fmtnorec2lem - Mathbox for Alexander van der Vekens

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Theorem fmtnorec2lem 44059

Description: Lemma for fmtnorec2 44060 (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 11925	. . . . . 6 ⊢ (𝑦 ∈ ℕ₀ → (𝑦 + 1) ∈ ℕ₀)
2		peano2nn0 11925	. . . . . 6 ⊢ ((𝑦 + 1) ∈ ℕ₀ → ((𝑦 + 1) + 1) ∈ ℕ₀)
3		fmtno 44046	. . . . . 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 11703	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ₀ → 2 ∈ ℂ)
6	5, 1	expp1d 13507	. . . . . . . 8 ⊢ (𝑦 ∈ ℕ₀ → (2↑((𝑦 + 1) + 1)) = ((2↑(𝑦 + 1)) · 2))
7	6	oveq2d 7151	. . . . . . 7 ⊢ (𝑦 ∈ ℕ₀ → (2↑(2↑((𝑦 + 1) + 1))) = (2↑((2↑(𝑦 + 1)) · 2)))
8		2nn0 11902	. . . . . . . . . . . 12 ⊢ 2 ∈ ℕ₀
9	8	a1i 11	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ₀ → 2 ∈ ℕ₀)
10	9, 1	nn0expcld 13603	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ₀ → (2↑(𝑦 + 1)) ∈ ℕ₀)
11	9, 10	nn0expcld 13603	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ₀ → (2↑(2↑(𝑦 + 1))) ∈ ℕ₀)
12	11	nn0cnd 11945	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ₀ → (2↑(2↑(𝑦 + 1))) ∈ ℂ)
13	12	sqvald 13503	. . . . . . . 8 ⊢ (𝑦 ∈ ℕ₀ → ((2↑(2↑(𝑦 + 1)))↑2) = ((2↑(2↑(𝑦 + 1))) · (2↑(2↑(𝑦 + 1)))))
14	5, 9, 10	expmuld 13509	. . . . . . . 8 ⊢ (𝑦 ∈ ℕ₀ → (2↑((2↑(𝑦 + 1)) · 2)) = ((2↑(2↑(𝑦 + 1)))↑2))
15		fmtnom1nn 44049	. . . . . . . . . 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 7153	. . . . . . . 8 ⊢ (𝑦 ∈ ℕ₀ → (((FermatNo‘(𝑦 + 1)) − 1) · ((FermatNo‘(𝑦 + 1)) − 1)) = ((2↑(2↑(𝑦 + 1))) · (2↑(2↑(𝑦 + 1)))))
18	13, 14, 17	3eqtr4d 2843	. . . . . . 7 ⊢ (𝑦 ∈ ℕ₀ → (2↑((2↑(𝑦 + 1)) · 2)) = (((FermatNo‘(𝑦 + 1)) − 1) · ((FermatNo‘(𝑦 + 1)) − 1)))
19	7, 18	eqtrd 2833	. . . . . 6 ⊢ (𝑦 ∈ ℕ₀ → (2↑(2↑((𝑦 + 1) + 1))) = (((FermatNo‘(𝑦 + 1)) − 1) · ((FermatNo‘(𝑦 + 1)) − 1)))
20	19	oveq1d 7150	. . . . 5 ⊢ (𝑦 ∈ ℕ₀ → ((2↑(2↑((𝑦 + 1) + 1))) + 1) = ((((FermatNo‘(𝑦 + 1)) − 1) · ((FermatNo‘(𝑦 + 1)) − 1)) + 1))
21	4, 20	eqtrd 2833	. . . 4 ⊢ (𝑦 ∈ ℕ₀ → (FermatNo‘((𝑦 + 1) + 1)) = ((((FermatNo‘(𝑦 + 1)) − 1) · ((FermatNo‘(𝑦 + 1)) − 1)) + 1))
22	21	adantr 484	. . 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 7142	. . . . . 6 ⊢ ((FermatNo‘(𝑦 + 1)) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2) → ((FermatNo‘(𝑦 + 1)) − 1) = ((∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2) − 1))
24	23	oveq1d 7150	. . . . 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 7150	. . . 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 485	. . 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 13336	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ₀ → (0...𝑦) ∈ Fin)
28		elfznn0 12995	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (0...𝑦) → 𝑛 ∈ ℕ₀)
29		fmtnonn 44048	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ₀ → (FermatNo‘𝑛) ∈ ℕ)
30	29	nncnd 11641	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ₀ → (FermatNo‘𝑛) ∈ ℂ)
31	28, 30	syl 17	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (0...𝑦) → (FermatNo‘𝑛) ∈ ℂ)
32	31	adantl 485	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ₀ ∧ 𝑛 ∈ (0...𝑦)) → (FermatNo‘𝑛) ∈ ℂ)
33	27, 32	fprodcl 15298	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ₀ → ∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) ∈ ℂ)
34		1cnd 10625	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ₀ → 1 ∈ ℂ)
35	33, 5, 34	addsubassd 11006	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ₀ → ((∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2) − 1) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + (2 − 1)))
36		2m1e1 11751	. . . . . . . . . 10 ⊢ (2 − 1) = 1
37	36	oveq2i 7146	. . . . . . . . 9 ⊢ (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + (2 − 1)) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 1)
38	35, 37	eqtrdi 2849	. . . . . . . 8 ⊢ (𝑦 ∈ ℕ₀ → ((∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2) − 1) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 1))
39	38	oveq1d 7150	. . . . . . 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 44048	. . . . . . . . . . 11 ⊢ ((𝑦 + 1) ∈ ℕ₀ → (FermatNo‘(𝑦 + 1)) ∈ ℕ)
41	1, 40	syl 17	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ₀ → (FermatNo‘(𝑦 + 1)) ∈ ℕ)
42	41	nncnd 11641	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ₀ → (FermatNo‘(𝑦 + 1)) ∈ ℂ)
43	42, 34	subcld 10986	. . . . . . . 8 ⊢ (𝑦 ∈ ℕ₀ → ((FermatNo‘(𝑦 + 1)) − 1) ∈ ℂ)
44	33, 42	muls1d 11089	. . . . . . . . 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 10648	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ₀ → (1 · ((FermatNo‘(𝑦 + 1)) − 1)) = ((FermatNo‘(𝑦 + 1)) − 1))
46	44, 45	oveq12d 7153	. . . . . . . 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 10657	. . . . . . 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 2833	. . . . . 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 484	. . . . 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 7150	. . . 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		eqcom 2805	. . . . . . 7 ⊢ ((FermatNo‘(𝑦 + 1)) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2) ↔ (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2) = (FermatNo‘(𝑦 + 1)))
52	42, 5, 33	subadd2d 11005	. . . . . . 7 ⊢ (𝑦 ∈ ℕ₀ → (((FermatNo‘(𝑦 + 1)) − 2) = ∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) ↔ (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2) = (FermatNo‘(𝑦 + 1))))
53	51, 52	bitr4id 293	. . . . . 6 ⊢ (𝑦 ∈ ℕ₀ → ((FermatNo‘(𝑦 + 1)) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2) ↔ ((FermatNo‘(𝑦 + 1)) − 2) = ∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛)))
54		oveq2 7143	. . . . . . . . . . 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 7150	. . . . . . . . . 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 7150	. . . . . . . . 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 2806	. . . . . . . 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 10650	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ₀ → (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) · (FermatNo‘(𝑦 + 1))) ∈ ℂ)
59	42, 5	subcld 10986	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ₀ → ((FermatNo‘(𝑦 + 1)) − 2) ∈ ℂ)
60	58, 59	subcld 10986	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ₀ → ((∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) · (FermatNo‘(𝑦 + 1))) − ((FermatNo‘(𝑦 + 1)) − 2)) ∈ ℂ)
61	60, 43, 34	addassd 10652	. . . . . . . . 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 12271	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℕ₀ ↔ 𝑦 ∈ (ℤ_≥‘0))
63	62	biimpi 219	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℕ₀ → 𝑦 ∈ (ℤ_≥‘0))
64		elfznn0 12995	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ (0...(𝑦 + 1)) → 𝑛 ∈ ℕ₀)
65	64, 30	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ (0...(𝑦 + 1)) → (FermatNo‘𝑛) ∈ ℂ)
66	65	adantl 485	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℕ₀ ∧ 𝑛 ∈ (0...(𝑦 + 1))) → (FermatNo‘𝑛) ∈ ℂ)
67		fveq2 6645	. . . . . . . . . . . . . 14 ⊢ (𝑛 = (𝑦 + 1) → (FermatNo‘𝑛) = (FermatNo‘(𝑦 + 1)))
68	63, 66, 67	fprodp1 15315	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℕ₀ → ∏𝑛 ∈ (0...(𝑦 + 1))(FermatNo‘𝑛) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) · (FermatNo‘(𝑦 + 1))))
69	68	eqcomd 2804	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℕ₀ → (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) · (FermatNo‘(𝑦 + 1))) = ∏𝑛 ∈ (0...(𝑦 + 1))(FermatNo‘𝑛))
70	69	oveq1d 7150	. . . . . . . . . . 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 11054	. . . . . . . . . . . 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 7153	. . . . . . . . . 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 13336	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℕ₀ → (0...(𝑦 + 1)) ∈ Fin)
75	74, 66	fprodcl 15298	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ₀ → ∏𝑛 ∈ (0...(𝑦 + 1))(FermatNo‘𝑛) ∈ ℂ)
76	75, 59, 42	subadd23d 11008	. . . . . . . . . 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 2833	. . . . . . . . 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 10991	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ₀ → ((FermatNo‘(𝑦 + 1)) − ((FermatNo‘(𝑦 + 1)) − 2)) = 2)
79	78	oveq2d 7151	. . . . . . . . 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 2837	. . . . . . . 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 2855	. . . . . . 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 416	. . . . . 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 243	. . . . 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 410	. . . 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 2833	. . 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 2837	. 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 416	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 399 = wceq 1538 ∈ wcel 2111 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 0cc0 10526 1c1 10527 + caddc 10529 · cmul 10531 − cmin 10859 ℕcn 11625 2c2 11680 ℕ₀cn0 11885 ℤ_≥cuz 12231 ...cfz 12885 ↑cexp 13425 ∏cprod 15251 FermatNocfmtno 44044

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-fz 12886 df-fzo 13029 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-prod 15252 df-fmtno 44045

This theorem is referenced by: fmtnorec2 44060

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