fmtnorec2lem - Mathbox for Alexander van der Vekens

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Theorem fmtnorec2lem 39992

Description: Lemma for fmtnorec2 39993 (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 11210	. . . . . 6 ⊢ (𝑦 ∈ ℕ₀ → (𝑦 + 1) ∈ ℕ₀)
2		peano2nn0 11210	. . . . . 6 ⊢ ((𝑦 + 1) ∈ ℕ₀ → ((𝑦 + 1) + 1) ∈ ℕ₀)
3		fmtno 39979	. . . . . 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 10970	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ₀ → 2 ∈ ℂ)
6	5, 1	expp1d 12871	. . . . . . . 8 ⊢ (𝑦 ∈ ℕ₀ → (2↑((𝑦 + 1) + 1)) = ((2↑(𝑦 + 1)) · 2))
7	6	oveq2d 6565	. . . . . . 7 ⊢ (𝑦 ∈ ℕ₀ → (2↑(2↑((𝑦 + 1) + 1))) = (2↑((2↑(𝑦 + 1)) · 2)))
8		2nn0 11186	. . . . . . . . . . . 12 ⊢ 2 ∈ ℕ₀
9	8	a1i 11	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ₀ → 2 ∈ ℕ₀)
10	9, 1	nn0expcld 12893	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ₀ → (2↑(𝑦 + 1)) ∈ ℕ₀)
11	9, 10	nn0expcld 12893	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ₀ → (2↑(2↑(𝑦 + 1))) ∈ ℕ₀)
12	11	nn0cnd 11230	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ₀ → (2↑(2↑(𝑦 + 1))) ∈ ℂ)
13	12	sqvald 12867	. . . . . . . 8 ⊢ (𝑦 ∈ ℕ₀ → ((2↑(2↑(𝑦 + 1)))↑2) = ((2↑(2↑(𝑦 + 1))) · (2↑(2↑(𝑦 + 1)))))
14	5, 9, 10	expmuld 12873	. . . . . . . 8 ⊢ (𝑦 ∈ ℕ₀ → (2↑((2↑(𝑦 + 1)) · 2)) = ((2↑(2↑(𝑦 + 1)))↑2))
15		fmtnom1nn 39982	. . . . . . . . . 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 6567	. . . . . . . 8 ⊢ (𝑦 ∈ ℕ₀ → (((FermatNo‘(𝑦 + 1)) − 1) · ((FermatNo‘(𝑦 + 1)) − 1)) = ((2↑(2↑(𝑦 + 1))) · (2↑(2↑(𝑦 + 1)))))
18	13, 14, 17	3eqtr4d 2654	. . . . . . 7 ⊢ (𝑦 ∈ ℕ₀ → (2↑((2↑(𝑦 + 1)) · 2)) = (((FermatNo‘(𝑦 + 1)) − 1) · ((FermatNo‘(𝑦 + 1)) − 1)))
19	7, 18	eqtrd 2644	. . . . . 6 ⊢ (𝑦 ∈ ℕ₀ → (2↑(2↑((𝑦 + 1) + 1))) = (((FermatNo‘(𝑦 + 1)) − 1) · ((FermatNo‘(𝑦 + 1)) − 1)))
20	19	oveq1d 6564	. . . . 5 ⊢ (𝑦 ∈ ℕ₀ → ((2↑(2↑((𝑦 + 1) + 1))) + 1) = ((((FermatNo‘(𝑦 + 1)) − 1) · ((FermatNo‘(𝑦 + 1)) − 1)) + 1))
21	4, 20	eqtrd 2644	. . . 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 6556	. . . . . 6 ⊢ ((FermatNo‘(𝑦 + 1)) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2) → ((FermatNo‘(𝑦 + 1)) − 1) = ((∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2) − 1))
24	23	oveq1d 6564	. . . . 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 6564	. . . 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 12634	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ₀ → (0...𝑦) ∈ Fin)
28		elfznn0 12302	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (0...𝑦) → 𝑛 ∈ ℕ₀)
29		fmtnonn 39981	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ₀ → (FermatNo‘𝑛) ∈ ℕ)
30	29	nncnd 10913	. . . . . . . . . . . . 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 14521	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ₀ → ∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) ∈ ℂ)
34		1cnd 9935	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ₀ → 1 ∈ ℂ)
35	33, 5, 34	addsubassd 10291	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ₀ → ((∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2) − 1) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + (2 − 1)))
36		2m1e1 11012	. . . . . . . . . 10 ⊢ (2 − 1) = 1
37	36	oveq2i 6560	. . . . . . . . 9 ⊢ (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + (2 − 1)) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 1)
38	35, 37	syl6eq 2660	. . . . . . . 8 ⊢ (𝑦 ∈ ℕ₀ → ((∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2) − 1) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 1))
39	38	oveq1d 6564	. . . . . . 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 39981	. . . . . . . . . . 11 ⊢ ((𝑦 + 1) ∈ ℕ₀ → (FermatNo‘(𝑦 + 1)) ∈ ℕ)
41	1, 40	syl 17	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ₀ → (FermatNo‘(𝑦 + 1)) ∈ ℕ)
42	41	nncnd 10913	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ₀ → (FermatNo‘(𝑦 + 1)) ∈ ℂ)
43	42, 34	subcld 10271	. . . . . . . 8 ⊢ (𝑦 ∈ ℕ₀ → ((FermatNo‘(𝑦 + 1)) − 1) ∈ ℂ)
44	33, 42	muls1d 10370	. . . . . . . . 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 9937	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ₀ → (1 · ((FermatNo‘(𝑦 + 1)) − 1)) = ((FermatNo‘(𝑦 + 1)) − 1))
46	44, 45	oveq12d 6567	. . . . . . . 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 9946	. . . . . . 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 2644	. . . . . 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 6564	. . . 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 10290	. . . . . . 7 ⊢ (𝑦 ∈ ℕ₀ → (((FermatNo‘(𝑦 + 1)) − 2) = ∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) ↔ (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2) = (FermatNo‘(𝑦 + 1))))
52		eqcom 2617	. . . . . . 7 ⊢ ((FermatNo‘(𝑦 + 1)) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2) ↔ (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2) = (FermatNo‘(𝑦 + 1)))
53	51, 52	syl6rbbr 278	. . . . . 6 ⊢ (𝑦 ∈ ℕ₀ → ((FermatNo‘(𝑦 + 1)) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2) ↔ ((FermatNo‘(𝑦 + 1)) − 2) = ∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛)))
54		oveq2 6557	. . . . . . . . . . 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 6564	. . . . . . . . . 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 6564	. . . . . . . . 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 2618	. . . . . . . 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 9939	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ₀ → (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) · (FermatNo‘(𝑦 + 1))) ∈ ℂ)
59	42, 5	subcld 10271	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ₀ → ((FermatNo‘(𝑦 + 1)) − 2) ∈ ℂ)
60	58, 59	subcld 10271	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ₀ → ((∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) · (FermatNo‘(𝑦 + 1))) − ((FermatNo‘(𝑦 + 1)) − 2)) ∈ ℂ)
61	60, 43, 34	addassd 9941	. . . . . . . . 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 11601	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℕ₀ ↔ 𝑦 ∈ (ℤ_≥‘0))
63	62	biimpi 205	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℕ₀ → 𝑦 ∈ (ℤ_≥‘0))
64		elfznn0 12302	. . . . . . . . . . . . . . . 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 6103	. . . . . . . . . . . . . 14 ⊢ (𝑛 = (𝑦 + 1) → (FermatNo‘𝑛) = (FermatNo‘(𝑦 + 1)))
68	63, 66, 67	fprodp1 14538	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℕ₀ → ∏𝑛 ∈ (0...(𝑦 + 1))(FermatNo‘𝑛) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) · (FermatNo‘(𝑦 + 1))))
69	68	eqcomd 2616	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℕ₀ → (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) · (FermatNo‘(𝑦 + 1))) = ∏𝑛 ∈ (0...(𝑦 + 1))(FermatNo‘𝑛))
70	69	oveq1d 6564	. . . . . . . . . . 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 10334	. . . . . . . . . . . 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 6567	. . . . . . . . . 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 12634	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℕ₀ → (0...(𝑦 + 1)) ∈ Fin)
75	74, 66	fprodcl 14521	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ₀ → ∏𝑛 ∈ (0...(𝑦 + 1))(FermatNo‘𝑛) ∈ ℂ)
76	75, 59, 42	subadd23d 10293	. . . . . . . . . 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 2644	. . . . . . . . 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 10276	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ₀ → ((FermatNo‘(𝑦 + 1)) − ((FermatNo‘(𝑦 + 1)) − 2)) = 2)
79	78	oveq2d 6565	. . . . . . . . 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 2648	. . . . . . . 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 2666	. . . . . . 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 449	. . . . . 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 229	. . . . 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 444	. . . 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 2644	. . 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 2648	. 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 449	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 383 = wceq 1475 ∈ wcel 1977 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 − cmin 10145 ℕcn 10897 2c2 10947 ℕ₀cn0 11169 ℤ_≥cuz 11563 ...cfz 12197 ↑cexp 12722 ∏cprod 14474 FermatNocfmtno 39977

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-fz 12198 df-fzo 12335 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-prod 14475 df-fmtno 39978

This theorem is referenced by: fmtnorec2 39993

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