fmtnorec2lem - Mathbox for Alexander van der Vekens

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Theorem fmtnorec2lem 48596

Description: Lemma for fmtnorec2 48597 (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 12639	. . . . . 6 ⊢ (𝑦 ∈ ℕ₀ → (𝑦 + 1) ∈ ℕ₀)
2		peano2nn0 12639	. . . . . 6 ⊢ ((𝑦 + 1) ∈ ℕ₀ → ((𝑦 + 1) + 1) ∈ ℕ₀)
3		fmtno 48583	. . . . . 6 ⊢ (((𝑦 + 1) + 1) ∈ ℕ₀ → (FermatNo‘((𝑦 + 1) + 1)) = ((2↑(2↑((𝑦 + 1) + 1))) + 1))
4	1, 2, 3	3syl 19	. . . . 5 ⊢ (𝑦 ∈ ℕ₀ → (FermatNo‘((𝑦 + 1) + 1)) = ((2↑(2↑((𝑦 + 1) + 1))) + 1))
5		2cnd 12414	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ₀ → 2 ∈ ℂ)
6	5, 1	expp1d 14283	. . . . . . . 8 ⊢ (𝑦 ∈ ℕ₀ → (2↑((𝑦 + 1) + 1)) = ((2↑(𝑦 + 1)) · 2))
7	6	oveq2d 7434	. . . . . . 7 ⊢ (𝑦 ∈ ℕ₀ → (2↑(2↑((𝑦 + 1) + 1))) = (2↑((2↑(𝑦 + 1)) · 2)))
8		2nn0 12616	. . . . . . . . . . . 12 ⊢ 2 ∈ ℕ₀
9	8	a1i 11	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ₀ → 2 ∈ ℕ₀)
10	9, 1	nn0expcld 14383	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ₀ → (2↑(𝑦 + 1)) ∈ ℕ₀)
11	9, 10	nn0expcld 14383	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ₀ → (2↑(2↑(𝑦 + 1))) ∈ ℕ₀)
12	11	nn0cnd 12662	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ₀ → (2↑(2↑(𝑦 + 1))) ∈ ℂ)
13	12	sqvald 14279	. . . . . . . 8 ⊢ (𝑦 ∈ ℕ₀ → ((2↑(2↑(𝑦 + 1)))↑2) = ((2↑(2↑(𝑦 + 1))) · (2↑(2↑(𝑦 + 1)))))
14	5, 9, 10	expmuld 14285	. . . . . . . 8 ⊢ (𝑦 ∈ ℕ₀ → (2↑((2↑(𝑦 + 1)) · 2)) = ((2↑(2↑(𝑦 + 1)))↑2))
15		fmtnom1nn 48586	. . . . . . . . . 10 ⊢ ((𝑦 + 1) ∈ ℕ₀ → ((FermatNo‘(𝑦 + 1)) − 1) = (2↑(2↑(𝑦 + 1))))
16	1, 15	syl 18	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ₀ → ((FermatNo‘(𝑦 + 1)) − 1) = (2↑(2↑(𝑦 + 1))))
17	16, 16	oveq12d 7436	. . . . . . . 8 ⊢ (𝑦 ∈ ℕ₀ → (((FermatNo‘(𝑦 + 1)) − 1) · ((FermatNo‘(𝑦 + 1)) − 1)) = ((2↑(2↑(𝑦 + 1))) · (2↑(2↑(𝑦 + 1)))))
18	13, 14, 17	3eqtr4d 2806	. . . . . . 7 ⊢ (𝑦 ∈ ℕ₀ → (2↑((2↑(𝑦 + 1)) · 2)) = (((FermatNo‘(𝑦 + 1)) − 1) · ((FermatNo‘(𝑦 + 1)) − 1)))
19	7, 18	eqtrd 2796	. . . . . 6 ⊢ (𝑦 ∈ ℕ₀ → (2↑(2↑((𝑦 + 1) + 1))) = (((FermatNo‘(𝑦 + 1)) − 1) · ((FermatNo‘(𝑦 + 1)) − 1)))
20	19	oveq1d 7433	. . . . 5 ⊢ (𝑦 ∈ ℕ₀ → ((2↑(2↑((𝑦 + 1) + 1))) + 1) = ((((FermatNo‘(𝑦 + 1)) − 1) · ((FermatNo‘(𝑦 + 1)) − 1)) + 1))
21	4, 20	eqtrd 2796	. . . 4 ⊢ (𝑦 ∈ ℕ₀ → (FermatNo‘((𝑦 + 1) + 1)) = ((((FermatNo‘(𝑦 + 1)) − 1) · ((FermatNo‘(𝑦 + 1)) − 1)) + 1))
22	21	adantr 486	. . 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 7425	. . . . . 6 ⊢ ((FermatNo‘(𝑦 + 1)) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2) → ((FermatNo‘(𝑦 + 1)) − 1) = ((∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2) − 1))
24	23	oveq1d 7433	. . . . 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 7433	. . . 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 487	. . 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 14109	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ₀ → (0...𝑦) ∈ Fin)
28		elfznn0 13747	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (0...𝑦) → 𝑛 ∈ ℕ₀)
29		fmtnonn 48585	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ₀ → (FermatNo‘𝑛) ∈ ℕ)
30	29	nncnd 12344	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ₀ → (FermatNo‘𝑛) ∈ ℂ)
31	28, 30	syl 18	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (0...𝑦) → (FermatNo‘𝑛) ∈ ℂ)
32	31	adantl 487	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ₀ ∧ 𝑛 ∈ (0...𝑦)) → (FermatNo‘𝑛) ∈ ℂ)
33	27, 32	fprodcl 16112	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ₀ → ∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) ∈ ℂ)
34		1cnd 11295	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ₀ → 1 ∈ ℂ)
35	33, 5, 34	addsubassd 11682	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ₀ → ((∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2) − 1) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + (2 − 1)))
36		2m1e1 12460	. . . . . . . . . 10 ⊢ (2 − 1) = 1
37	36	oveq2i 7429	. . . . . . . . 9 ⊢ (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + (2 − 1)) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 1)
38	35, 37	eqtrdi 2812	. . . . . . . 8 ⊢ (𝑦 ∈ ℕ₀ → ((∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2) − 1) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 1))
39	38	oveq1d 7433	. . . . . . 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 48585	. . . . . . . . . . 11 ⊢ ((𝑦 + 1) ∈ ℕ₀ → (FermatNo‘(𝑦 + 1)) ∈ ℕ)
41	1, 40	syl 18	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ₀ → (FermatNo‘(𝑦 + 1)) ∈ ℕ)
42	41	nncnd 12344	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ₀ → (FermatNo‘(𝑦 + 1)) ∈ ℂ)
43	42, 34	subcld 11662	. . . . . . . 8 ⊢ (𝑦 ∈ ℕ₀ → ((FermatNo‘(𝑦 + 1)) − 1) ∈ ℂ)
44	33, 42	muls1d 11769	. . . . . . . . 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	mullidd 11320	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ₀ → (1 · ((FermatNo‘(𝑦 + 1)) − 1)) = ((FermatNo‘(𝑦 + 1)) − 1))
46	44, 45	oveq12d 7436	. . . . . . . 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 11329	. . . . . . 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 2796	. . . . . 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 486	. . . . 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 7433	. . . 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 2768	. . . . . . 7 ⊢ ((FermatNo‘(𝑦 + 1)) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2) ↔ (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2) = (FermatNo‘(𝑦 + 1)))
52	42, 5, 33	subadd2d 11681	. . . . . . 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 7426	. . . . . . . . . . 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 7433	. . . . . . . . . 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 7433	. . . . . . . . 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 2769	. . . . . . . 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 11322	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ₀ → (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) · (FermatNo‘(𝑦 + 1))) ∈ ℂ)
59	42, 5	subcld 11662	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ₀ → ((FermatNo‘(𝑦 + 1)) − 2) ∈ ℂ)
60	58, 59	subcld 11662	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ₀ → ((∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) · (FermatNo‘(𝑦 + 1))) − ((FermatNo‘(𝑦 + 1)) − 2)) ∈ ℂ)
61	60, 43, 34	addassd 11324	. . . . . . . . 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 12999	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℕ₀ ↔ 𝑦 ∈ (ℤ_≥‘0))
63	62	biimpi 219	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℕ₀ → 𝑦 ∈ (ℤ_≥‘0))
64		elfznn0 13747	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ (0...(𝑦 + 1)) → 𝑛 ∈ ℕ₀)
65	64, 30	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ (0...(𝑦 + 1)) → (FermatNo‘𝑛) ∈ ℂ)
66	65	adantl 487	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℕ₀ ∧ 𝑛 ∈ (0...(𝑦 + 1))) → (FermatNo‘𝑛) ∈ ℂ)
67		fveq2 6883	. . . . . . . . . . . . . 14 ⊢ (𝑛 = (𝑦 + 1) → (FermatNo‘𝑛) = (FermatNo‘(𝑦 + 1)))
68	63, 66, 67	fprodp1 16129	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℕ₀ → ∏𝑛 ∈ (0...(𝑦 + 1))(FermatNo‘𝑛) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) · (FermatNo‘(𝑦 + 1))))
69	68	eqcomd 2767	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℕ₀ → (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) · (FermatNo‘(𝑦 + 1))) = ∏𝑛 ∈ (0...(𝑦 + 1))(FermatNo‘𝑛))
70	69	oveq1d 7433	. . . . . . . . . . 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 11734	. . . . . . . . . . . 12 ⊢ ((FermatNo‘(𝑦 + 1)) ∈ ℂ → (((FermatNo‘(𝑦 + 1)) − 1) + 1) = (FermatNo‘(𝑦 + 1)))
72	42, 71	syl 18	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ₀ → (((FermatNo‘(𝑦 + 1)) − 1) + 1) = (FermatNo‘(𝑦 + 1)))
73	70, 72	oveq12d 7436	. . . . . . . . . 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 14109	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℕ₀ → (0...(𝑦 + 1)) ∈ Fin)
75	74, 66	fprodcl 16112	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ₀ → ∏𝑛 ∈ (0...(𝑦 + 1))(FermatNo‘𝑛) ∈ ℂ)
76	75, 59, 42	subadd23d 11684	. . . . . . . . . 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 2796	. . . . . . . . 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 11667	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ₀ → ((FermatNo‘(𝑦 + 1)) − ((FermatNo‘(𝑦 + 1)) − 2)) = 2)
79	78	oveq2d 7434	. . . . . . . . 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 2800	. . . . . . . 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 2818	. . . . . . 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 418	. . . . . 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 412	. . . 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 2796	. . 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 2800	. 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 418	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

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ‘cfv 6537 (class class class)co 7418 ℂcc 11191 0cc0 11193 1c1 11194 + caddc 11196 · cmul 11198 − cmin 11534 ℕcn 12328 2c2 12390 ℕ₀cn0 12599 ℤ_≥cuz 12958 ...cfz 13632 ↑cexp 14197 ∏cprod 16065 FermatNocfmtno 48581

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 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7749 ax-inf2 9635 ax-cnex 11249 ax-resscn 11250 ax-1cn 11251 ax-icn 11252 ax-addcl 11253 ax-addrcl 11254 ax-mulcl 11255 ax-mulrcl 11256 ax-mulcom 11257 ax-addass 11258 ax-mulass 11259 ax-distr 11260 ax-i2m1 11261 ax-1ne0 11262 ax-1rid 11263 ax-rnegex 11264 ax-rrecex 11265 ax-cnre 11266 ax-pre-lttri 11267 ax-pre-lttrn 11268 ax-pre-ltadd 11269 ax-pre-mulgt0 11270 ax-pre-sup 11271

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 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 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 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7375 df-ov 7421 df-oprab 7422 df-mpo 7423 df-om 7876 df-1st 7999 df-2nd 8000 df-frecs 8292 df-wrecs 8323 df-recs 8372 df-rdg 8411 df-1o 8469 df-er 8710 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-sup 9427 df-oi 9497 df-card 10013 df-pnf 11338 df-mnf 11339 df-xr 11340 df-ltxr 11341 df-le 11342 df-sub 11536 df-neg 11537 df-div 11967 df-nn 12329 df-2 12398 df-3 12399 df-n0 12600 df-z 12687 df-uz 12959 df-rp 13114 df-fz 13633 df-fzo 13782 df-seq 14138 df-exp 14198 df-hash 14468 df-cj 15259 df-re 15260 df-im 15261 df-sqrt 15395 df-abs 15396 df-clim 15648 df-prod 16066 df-fmtno 48582

This theorem is used by: fmtnorec2 48597

W3C validator