fmtnorec2lem - Mathbox for Alexander van der Vekens

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Theorem fmtnorec2lem 45034

Description: Lemma for fmtnorec2 45035 (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 12301	. . . . . 6 ⊢ (𝑦 ∈ ℕ₀ → (𝑦 + 1) ∈ ℕ₀)
2		peano2nn0 12301	. . . . . 6 ⊢ ((𝑦 + 1) ∈ ℕ₀ → ((𝑦 + 1) + 1) ∈ ℕ₀)
3		fmtno 45021	. . . . . 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 12079	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ₀ → 2 ∈ ℂ)
6	5, 1	expp1d 13893	. . . . . . . 8 ⊢ (𝑦 ∈ ℕ₀ → (2↑((𝑦 + 1) + 1)) = ((2↑(𝑦 + 1)) · 2))
7	6	oveq2d 7311	. . . . . . 7 ⊢ (𝑦 ∈ ℕ₀ → (2↑(2↑((𝑦 + 1) + 1))) = (2↑((2↑(𝑦 + 1)) · 2)))
8		2nn0 12278	. . . . . . . . . . . 12 ⊢ 2 ∈ ℕ₀
9	8	a1i 11	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ₀ → 2 ∈ ℕ₀)
10	9, 1	nn0expcld 13989	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ₀ → (2↑(𝑦 + 1)) ∈ ℕ₀)
11	9, 10	nn0expcld 13989	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ₀ → (2↑(2↑(𝑦 + 1))) ∈ ℕ₀)
12	11	nn0cnd 12323	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ₀ → (2↑(2↑(𝑦 + 1))) ∈ ℂ)
13	12	sqvald 13889	. . . . . . . 8 ⊢ (𝑦 ∈ ℕ₀ → ((2↑(2↑(𝑦 + 1)))↑2) = ((2↑(2↑(𝑦 + 1))) · (2↑(2↑(𝑦 + 1)))))
14	5, 9, 10	expmuld 13895	. . . . . . . 8 ⊢ (𝑦 ∈ ℕ₀ → (2↑((2↑(𝑦 + 1)) · 2)) = ((2↑(2↑(𝑦 + 1)))↑2))
15		fmtnom1nn 45024	. . . . . . . . . 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 7313	. . . . . . . 8 ⊢ (𝑦 ∈ ℕ₀ → (((FermatNo‘(𝑦 + 1)) − 1) · ((FermatNo‘(𝑦 + 1)) − 1)) = ((2↑(2↑(𝑦 + 1))) · (2↑(2↑(𝑦 + 1)))))
18	13, 14, 17	3eqtr4d 2783	. . . . . . 7 ⊢ (𝑦 ∈ ℕ₀ → (2↑((2↑(𝑦 + 1)) · 2)) = (((FermatNo‘(𝑦 + 1)) − 1) · ((FermatNo‘(𝑦 + 1)) − 1)))
19	7, 18	eqtrd 2773	. . . . . 6 ⊢ (𝑦 ∈ ℕ₀ → (2↑(2↑((𝑦 + 1) + 1))) = (((FermatNo‘(𝑦 + 1)) − 1) · ((FermatNo‘(𝑦 + 1)) − 1)))
20	19	oveq1d 7310	. . . . 5 ⊢ (𝑦 ∈ ℕ₀ → ((2↑(2↑((𝑦 + 1) + 1))) + 1) = ((((FermatNo‘(𝑦 + 1)) − 1) · ((FermatNo‘(𝑦 + 1)) − 1)) + 1))
21	4, 20	eqtrd 2773	. . . 4 ⊢ (𝑦 ∈ ℕ₀ → (FermatNo‘((𝑦 + 1) + 1)) = ((((FermatNo‘(𝑦 + 1)) − 1) · ((FermatNo‘(𝑦 + 1)) − 1)) + 1))
22	21	adantr 480	. . 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 7302	. . . . . 6 ⊢ ((FermatNo‘(𝑦 + 1)) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2) → ((FermatNo‘(𝑦 + 1)) − 1) = ((∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2) − 1))
24	23	oveq1d 7310	. . . . 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 7310	. . . 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 481	. . 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 13721	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ₀ → (0...𝑦) ∈ Fin)
28		elfznn0 13377	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (0...𝑦) → 𝑛 ∈ ℕ₀)
29		fmtnonn 45023	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ₀ → (FermatNo‘𝑛) ∈ ℕ)
30	29	nncnd 12017	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ₀ → (FermatNo‘𝑛) ∈ ℂ)
31	28, 30	syl 17	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (0...𝑦) → (FermatNo‘𝑛) ∈ ℂ)
32	31	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ₀ ∧ 𝑛 ∈ (0...𝑦)) → (FermatNo‘𝑛) ∈ ℂ)
33	27, 32	fprodcl 15690	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ₀ → ∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) ∈ ℂ)
34		1cnd 10998	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ₀ → 1 ∈ ℂ)
35	33, 5, 34	addsubassd 11380	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ₀ → ((∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2) − 1) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + (2 − 1)))
36		2m1e1 12127	. . . . . . . . . 10 ⊢ (2 − 1) = 1
37	36	oveq2i 7306	. . . . . . . . 9 ⊢ (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + (2 − 1)) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 1)
38	35, 37	eqtrdi 2789	. . . . . . . 8 ⊢ (𝑦 ∈ ℕ₀ → ((∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2) − 1) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 1))
39	38	oveq1d 7310	. . . . . . 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 45023	. . . . . . . . . . 11 ⊢ ((𝑦 + 1) ∈ ℕ₀ → (FermatNo‘(𝑦 + 1)) ∈ ℕ)
41	1, 40	syl 17	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ₀ → (FermatNo‘(𝑦 + 1)) ∈ ℕ)
42	41	nncnd 12017	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ₀ → (FermatNo‘(𝑦 + 1)) ∈ ℂ)
43	42, 34	subcld 11360	. . . . . . . 8 ⊢ (𝑦 ∈ ℕ₀ → ((FermatNo‘(𝑦 + 1)) − 1) ∈ ℂ)
44	33, 42	muls1d 11463	. . . . . . . . 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 11021	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ₀ → (1 · ((FermatNo‘(𝑦 + 1)) − 1)) = ((FermatNo‘(𝑦 + 1)) − 1))
46	44, 45	oveq12d 7313	. . . . . . . 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 11030	. . . . . . 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 2773	. . . . . 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 480	. . . . 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 7310	. . . 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 2740	. . . . . . 7 ⊢ ((FermatNo‘(𝑦 + 1)) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2) ↔ (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2) = (FermatNo‘(𝑦 + 1)))
52	42, 5, 33	subadd2d 11379	. . . . . . 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 289	. . . . . 6 ⊢ (𝑦 ∈ ℕ₀ → ((FermatNo‘(𝑦 + 1)) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2) ↔ ((FermatNo‘(𝑦 + 1)) − 2) = ∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛)))
54		oveq2 7303	. . . . . . . . . . 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 7310	. . . . . . . . . 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 7310	. . . . . . . . 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 2741	. . . . . . . 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 11023	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ₀ → (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) · (FermatNo‘(𝑦 + 1))) ∈ ℂ)
59	42, 5	subcld 11360	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ₀ → ((FermatNo‘(𝑦 + 1)) − 2) ∈ ℂ)
60	58, 59	subcld 11360	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ₀ → ((∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) · (FermatNo‘(𝑦 + 1))) − ((FermatNo‘(𝑦 + 1)) − 2)) ∈ ℂ)
61	60, 43, 34	addassd 11025	. . . . . . . . 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 12651	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℕ₀ ↔ 𝑦 ∈ (ℤ_≥‘0))
63	62	biimpi 215	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℕ₀ → 𝑦 ∈ (ℤ_≥‘0))
64		elfznn0 13377	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ (0...(𝑦 + 1)) → 𝑛 ∈ ℕ₀)
65	64, 30	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ (0...(𝑦 + 1)) → (FermatNo‘𝑛) ∈ ℂ)
66	65	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℕ₀ ∧ 𝑛 ∈ (0...(𝑦 + 1))) → (FermatNo‘𝑛) ∈ ℂ)
67		fveq2 6792	. . . . . . . . . . . . . 14 ⊢ (𝑛 = (𝑦 + 1) → (FermatNo‘𝑛) = (FermatNo‘(𝑦 + 1)))
68	63, 66, 67	fprodp1 15707	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℕ₀ → ∏𝑛 ∈ (0...(𝑦 + 1))(FermatNo‘𝑛) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) · (FermatNo‘(𝑦 + 1))))
69	68	eqcomd 2739	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℕ₀ → (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) · (FermatNo‘(𝑦 + 1))) = ∏𝑛 ∈ (0...(𝑦 + 1))(FermatNo‘𝑛))
70	69	oveq1d 7310	. . . . . . . . . . 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 11428	. . . . . . . . . . . 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 7313	. . . . . . . . . 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 13721	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℕ₀ → (0...(𝑦 + 1)) ∈ Fin)
75	74, 66	fprodcl 15690	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ₀ → ∏𝑛 ∈ (0...(𝑦 + 1))(FermatNo‘𝑛) ∈ ℂ)
76	75, 59, 42	subadd23d 11382	. . . . . . . . . 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 2773	. . . . . . . . 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 11365	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ₀ → ((FermatNo‘(𝑦 + 1)) − ((FermatNo‘(𝑦 + 1)) − 2)) = 2)
79	78	oveq2d 7311	. . . . . . . . 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 2777	. . . . . . . 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 2795	. . . . . . 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 412	. . . . . 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 239	. . . . 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 406	. . . 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 2773	. . 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 2777	. 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 412	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 395 = wceq 1537 ∈ wcel 2101 ‘cfv 6447 (class class class)co 7295 ℂcc 10897 0cc0 10899 1c1 10900 + caddc 10902 · cmul 10904 − cmin 11233 ℕcn 12001 2c2 12056 ℕ₀cn0 12261 ℤ_≥cuz 12610 ...cfz 13267 ↑cexp 13810 ∏cprod 15643 FermatNocfmtno 45019

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2103 ax-9 2111 ax-10 2132 ax-11 2149 ax-12 2166 ax-ext 2704 ax-rep 5212 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7608 ax-inf2 9427 ax-cnex 10955 ax-resscn 10956 ax-1cn 10957 ax-icn 10958 ax-addcl 10959 ax-addrcl 10960 ax-mulcl 10961 ax-mulrcl 10962 ax-mulcom 10963 ax-addass 10964 ax-mulass 10965 ax-distr 10966 ax-i2m1 10967 ax-1ne0 10968 ax-1rid 10969 ax-rnegex 10970 ax-rrecex 10971 ax-cnre 10972 ax-pre-lttri 10973 ax-pre-lttrn 10974 ax-pre-ltadd 10975 ax-pre-mulgt0 10976 ax-pre-sup 10977

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2063 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2884 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3222 df-reu 3223 df-rab 3224 df-v 3436 df-sbc 3719 df-csb 3835 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3908 df-nul 4260 df-if 4463 df-pw 4538 df-sn 4565 df-pr 4567 df-op 4571 df-uni 4842 df-int 4883 df-iun 4929 df-br 5078 df-opab 5140 df-mpt 5161 df-tr 5195 df-id 5491 df-eprel 5497 df-po 5505 df-so 5506 df-fr 5546 df-se 5547 df-we 5548 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-pred 6206 df-ord 6273 df-on 6274 df-lim 6275 df-suc 6276 df-iota 6399 df-fun 6449 df-fn 6450 df-f 6451 df-f1 6452 df-fo 6453 df-f1o 6454 df-fv 6455 df-isom 6456 df-riota 7252 df-ov 7298 df-oprab 7299 df-mpo 7300 df-om 7733 df-1st 7851 df-2nd 7852 df-frecs 8117 df-wrecs 8148 df-recs 8222 df-rdg 8261 df-1o 8317 df-er 8518 df-en 8754 df-dom 8755 df-sdom 8756 df-fin 8757 df-sup 9229 df-oi 9297 df-card 9725 df-pnf 11039 df-mnf 11040 df-xr 11041 df-ltxr 11042 df-le 11043 df-sub 11235 df-neg 11236 df-div 11661 df-nn 12002 df-2 12064 df-3 12065 df-n0 12262 df-z 12348 df-uz 12611 df-rp 12759 df-fz 13268 df-fzo 13411 df-seq 13750 df-exp 13811 df-hash 14073 df-cj 14838 df-re 14839 df-im 14840 df-sqrt 14974 df-abs 14975 df-clim 15225 df-prod 15644 df-fmtno 45020

This theorem is referenced by: fmtnorec2 45035

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