∈ ℕ₀ ∧ (FermatNo‘(𝑦 + 1)) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2)) → ((((∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) · (FermatNo‘(𝑦 + 1))) − ∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛)) + ((FermatNo‘(𝑦 + 1)) − 1)) + 1) = (∏𝑛 ∈ (0...(𝑦 + 1))(FermatNo‘𝑛) + 2))
85	50, 84	eqtrd 2814	. . 3 ⊢ ((𝑦 ∈ ℕ₀ ∧ (FermatNo‘(𝑦 + 1)) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2)) → ((((∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2) − 1) · ((FermatNo‘(𝑦 + 1)) − 1)) + 1) = (∏𝑛 ∈ (0...(𝑦 + 1))(FermatNo‘𝑛) + 2))
86	22, 26, 85	3eqtrd 2818	. 2 ⊢ ((𝑦 ∈ ℕ₀ ∧ (FermatNo‘(𝑦 + 1)) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2)) → (FermatNo‘((𝑦 + 1) + 1)) = (∏𝑛 ∈ (0...(𝑦 + 1))(FermatNo‘𝑛) + 2))
87	86	ex 403	1 ⊢ (𝑦 ∈ ℕ₀ → ((FermatNo‘(𝑦 + 1)) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2) → (FermatNo‘((𝑦 + 1) + 1)) = (∏𝑛 ∈ (0...(𝑦 + 1))(FermatNo‘𝑛) + 2)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ‘cfv 6137 (class class class)co 6924 ℂcc 10272 0cc0 10274 1c1 10275 + caddc 10277 · cmul 10279 − cmin 10608 ℕcn 11378 2c2 11434 ℕ₀cn0 11646 ℤ_≥cuz 11996 ...cfz 12647 ↑cexp 13182 ∏cprod 15042 FermatNocfmtno 42470

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-inf2 8837 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 ax-pre-sup 10352

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-se 5317 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-isom 6146 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-1st 7447 df-2nd 7448 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-oadd 7849 df-er 8028 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-sup 8638 df-oi 8706 df-card 9100 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-div 11035 df-nn 11379 df-2 11442 df-3 11443 df-n0 11647 df-z 11733 df-uz 11997 df-rp 12142 df-fz 12648 df-fzo 12789 df-seq 13124 df-exp 13183 df-hash 13440 df-cj 14250 df-re 14251 df-im 14252 df-sqrt 14386 df-abs 14387 df-clim 14631 df-prod 15043 df-fmtno 42471

This theorem is referenced by: fmtnorec2 42486

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